std and expm1 (#973)

* std and expm1

* actually add expm1

* fix linux

* fix vjp

* relax tol for linux test

* Add it to the compilable primitives

---------

Co-authored-by: Angelos Katharopoulos <a_katharopoulos@apple.com>

mlx/backend/metal/kernels/CMakeLists.txt

@@ -7,6 +7,7 @@ set(
   ${CMAKE_CURRENT_SOURCE_DIR}/complex.h
   ${CMAKE_CURRENT_SOURCE_DIR}/defines.h
   ${CMAKE_CURRENT_SOURCE_DIR}/erf.h
+  ${CMAKE_CURRENT_SOURCE_DIR}/expm1f.h
   ${CMAKE_CURRENT_SOURCE_DIR}/indexing.h
   ${CMAKE_CURRENT_SOURCE_DIR}/unary.h
   ${CMAKE_CURRENT_SOURCE_DIR}/utils.h

mlx/backend/metal/kernels/expm1f.h

@@ -0,0 +1,89 @@
+// Copyright © 2023 Apple Inc.
+
+#pragma once
+
+#include <metal_math>
+
+// Original license copied below:
+//  Copyright (c) 2015-2023 Norbert Juffa
+//  All rights reserved.
+//
+//  Redistribution and use in source and binary forms, with or without
+//  modification, are permitted provided that the following conditions
+//  are met:
+//
+//  1. Redistributions of source code must retain the above copyright
+//     notice, this list of conditions and the following disclaimer.
+//
+//  2. Redistributions in binary form must reproduce the above copyright
+//     notice, this list of conditions and the following disclaimer in the
+//     documentation and/or other materials provided with the distribution.
+//
+//  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+//  "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+//  LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+//  A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+//  HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+//  SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+//  LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+//  DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+//  THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+//  (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+//  OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+/* Compute exponential base e minus 1. Maximum ulp error = 0.997458
+
+   i = rint(a/log(2)), f = a-i*log(2). Then expm1(a) = 2**i * (expm1(f)+1) - 1.
+   Compute r = expm1(f). Then expm1(a)= 2 * (0.5 * 2**i * r + 0.5 * 2**i - 0.5).
+   With t = 0.5*2**i, expm1(a) = 2*(r * t + t-0.5). However, for best accuracy,
+   when i == 1, expm1(a)= 2*(r + 0.5), and when i == 0, expm1(a) = r.
+
+   NOTE: Scale factor b is only applied if i < 0 or i > 1 (should be power of 2)
+*/
+float expm1f_scaled_unchecked(float a, float b) {
+  float f, j, r, s, t, u, v, x, y;
+  int i;
+
+  // exp(a) = 2**i * exp(f); i = rintf (a / log(2))
+  j = fma(1.442695f, a, 12582912.f); // 0x1.715476p0, 0x1.8p23
+  j = j - 12582912.0f; // 0x1.8p23
+  i = (int)j;
+  f = fma(j, -6.93145752e-1f, a);
+
+  // approximate r = exp(f)-1 on interval [-log(2)/2, +log(2)/2]
+  s = f * f;
+  if (a == 0.0f)
+    s = a; // ensure -0 is passed through
+  // err = 0.997458  ulp1 = 11081805
+  r = 1.97350979e-4f; // 0x1.9de000p-13
+  r = fma(r, f, 1.39309070e-3f); // 0x1.6d30bcp-10
+  r = fma(r, f, 8.33343994e-3f); // 0x1.1111f6p-7
+  r = fma(r, f, 4.16668020e-2f); // 0x1.55559ep-5
+  r = fma(r, f, 1.66666716e-1f); // 0x1.55555cp-3
+  r = fma(r, f, 4.99999970e-1f); // 0x1.fffffep-2
+  u = (j == 1) ? (f + 0.5f) : f;
+  v = fma(r, s, u);
+  s = 0.5f * b;
+  t = ldexp(s, i);
+  y = t - s;
+  x = (t - y) - s; // double-float canonicalization of difference
+  r = fma(v, t, x) + y;
+  r = r + r;
+  if (j == 0)
+    r = v;
+  if (j == 1)
+    r = v + v;
+  return r;
+}
+
+/* Compute exponential base e minus 1. max ulp err = 0.99746 */
+float expm1f(float a) {
+  float r;
+
+  r = expm1f_scaled_unchecked(a, 1.0f);
+  /* handle severe overflow and underflow */
+  if (abs(a - 1.0f) > 88.0f) {
+    r = fma(r, r, -1.0f);
+  }
+  return r;
+}

mlx/backend/metal/kernels/unary.h

@@ -7,6 +7,7 @@
 
 #include "mlx/backend/metal/kernels/bf16.h"
 #include "mlx/backend/metal/kernels/erf.h"
+#include "mlx/backend/metal/kernels/expm1f.h"
 #include "mlx/backend/metal/kernels/utils.h"
 
 namespace {
@@ -183,6 +184,13 @@ struct Exp {
   }
 };
 
+struct Expm1 {
+  template <typename T>
+  T operator()(T x) {
+    return static_cast<T>(expm1f(static_cast<float>(x)));
+  };
+};
+
 struct Floor {
   template <typename T>
   T operator()(T x) {

mlx/backend/metal/kernels/unary.metal

@@ -71,6 +71,7 @@ instantiate_unary_types(ceil, Ceil)
 instantiate_unary_float(cos, Cos)
 instantiate_unary_float(cosh, Cosh)
 instantiate_unary_float(exp, Exp)
+instantiate_unary_float(expm1, Expm1)
 instantiate_unary_types(floor, Floor)
 instantiate_unary_float(log, Log)
 instantiate_unary_float(log2, Log2)

mlx/backend/metal/primitives.cpp

@@ -615,6 +615,10 @@ void Exp::eval_gpu(const std::vector<array>& inputs, array& out) {
   unary_op(inputs, out, "exp");
 }
 
+void Expm1::eval_gpu(const std::vector<array>& inputs, array& out) {
+  unary_op(inputs, out, "expm1");
+}
+
 void Full::eval_gpu(const std::vector<array>& inputs, array& out) {
   auto in = inputs[0];
   CopyType ctype;