Kahan's exp quotient

Percentage Accurate: 52.6% → 100.0%

Time: 6.4s

Alternatives: 14

Speedup: 8.8×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x} - 1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))

C

double code(double x) {
	return (exp(x) - 1.0) / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (exp(x) - 1.0d0) / x
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 1.0) / x;
}

Python

def code(x):
	return (math.exp(x) - 1.0) / x

Julia

function code(x)
	return Float64(Float64(exp(x) - 1.0) / x)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 1.0) / x;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision]

Initial Program: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x} - 1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))

C

double code(double x) {
	return (exp(x) - 1.0) / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (exp(x) - 1.0d0) / x
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 1.0) / x;
}

Python

def code(x):
	return (math.exp(x) - 1.0) / x

Julia

function code(x)
	return Float64(Float64(exp(x) - 1.0) / x)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 1.0) / x;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{expm1}\left(x\right)}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (expm1 x) x))

C

double code(double x) {
	return expm1(x) / x;
}

Java

public static double code(double x) {
	return Math.expm1(x) / x;
}

Python

def code(x):
	return math.expm1(x) / x

Julia

function code(x)
	return Float64(expm1(x) / x)
end

Wolfram

code[x_] := N[(N[(Exp[x] - 1), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{x} - 1}}{x} \]

   2. unpow1 N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{1}} - 1}{x} \]

   3. metadata-eval N/A

      \[\leadsto \frac{{\left(e^{x}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} - 1}{x} \]

   4. sqrt-pow1 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{{\left(e^{x}\right)}^{2}}} - 1}{x} \]

   5. pow2 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{e^{x} \cdot e^{x}}} - 1}{x} \]

   6. rem-sqrt-square-rev N/A

      \[\leadsto \frac{\color{blue}{\left|e^{x}\right|} - 1}{x} \]

   7. rem-sqrt-square-rev N/A

      \[\leadsto \frac{\color{blue}{\sqrt{e^{x} \cdot e^{x}}} - 1}{x} \]

   8. pow2 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{{\left(e^{x}\right)}^{2}}} - 1}{x} \]

   9. sqrt-pow1 N/A

      \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{\left(\frac{2}{2}\right)}} - 1}{x} \]

   10. metadata-eval N/A

       \[\leadsto \frac{{\left(e^{x}\right)}^{\color{blue}{1}} - 1}{x} \]

   11. unpow1 N/A

       \[\leadsto \frac{\color{blue}{e^{x}} - 1}{x} \]

   12. lift-exp.f64 N/A

       \[\leadsto \frac{\color{blue}{e^{x}} - 1}{x} \]

   13. lower-expm1.f64 100.0

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{x} \]

4. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{x} \]
5. Add Preprocessing

Alternative 2: 72.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x, 0.5\right)\\ \frac{\mathsf{fma}\left(\sqrt{t\_0}, \sqrt{\left(x \cdot x\right) \cdot \left(\left(t\_0 \cdot x\right) \cdot x\right)}, x\right)}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma x 0.041666666666666664 0.16666666666666666) x 0.5)))
   (/ (fma (sqrt t_0) (sqrt (* (* x x) (* (* t_0 x) x))) x) x)))

C

double code(double x) {
	double t_0 = fma(fma(x, 0.041666666666666664, 0.16666666666666666), x, 0.5);
	return fma(sqrt(t_0), sqrt(((x * x) * ((t_0 * x) * x))), x) / x;
}

Julia

function code(x)
	t_0 = fma(fma(x, 0.041666666666666664, 0.16666666666666666), x, 0.5)
	return Float64(fma(sqrt(t_0), sqrt(Float64(Float64(x * x) * Float64(Float64(t_0 * x) * x))), x) / x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision]}, N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}}{x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}}{x} \]

5. Applied rewrites 74.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}}{x} \]
6. Step-by-step derivation

7. Applied rewrites 74.0%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(0.5, x, 1\right)\right) \cdot x}{x} \]
8. Step-by-step derivation

9. Applied rewrites 75.5%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x, 0.5\right)}, \color{blue}{\sqrt{\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x, 0.5\right) \cdot x\right) \cdot x\right)}}, x\right)}{x} \]
10. Add Preprocessing

Alternative 3: 71.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{0.5}, \sqrt{\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x, 0.5\right) \cdot x\right) \cdot x\right)}, x\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma
   (sqrt 0.5)
   (sqrt
    (*
     (* x x)
     (* (* (fma (fma x 0.041666666666666664 0.16666666666666666) x 0.5) x) x)))
   x)
  x))

C

double code(double x) {
	return fma(sqrt(0.5), sqrt(((x * x) * ((fma(fma(x, 0.041666666666666664, 0.16666666666666666), x, 0.5) * x) * x))), x) / x;
}

Julia

function code(x)
	return Float64(fma(sqrt(0.5), sqrt(Float64(Float64(x * x) * Float64(Float64(fma(fma(x, 0.041666666666666664, 0.16666666666666666), x, 0.5) * x) * x))), x) / x)
end

Wolfram

code[x_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(N[(N[(N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}}{x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}}{x} \]

5. Applied rewrites 74.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}}{x} \]
6. Step-by-step derivation

7. Applied rewrites 74.0%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(0.5, x, 1\right)\right) \cdot x}{x} \]
8. Step-by-step derivation

9. Applied rewrites 75.5%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x, 0.5\right)}, \color{blue}{\sqrt{\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x, 0.5\right) \cdot x\right) \cdot x\right)}}, x\right)}{x} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{2}}, \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{1}{6}\right), x, \frac{1}{2}\right) \cdot x\right) \cdot x\right)}}, x\right)}{x} \]
11. Step-by-step derivation

12. Applied rewrites 75.0%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{0.5}, \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x, 0.5\right) \cdot x\right) \cdot x\right)}}, x\right)}{x} \]
13. Add Preprocessing

Alternative 4: 70.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(0.5, x, 1\right)\right) \cdot x}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (fma
    (* (fma 0.041666666666666664 x 0.16666666666666666) x)
    x
    (fma 0.5 x 1.0))
   x)
  x))

C

double code(double x) {
	return (fma((fma(0.041666666666666664, x, 0.16666666666666666) * x), x, fma(0.5, x, 1.0)) * x) / x;
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(fma(0.041666666666666664, x, 0.16666666666666666) * x), x, fma(0.5, x, 1.0)) * x) / x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x + N[(0.5 * x + 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}}{x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}}{x} \]

5. Applied rewrites 74.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}}{x} \]
6. Step-by-step derivation

7. Applied rewrites 74.0%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x, x, \mathsf{fma}\left(0.5, x, 1\right)\right) \cdot x}{x} \]
8. Add Preprocessing

Alternative 5: 70.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0)
   x)
  x))

C

double code(double x) {
	return (fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) / x;
}

Julia

function code(x)
	return Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) / x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}}{x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}}{x} \]

5. Applied rewrites 74.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}}{x} \]
6. Add Preprocessing

Alternative 6: 69.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \cdot x}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (fma (* (* x x) 0.041666666666666664) x 1.0) x) x))

C

double code(double x) {
	return (fma(((x * x) * 0.041666666666666664), x, 1.0) * x) / x;
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(Float64(x * x) * 0.041666666666666664), x, 1.0) * x) / x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}}{x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right) \cdot x}}{x} \]

5. Applied rewrites 74.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x}}{x} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, x, 1\right) \cdot x}{x} \]
7. Step-by-step derivation

8. Applied rewrites 73.2%

   \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \cdot x}{x} \]
9. Add Preprocessing

Alternative 7: 68.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.2)
   (fma (fma 0.16666666666666666 x 0.5) x 1.0)
   (* (* (fma 0.041666666666666664 x 0.16666666666666666) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 4.2) {
		tmp = fma(fma(0.16666666666666666, x, 0.5), x, 1.0);
	} else {
		tmp = (fma(0.041666666666666664, x, 0.16666666666666666) * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.2)
		tmp = fma(fma(0.16666666666666666, x, 0.5), x, 1.0);
	else
		tmp = Float64(Float64(fma(0.041666666666666664, x, 0.16666666666666666) * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.2], N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.20000000000000018

   1. Initial program 29.1%

      \[\frac{e^{x} - 1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} + 1 \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)}\right)\right) + 1 \]

      6. remove-double-neg N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)} \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right) \]

      10. lower-fma.f64 75.4

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)} \]

   if 4.20000000000000018 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1} \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)} + 1 \]

      5. remove-double-neg N/A

         \[\leadsto \color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right) + 1 \]

      6. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) \cdot x} + 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x, 1\right)} \]

      9. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}, x, 1\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}, x, 1\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x, 1\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)} + \frac{1}{2}, x, 1\right) \]

      13. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) + \frac{1}{2}, x, 1\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)} + \frac{1}{2}, x, 1\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right) \cdot x} + \frac{1}{2}, x, 1\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{24} \cdot x, x, \frac{1}{2}\right)}, x, 1\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, x, \frac{1}{2}\right), x, 1\right) \]

      18. lower-fma.f64 63.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right), x, 1\right) \]

   5. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right), x, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 63.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x, x, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.6%

       \[\leadsto \left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0))

C

double code(double x) {
	return fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0);
}

Julia

function code(x)
	return fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0)
end

Wolfram

code[x_] := N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision]

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]

   2. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1} \]

   4. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)} + 1 \]

   5. remove-double-neg N/A

      \[\leadsto \color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right) + 1 \]

   6. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} + 1 \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) \cdot x} + 1 \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x, 1\right)} \]

   9. fp-cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}, x, 1\right) \]

   10. fp-cancel-sub-sign-inv N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}, x, 1\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x, 1\right) \]

   12. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)} + \frac{1}{2}, x, 1\right) \]

   13. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) + \frac{1}{2}, x, 1\right) \]

   14. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)} + \frac{1}{2}, x, 1\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right) \cdot x} + \frac{1}{2}, x, 1\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{24} \cdot x, x, \frac{1}{2}\right)}, x, 1\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, x, \frac{1}{2}\right), x, 1\right) \]

   18. lower-fma.f64 72.6

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right), x, 1\right) \]

5. Applied rewrites 72.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right)} \]
6. Add Preprocessing

Alternative 9: 64.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35) 1.0 (* (fma 0.16666666666666666 x 0.5) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35) {
		tmp = 1.0;
	} else {
		tmp = fma(0.16666666666666666, x, 0.5) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35)
		tmp = 1.0;
	else
		tmp = Float64(fma(0.16666666666666666, x, 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35], 1.0, N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3500000000000001

   1. Initial program 29.1%

      \[\frac{e^{x} - 1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{1} \]

   if 1.3500000000000001 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} + 1 \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)}\right)\right) + 1 \]

      6. remove-double-neg N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)} \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right) \]

      10. lower-fma.f64 46.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \]

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{x}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.2%

      \[\leadsto \mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.5) 1.0 (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = 1.0d0
    else
        tmp = (x * x) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.5:
		tmp = 1.0
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.5)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * x) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = 1.0;
	else
		tmp = (x * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.5], 1.0, N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 29.1%

      \[\frac{e^{x} - 1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{1} \]

   if 2.5 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - 1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} + 1 \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)}\right)\right) + 1 \]

      6. remove-double-neg N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)} \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right) \]

      10. lower-fma.f64 46.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \]

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{{x}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.2%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{0.16666666666666666} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* (* x x) 0.041666666666666664) x 1.0))

C

double code(double x) {
	return fma(((x * x) * 0.041666666666666664), x, 1.0);
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * 0.041666666666666664), x, 1.0)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * x + 1.0), $MachinePrecision]

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]

   2. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) + 1} \]

   4. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right)} + 1 \]

   5. remove-double-neg N/A

      \[\leadsto \color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}\right) + 1 \]

   6. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} + 1 \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right) \cdot x} + 1 \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right), x, 1\right)} \]

   9. fp-cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}, x, 1\right) \]

   10. fp-cancel-sub-sign-inv N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}, x, 1\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right) + \frac{1}{2}}, x, 1\right) \]

   12. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)} + \frac{1}{2}, x, 1\right) \]

   13. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}\right)\right) + \frac{1}{2}, x, 1\right) \]

   14. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)} + \frac{1}{2}, x, 1\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right) \cdot x} + \frac{1}{2}, x, 1\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{24} \cdot x, x, \frac{1}{2}\right)}, x, 1\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot x + \frac{1}{6}}, x, \frac{1}{2}\right), x, 1\right) \]

   18. lower-fma.f64 72.6

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right), x, 1\right) \]

5. Applied rewrites 72.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, x, 1\right) \]
7. Step-by-step derivation

8. Applied rewrites 71.8%

   \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \]
9. Add Preprocessing

Alternative 12: 64.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (fma 0.16666666666666666 x 0.5) x 1.0))

C

double code(double x) {
	return fma(fma(0.16666666666666666, x, 0.5), x, 1.0);
}

Julia

function code(x)
	return fma(fma(0.16666666666666666, x, 0.5), x, 1.0)
end

Wolfram

code[x_] := N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision]

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]

   2. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1} \]

   4. distribute-lft-neg-out N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} + 1 \]

   5. distribute-lft-neg-out N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)}\right)\right) + 1 \]

   6. remove-double-neg N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)} + 1 \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1 \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)} \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right) \]

   10. lower-fma.f64 68.6

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \]

5. Applied rewrites 68.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)} \]
6. Add Preprocessing

Alternative 13: 51.8% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma 0.5 x 1.0))

C

double code(double x) {
	return fma(0.5, x, 1.0);
}

Julia

function code(x)
	return fma(0.5, x, 1.0)
end

Wolfram

code[x_] := N[(0.5 * x + 1.0), $MachinePrecision]

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + 1} \]

   2. lower-fma.f64 58.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, 1\right)} \]

5. Applied rewrites 58.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, 1\right)} \]
6. Add Preprocessing

Alternative 14: 51.6% accurate, 115.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 45.7%

   \[\frac{e^{x} - 1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 58.3%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 52.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - 1\\ \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{t\_0}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) 1.0)))
   (if (and (< x 1.0) (> x -1.0)) (/ t_0 (log (exp x))) (/ t_0 x))))

C

double code(double x) {
	double t_0 = exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / log(exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x) - 1.0d0
    if ((x < 1.0d0) .and. (x > (-1.0d0))) then
        tmp = t_0 / log(exp(x))
    else
        tmp = t_0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / Math.log(Math.exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - 1.0
	tmp = 0
	if (x < 1.0) and (x > -1.0):
		tmp = t_0 / math.log(math.exp(x))
	else:
		tmp = t_0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(exp(x) - 1.0)
	tmp = 0.0
	if ((x < 1.0) && (x > -1.0))
		tmp = Float64(t_0 / log(exp(x)));
	else
		tmp = Float64(t_0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(x) - 1.0;
	tmp = 0.0;
	if ((x < 1.0) && (x > -1.0))
		tmp = t_0 / log(exp(x));
	else
		tmp = t_0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[And[Less[x, 1.0], Greater[x, -1.0]], N[(t$95$0 / N[Log[N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024360 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :alt
  (! :herbie-platform default (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x)))

  (/ (- (exp x) 1.0) x))