expq2 (section 3.11)

Percentage Accurate: 38.3% → 100.0%

Time: 6.1s

Alternatives: 17

Speedup: 2.1×

Specification

\[710 > x\]

Math

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

FPCore

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

C

double code(double x) {
	return exp(x) / (exp(x) - 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 = exp(x) / (exp(x) - 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return exp(x) / (exp(x) - 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 = exp(x) / (exp(x) - 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(e^{-x} \cdot \mathsf{expm1}\left(x\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (* (exp (- x)) (expm1 x)) -1.0))

C

double code(double x) {
	return pow((exp(-x) * expm1(x)), -1.0);
}

Java

public static double code(double x) {
	return Math.pow((Math.exp(-x) * Math.expm1(x)), -1.0);
}

Python

def code(x):
	return math.pow((math.exp(-x) * math.expm1(x)), -1.0)

Julia

function code(x)
	return Float64(exp(Float64(-x)) * expm1(x)) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[Exp[(-x)], $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 38.6%

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

   1. lift-/.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 38.6

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

   12. lift--.f64 N/A

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

   13. unpow1 N/A

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

   14. metadata-eval N/A

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

   15. sqrt-pow1 N/A

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

   16. pow2 N/A

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

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

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

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

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

   19. pow2 N/A

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

   20. sqrt-pow1 N/A

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

   21. metadata-eval N/A

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

   22. unpow1 N/A

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

   23. lift-exp.f64 N/A

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

   24. lower-expm1.f64 100.0

       \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

5. Final simplification 100.0%

   \[\leadsto {\left(e^{-x} \cdot \mathsf{expm1}\left(x\right)\right)}^{-1} \]
6. Add Preprocessing

Alternative 2: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;{\left(-0.5 \cdot \left(x \cdot x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (pow (* -0.5 (* x x)) -1.0)
   (fma 0.08333333333333333 x (- (pow x -1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = pow((-0.5 * (x * x)), -1.0);
	} else {
		tmp = fma(0.08333333333333333, x, (pow(x, -1.0) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.0)
		tmp = Float64(-0.5 * Float64(x * x)) ^ -1.0;
	else
		tmp = fma(0.08333333333333333, x, Float64((x ^ -1.0) - -0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[Power[N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(0.08333333333333333 * x + N[(N[Power[x, -1.0], $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{{x}^{3} \cdot \color{blue}{\left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{x}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1}{\frac{-1}{2} \cdot \left(x \cdot x\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 38.5%

       \[\leadsto \frac{1}{-0.5 \cdot \left(x \cdot x\right)} \]

   if 0.0 < (exp.f64 x)

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{1 + \color{blue}{\left(x \cdot \frac{1}{2} + x \cdot \left(\frac{1}{12} \cdot x\right)\right)}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{1 + \left(\color{blue}{\frac{1}{2} \cdot x} + x \cdot \left(\frac{1}{12} \cdot x\right)\right)}{x} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{12} \cdot x\right)}}{x} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x} + \frac{x \cdot \left(\frac{1}{12} \cdot x\right)}{x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot x}}{x} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot \frac{x}{x}} \]

      7. *-inverses N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{12} \cdot x\right) \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \color{blue}{\frac{1}{12} \cdot x} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \frac{1 + \frac{1}{2} \cdot x}{x}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1 + \color{blue}{x \cdot \frac{1}{2}}}{x}\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{\color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}}}{x}\right) \]

      13. div-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \color{blue}{\frac{1}{x} - \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}}{x}}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\frac{1}{2}}{x}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\frac{1}{2}}{x}\right)\right)}\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \frac{1}{2}}{x}}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot x}}{x}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{x \cdot \frac{1}{2}}}{x}\right)\right)\right) \]

      19. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{x \cdot \frac{1}{2}}{\color{blue}{x \cdot 1}}\right)\right)\right) \]

      20. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{2}}{1}}\right)\right)\right) \]

      21. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{1} \cdot \frac{\frac{1}{2}}{1}\right)\right)\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(1 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right) \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} - -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;{\left(-0.5 \cdot \left(x \cdot x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. lift--.f64 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

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

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

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

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

   8. pow2 N/A

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

   9. sqrt-pow1 N/A

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

   10. metadata-eval N/A

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

   11. unpow1 N/A

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

   12. lift-exp.f64 N/A

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

   13. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 4: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(-0.041666666666666664, x, 0.16666666666666666\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)}^{-1}\\ \mathbf{elif}\;x \leq -3.6:\\ \;\;\;\;{\left(\frac{t\_0 \cdot t\_0 - x \cdot x}{t\_0 - x}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (- (* (* -0.041666666666666664 x) x) 0.5) x) x)))
   (if (<= x -2.6e+77)
     (pow
      (* (* (fma -0.041666666666666664 x 0.16666666666666666) (* x x)) x)
      -1.0)
     (if (<= x -3.6)
       (pow (/ (- (* t_0 t_0) (* x x)) (- t_0 x)) -1.0)
       (fma
        (fma (* x x) -0.001388888888888889 0.08333333333333333)
        x
        (- (pow x -1.0) -0.5))))))

C

double code(double x) {
	double t_0 = ((((-0.041666666666666664 * x) * x) - 0.5) * x) * x;
	double tmp;
	if (x <= -2.6e+77) {
		tmp = pow(((fma(-0.041666666666666664, x, 0.16666666666666666) * (x * x)) * x), -1.0);
	} else if (x <= -3.6) {
		tmp = pow((((t_0 * t_0) - (x * x)) / (t_0 - x)), -1.0);
	} else {
		tmp = fma(fma((x * x), -0.001388888888888889, 0.08333333333333333), x, (pow(x, -1.0) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(Float64(-0.041666666666666664 * x) * x) - 0.5) * x) * x)
	tmp = 0.0
	if (x <= -2.6e+77)
		tmp = Float64(Float64(fma(-0.041666666666666664, x, 0.16666666666666666) * Float64(x * x)) * x) ^ -1.0;
	elseif (x <= -3.6)
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(x * x)) / Float64(t_0 - x)) ^ -1.0;
	else
		tmp = fma(fma(Float64(x * x), -0.001388888888888889, 0.08333333333333333), x, Float64((x ^ -1.0) - -0.5));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(N[(-0.041666666666666664 * x), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.6e+77], N[Power[N[(N[(N[(-0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[x, -3.6], N[Power[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.08333333333333333), $MachinePrecision] * x + N[(N[Power[x, -1.0], $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6000000000000002e77

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 100.0%

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

   if -2.6000000000000002e77 < x < -3.60000000000000009

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 5.2

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

   7. Applied rewrites 5.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 5.2%

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

   12. Applied rewrites 46.3%

       \[\leadsto \frac{1}{\frac{\left(\left(\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x\right) \cdot x\right) - x \cdot x}{\color{blue}{\left(\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x\right) \cdot x - x}}} \]

   if -3.60000000000000009 < x

   1. Initial program 6.5%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 6.5

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, \frac{1}{x} - -0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(-0.041666666666666664, x, 0.16666666666666666\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)}^{-1}\\ \mathbf{elif}\;x \leq -3.6:\\ \;\;\;\;{\left(\frac{\left(\left(\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x\right) \cdot x\right) - x \cdot x}{\left(\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x\right) \cdot x - x}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{x} \cdot \left({x}^{-1} - 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (exp x) (- (pow x -1.0) 0.5)))

C

double code(double x) {
	return exp(x) * (pow(x, -1.0) - 0.5);
}

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) * ((x ** (-1.0d0)) - 0.5d0)
end function

Java

public static double code(double x) {
	return Math.exp(x) * (Math.pow(x, -1.0) - 0.5);
}

Python

def code(x):
	return math.exp(x) * (math.pow(x, -1.0) - 0.5)

Julia

function code(x)
	return Float64(exp(x) * Float64((x ^ -1.0) - 0.5))
end

MATLAB

function tmp = code(x)
	tmp = exp(x) * ((x ^ -1.0) - 0.5);
end

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. lift-/.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 38.6

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

   7. lift--.f64 N/A

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

   8. unpow1 N/A

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

   9. metadata-eval N/A

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

   10. sqrt-pow1 N/A

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

   11. pow2 N/A

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

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

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

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

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

   14. pow2 N/A

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

   15. sqrt-pow1 N/A

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

   16. metadata-eval N/A

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

   17. unpow1 N/A

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

   18. lift-exp.f64 N/A

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

   19. lower-expm1.f64 100.0

       \[\leadsto e^{x} \cdot {\color{blue}{\left(\mathsf{expm1}\left(x\right)\right)}}^{-1} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{e^{x} \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{-1}} \]

5. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

      \[\leadsto e^{x} \cdot \frac{1 - \color{blue}{\frac{1}{2}} \cdot x}{x} \]

   3. div-sub N/A

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

   4. *-rgt-identity N/A

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

   5. associate-*r* N/A

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

   6. associate-*r/ N/A

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

   7. associate-*l* N/A

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

   8. rgt-mult-inverse N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 98.7

       \[\leadsto e^{x} \cdot \left(\color{blue}{\frac{1}{x}} - 0.5\right) \]

7. Applied rewrites 98.7%

   \[\leadsto e^{x} \cdot \color{blue}{\left(\frac{1}{x} - 0.5\right)} \]

8. Final simplification 98.7%

   \[\leadsto e^{x} \cdot \left({x}^{-1} - 0.5\right) \]
9. Add Preprocessing

Alternative 6: 94.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x\\ \mathbf{if}\;x \leq -2 \cdot 10^{+103}:\\ \;\;\;\;{\left(\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 - t\_0 \cdot t\_0}{1 - t\_0} \cdot x\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (- (* (* -0.041666666666666664 x) x) 0.5) x)))
   (if (<= x -2e+103)
     (pow (* (* (* 0.16666666666666666 x) x) x) -1.0)
     (pow (* (/ (- 1.0 (* t_0 t_0)) (- 1.0 t_0)) x) -1.0))))

C

double code(double x) {
	double t_0 = (((-0.041666666666666664 * x) * x) - 0.5) * x;
	double tmp;
	if (x <= -2e+103) {
		tmp = pow((((0.16666666666666666 * x) * x) * x), -1.0);
	} else {
		tmp = pow((((1.0 - (t_0 * t_0)) / (1.0 - t_0)) * x), -1.0);
	}
	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 = ((((-0.041666666666666664d0) * x) * x) - 0.5d0) * x
    if (x <= (-2d+103)) then
        tmp = (((0.16666666666666666d0 * x) * x) * x) ** (-1.0d0)
    else
        tmp = (((1.0d0 - (t_0 * t_0)) / (1.0d0 - t_0)) * x) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (((-0.041666666666666664 * x) * x) - 0.5) * x;
	double tmp;
	if (x <= -2e+103) {
		tmp = Math.pow((((0.16666666666666666 * x) * x) * x), -1.0);
	} else {
		tmp = Math.pow((((1.0 - (t_0 * t_0)) / (1.0 - t_0)) * x), -1.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (((-0.041666666666666664 * x) * x) - 0.5) * x
	tmp = 0
	if x <= -2e+103:
		tmp = math.pow((((0.16666666666666666 * x) * x) * x), -1.0)
	else:
		tmp = math.pow((((1.0 - (t_0 * t_0)) / (1.0 - t_0)) * x), -1.0)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(-0.041666666666666664 * x) * x) - 0.5) * x)
	tmp = 0.0
	if (x <= -2e+103)
		tmp = Float64(Float64(Float64(0.16666666666666666 * x) * x) * x) ^ -1.0;
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(1.0 - t_0)) * x) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (((-0.041666666666666664 * x) * x) - 0.5) * x;
	tmp = 0.0;
	if (x <= -2e+103)
		tmp = (((0.16666666666666666 * x) * x) * x) ^ -1.0;
	else
		tmp = (((1.0 - (t_0 * t_0)) / (1.0 - t_0)) * x) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(-0.041666666666666664 * x), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2e+103], N[Power[N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2e103

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x} \]

   if -2e103 < x

   1. Initial program 25.5%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 25.5

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 83.5

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

   7. Applied rewrites 83.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 83.1%

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

   12. Applied rewrites 87.5%

       \[\leadsto \frac{1}{\frac{1 - \left(\left(-\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right)\right) \cdot x\right) \cdot \left(\left(-\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right)\right) \cdot x\right)}{1 + \left(-\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right)\right) \cdot x} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+103}:\\ \;\;\;\;{\left(\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 - \left(\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x\right) \cdot \left(\left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x\right)}{1 - \left(\left(-0.041666666666666664 \cdot x\right) \cdot x - 0.5\right) \cdot x} \cdot x\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(-0.041666666666666664, x, 0.16666666666666666\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.7)
   (pow
    (* (* (fma -0.041666666666666664 x 0.16666666666666666) (* x x)) x)
    -1.0)
   (fma
    (fma (* x x) -0.001388888888888889 0.08333333333333333)
    x
    (- (pow x -1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -3.7) {
		tmp = pow(((fma(-0.041666666666666664, x, 0.16666666666666666) * (x * x)) * x), -1.0);
	} else {
		tmp = fma(fma((x * x), -0.001388888888888889, 0.08333333333333333), x, (pow(x, -1.0) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.7)
		tmp = Float64(Float64(fma(-0.041666666666666664, x, 0.16666666666666666) * Float64(x * x)) * x) ^ -1.0;
	else
		tmp = fma(fma(Float64(x * x), -0.001388888888888889, 0.08333333333333333), x, Float64((x ^ -1.0) - -0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.7], N[Power[N[(N[(N[(-0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.08333333333333333), $MachinePrecision] * x + N[(N[Power[x, -1.0], $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000002

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 63.4%

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

   if -3.7000000000000002 < x

   1. Initial program 6.5%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 6.5

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, \frac{1}{x} - -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(-0.041666666666666664, x, 0.16666666666666666\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75:\\ \;\;\;\;\frac{e^{x}}{\left(1 + x\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.75)
   (/ (exp x) (- (+ 1.0 x) 1.0))
   (fma
    (fma (* x x) -0.001388888888888889 0.08333333333333333)
    x
    (- (pow x -1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -3.75) {
		tmp = exp(x) / ((1.0 + x) - 1.0);
	} else {
		tmp = fma(fma((x * x), -0.001388888888888889, 0.08333333333333333), x, (pow(x, -1.0) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.75)
		tmp = Float64(exp(x) / Float64(Float64(1.0 + x) - 1.0));
	else
		tmp = fma(fma(Float64(x * x), -0.001388888888888889, 0.08333333333333333), x, Float64((x ^ -1.0) - -0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.75], N[(N[Exp[x], $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.08333333333333333), $MachinePrecision] * x + N[(N[Power[x, -1.0], $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.75

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -3.75 < x

   1. Initial program 6.5%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 6.5

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, \frac{1}{x} - -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75:\\ \;\;\;\;\frac{e^{x}}{\left(1 + x\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. lift-/.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 38.6

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

   12. lift--.f64 N/A

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

   13. unpow1 N/A

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

   14. metadata-eval N/A

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

   15. sqrt-pow1 N/A

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

   16. pow2 N/A

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

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

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

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

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

   19. pow2 N/A

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

   20. sqrt-pow1 N/A

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

   21. metadata-eval N/A

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

   22. unpow1 N/A

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

   23. lift-exp.f64 N/A

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

   24. lower-expm1.f64 100.0

       \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. remove-double-neg N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 86.4

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

7. Applied rewrites 86.4%

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

8. Final simplification 86.4%

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

Alternative 10: 91.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (pow (* (fma (- (* (* x x) -0.041666666666666664) 0.5) x 1.0) x) -1.0))

C

double code(double x) {
	return pow((fma((((x * x) * -0.041666666666666664) - 0.5), x, 1.0) * x), -1.0);
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(Float64(x * x) * -0.041666666666666664) - 0.5), x, 1.0) * x) ^ -1.0
end

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. lift-/.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 38.6

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

   12. lift--.f64 N/A

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

   13. unpow1 N/A

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

   14. metadata-eval N/A

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

   15. sqrt-pow1 N/A

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

   16. pow2 N/A

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

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

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

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

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

   19. pow2 N/A

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

   20. sqrt-pow1 N/A

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

   21. metadata-eval N/A

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

   22. unpow1 N/A

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

   23. lift-exp.f64 N/A

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

   24. lower-expm1.f64 100.0

       \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. remove-double-neg N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 86.4

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

7. Applied rewrites 86.4%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 86.1%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 86.1%

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

14. Final simplification 86.1%

    \[\leadsto {\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.041666666666666664 - 0.5, x, 1\right) \cdot x\right)}^{-1} \]
15. Add Preprocessing

Alternative 11: 89.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.16666666666666666, x, -0.5\right) \cdot \left(x \cdot x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.5)
   (pow (* (fma 0.16666666666666666 x -0.5) (* x x)) -1.0)
   (fma 0.08333333333333333 x (- (pow x -1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = pow((fma(0.16666666666666666, x, -0.5) * (x * x)), -1.0);
	} else {
		tmp = fma(0.08333333333333333, x, (pow(x, -1.0) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.5)
		tmp = Float64(fma(0.16666666666666666, x, -0.5) * Float64(x * x)) ^ -1.0;
	else
		tmp = fma(0.08333333333333333, x, Float64((x ^ -1.0) - -0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.5], N[Power[N[(N[(0.16666666666666666 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(0.08333333333333333 * x + N[(N[Power[x, -1.0], $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{{x}^{3} \cdot \color{blue}{\left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{x}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.8%

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

   if -3.5 < x

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{1 + \color{blue}{\left(x \cdot \frac{1}{2} + x \cdot \left(\frac{1}{12} \cdot x\right)\right)}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{1 + \left(\color{blue}{\frac{1}{2} \cdot x} + x \cdot \left(\frac{1}{12} \cdot x\right)\right)}{x} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{12} \cdot x\right)}}{x} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x} + \frac{x \cdot \left(\frac{1}{12} \cdot x\right)}{x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot x}}{x} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot \frac{x}{x}} \]

      7. *-inverses N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{12} \cdot x\right) \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \color{blue}{\frac{1}{12} \cdot x} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \frac{1 + \frac{1}{2} \cdot x}{x}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1 + \color{blue}{x \cdot \frac{1}{2}}}{x}\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{\color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}}}{x}\right) \]

      13. div-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \color{blue}{\frac{1}{x} - \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}}{x}}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\frac{1}{2}}{x}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\frac{1}{2}}{x}\right)\right)}\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \frac{1}{2}}{x}}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot x}}{x}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{x \cdot \frac{1}{2}}}{x}\right)\right)\right) \]

      19. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{x \cdot \frac{1}{2}}{\color{blue}{x \cdot 1}}\right)\right)\right) \]

      20. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{2}}{1}}\right)\right)\right) \]

      21. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{1} \cdot \frac{\frac{1}{2}}{1}\right)\right)\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(1 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right) \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} - -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.16666666666666666, x, -0.5\right) \cdot \left(x \cdot x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;{\left(\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (pow (* (* (* 0.16666666666666666 x) x) x) -1.0)
   (fma 0.08333333333333333 x (- (pow x -1.0) -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = pow((((0.16666666666666666 * x) * x) * x), -1.0);
	} else {
		tmp = fma(0.08333333333333333, x, (pow(x, -1.0) - -0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(Float64(Float64(0.16666666666666666 * x) * x) * x) ^ -1.0;
	else
		tmp = fma(0.08333333333333333, x, Float64((x ^ -1.0) - -0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[Power[N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision], N[(0.08333333333333333 * x + N[(N[Power[x, -1.0], $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

      12. lift--.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow1 N/A

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

      16. pow2 N/A

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

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

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

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

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

      19. pow2 N/A

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

      20. sqrt-pow1 N/A

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

      21. metadata-eval N/A

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

      22. unpow1 N/A

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

      23. lift-exp.f64 N/A

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

      24. lower-expm1.f64 100.0

          \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 53.8%

       \[\leadsto \frac{1}{\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x} \]

   if -4.20000000000000018 < x

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{1 + \color{blue}{\left(x \cdot \frac{1}{2} + x \cdot \left(\frac{1}{12} \cdot x\right)\right)}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{1 + \left(\color{blue}{\frac{1}{2} \cdot x} + x \cdot \left(\frac{1}{12} \cdot x\right)\right)}{x} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{12} \cdot x\right)}}{x} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x} + \frac{x \cdot \left(\frac{1}{12} \cdot x\right)}{x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot x}}{x} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot \frac{x}{x}} \]

      7. *-inverses N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{12} \cdot x\right) \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \color{blue}{\frac{1}{12} \cdot x} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \frac{1 + \frac{1}{2} \cdot x}{x}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1 + \color{blue}{x \cdot \frac{1}{2}}}{x}\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{\color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}}}{x}\right) \]

      13. div-sub N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \color{blue}{\frac{1}{x} - \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}}{x}}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\frac{1}{2}}{x}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\frac{1}{2}}{x}\right)\right)}\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \frac{1}{2}}{x}}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot x}}{x}\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{x \cdot \frac{1}{2}}}{x}\right)\right)\right) \]

      19. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{x \cdot \frac{1}{2}}{\color{blue}{x \cdot 1}}\right)\right)\right) \]

      20. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{2}}{1}}\right)\right)\right) \]

      21. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{1} \cdot \frac{\frac{1}{2}}{1}\right)\right)\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(1 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right) \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} - -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;{\left(\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (pow (* (fma (fma 0.16666666666666666 x -0.5) x 1.0) x) -1.0))

C

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

Julia

function code(x)
	return Float64(fma(fma(0.16666666666666666, x, -0.5), x, 1.0) * x) ^ -1.0
end

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. lift-/.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 38.6

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

   12. lift--.f64 N/A

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

   13. unpow1 N/A

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

   14. metadata-eval N/A

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

   15. sqrt-pow1 N/A

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

   16. pow2 N/A

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

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

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

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

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

   19. pow2 N/A

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

   20. sqrt-pow1 N/A

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

   21. metadata-eval N/A

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

   22. unpow1 N/A

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

   23. lift-exp.f64 N/A

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

   24. lower-expm1.f64 100.0

       \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 83.0

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

7. Applied rewrites 83.0%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{{x}^{3} \cdot \color{blue}{\left(\frac{1}{6} - \frac{1}{2} \cdot \frac{1}{x}\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 20.9%

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

11. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

13. Applied rewrites 83.0%

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

14. Final simplification 83.0%

    \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, -0.5\right), x, 1\right) \cdot x\right)}^{-1} \]
15. Add Preprocessing

Alternative 14: 83.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (pow (* (fma -0.5 x 1.0) x) -1.0))

C

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

Julia

function code(x)
	return Float64(fma(-0.5, x, 1.0) * x) ^ -1.0
end

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. lift-/.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 38.6

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

   12. lift--.f64 N/A

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

   13. unpow1 N/A

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

   14. metadata-eval N/A

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

   15. sqrt-pow1 N/A

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

   16. pow2 N/A

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

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

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

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

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

   19. pow2 N/A

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

   20. sqrt-pow1 N/A

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

   21. metadata-eval N/A

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

   22. unpow1 N/A

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

   23. lift-exp.f64 N/A

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

   24. lower-expm1.f64 100.0

       \[\leadsto \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{e^{-x} \cdot \mathsf{expm1}\left(x\right)}} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 77.5

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

7. Applied rewrites 77.5%

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

8. Final simplification 77.5%

   \[\leadsto {\left(\mathsf{fma}\left(-0.5, x, 1\right) \cdot x\right)}^{-1} \]
9. Add Preprocessing

Alternative 15: 66.3% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (pow x -1.0))

C

double code(double x) {
	return pow(x, -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 = x ** (-1.0d0)
end function

Java

public static double code(double x) {
	return Math.pow(x, -1.0);
}

Python

def code(x):
	return math.pow(x, -1.0)

Julia

function code(x)
	return x ^ -1.0
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -1.0], $MachinePrecision]

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 65.6

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

5. Applied rewrites 65.6%

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

6. Final simplification 65.6%

   \[\leadsto {x}^{-1} \]
7. Add Preprocessing

Alternative 16: 3.4% accurate, 35.8× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.08333333333333333 x))

C

double code(double x) {
	return 0.08333333333333333 * 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 = 0.08333333333333333d0 * x
end function

Java

public static double code(double x) {
	return 0.08333333333333333 * x;
}

Python

def code(x):
	return 0.08333333333333333 * x

Julia

function code(x)
	return Float64(0.08333333333333333 * x)
end

MATLAB

function tmp = code(x)
	tmp = 0.08333333333333333 * x;
end

Wolfram

code[x_] := N[(0.08333333333333333 * x), $MachinePrecision]

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
4. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \frac{1 + \color{blue}{\left(x \cdot \frac{1}{2} + x \cdot \left(\frac{1}{12} \cdot x\right)\right)}}{x} \]

   2. *-commutative N/A

      \[\leadsto \frac{1 + \left(\color{blue}{\frac{1}{2} \cdot x} + x \cdot \left(\frac{1}{12} \cdot x\right)\right)}{x} \]

   3. associate-+r+ N/A

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(\frac{1}{12} \cdot x\right)}}{x} \]

   4. div-add N/A

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x} + \frac{x \cdot \left(\frac{1}{12} \cdot x\right)}{x}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \frac{\color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot x}}{x} \]

   6. associate-/l* N/A

      \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot \frac{x}{x}} \]

   7. *-inverses N/A

      \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \left(\frac{1}{12} \cdot x\right) \cdot \color{blue}{1} \]

   8. *-rgt-identity N/A

      \[\leadsto \frac{1 + \frac{1}{2} \cdot x}{x} + \color{blue}{\frac{1}{12} \cdot x} \]

   9. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \frac{1 + \frac{1}{2} \cdot x}{x}} \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1 + \color{blue}{x \cdot \frac{1}{2}}}{x}\right) \]

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

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{\color{blue}{1 - \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}}}{x}\right) \]

   13. div-sub N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \color{blue}{\frac{1}{x} - \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}}{x}}\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\frac{1}{2}}{x}}\right) \]

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

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\frac{1}{2}}{x}\right)\right)}\right) \]

   16. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \frac{1}{2}}{x}}\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot x}}{x}\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{\color{blue}{x \cdot \frac{1}{2}}}{x}\right)\right)\right) \]

   19. *-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\frac{x \cdot \frac{1}{2}}{\color{blue}{x \cdot 1}}\right)\right)\right) \]

   20. times-frac N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{2}}{1}}\right)\right)\right) \]

   21. *-inverses N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{1} \cdot \frac{\frac{1}{2}}{1}\right)\right)\right) \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(1 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x} - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right) \]

5. Applied rewrites 65.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} - -0.5\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{12} \cdot \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 3.4%

   \[\leadsto 0.08333333333333333 \cdot \color{blue}{x} \]

9. Final simplification 3.4%

   \[\leadsto 0.08333333333333333 \cdot x \]
10. Add Preprocessing

Alternative 17: 3.2% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

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 = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

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

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

   3. div-sub N/A

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

   4. associate-*r/ N/A

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

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

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. *-rgt-identity N/A

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

   10. times-frac N/A

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

   11. *-inverses N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval 65.4

       \[\leadsto \frac{1}{x} - \color{blue}{-0.5} \]

5. Applied rewrites 65.4%

   \[\leadsto \color{blue}{\frac{1}{x} - -0.5} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{2} \]
7. Step-by-step derivation

8. Applied rewrites 3.4%

   \[\leadsto 0.5 \]

9. Final simplification 3.4%

   \[\leadsto 0.5 \]
10. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{expm1}\left(-x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0) (expm1 (- x))))

C

double code(double x) {
	return -1.0 / expm1(-x);
}

Java

public static double code(double x) {
	return -1.0 / Math.expm1(-x);
}

Python

def code(x):
	return -1.0 / math.expm1(-x)

Julia

function code(x)
	return Float64(Float64(-1.0) / expm1(Float64(-x)))
end

Wolfram

code[x_] := N[((-1.0) / N[(Exp[(-x)] - 1), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024346 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :alt
  (! :herbie-platform default (/ (- 1) (expm1 (- x))))

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