expq2 (section 3.11)

Percentage Accurate: 38.2% → 100.0%

Time: 5.7s

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: 38.2% 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 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 35.9

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

   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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{e^{x}}{\left(1 + x\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.001388888888888889, \mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.8)
   (/ (exp x) (- (+ 1.0 x) 1.0))
   (fma
    (pow x 3.0)
    -0.001388888888888889
    (fma 0.08333333333333333 x (- (pow x -1.0) -0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.8) {
		tmp = exp(x) / ((1.0 + x) - 1.0);
	} else {
		tmp = fma(pow(x, 3.0), -0.001388888888888889, fma(0.08333333333333333, x, (pow(x, -1.0) - -0.5)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.8)
		tmp = Float64(exp(x) / Float64(Float64(1.0 + x) - 1.0));
	else
		tmp = fma((x ^ 3.0), -0.001388888888888889, fma(0.08333333333333333, x, Float64((x ^ -1.0) - -0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.8], N[(N[Exp[x], $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 3.0], $MachinePrecision] * -0.001388888888888889 + N[(0.08333333333333333 * x + N[(N[Power[x, -1.0], $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998

   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 -3.7999999999999998 < x

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{e^{x}}{\left(1 + x\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.001388888888888889, \mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.16666666666666666, x, -0.5\right) \cdot \left(x \cdot x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (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 (exp(x) <= 0.0) {
		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 (exp(x) <= 0.0)
		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[N[Exp[x], $MachinePrecision], 0.0], 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 (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. fp-cancel-sign-sub-inv N/A

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. remove-double-neg N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 68.4

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

   7. Applied rewrites 68.4%

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

   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 68.4%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 6.3%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.16666666666666666, x, -0.5\right) \cdot \left(x \cdot x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;{\left(\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (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 (exp(x) <= 0.0) {
		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 (exp(x) <= 0.0)
		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[N[Exp[x], $MachinePrecision], 0.0], 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 (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. fp-cancel-sign-sub-inv N/A

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. remove-double-neg N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 68.4

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

   7. Applied rewrites 68.4%

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

   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 68.4%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 6.3%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 99.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;{\left(\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 6: 93.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(\frac{0.027777777777777776 - \left(0.041666666666666664 \cdot x\right) \cdot \left(0.041666666666666664 \cdot x\right)}{0.16666666666666666} \cdot x - 0.5, x, 1\right) \cdot x\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (*
   (fma
    (-
     (*
      (/
       (-
        0.027777777777777776
        (* (* 0.041666666666666664 x) (* 0.041666666666666664 x)))
       0.16666666666666666)
      x)
     0.5)
    x
    1.0)
   x)
  -1.0))

C

double code(double x) {
	return pow((fma(((((0.027777777777777776 - ((0.041666666666666664 * x) * (0.041666666666666664 * x))) / 0.16666666666666666) * x) - 0.5), x, 1.0) * x), -1.0);
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(Float64(Float64(0.027777777777777776 - Float64(Float64(0.041666666666666664 * x) * Float64(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[(N[(0.027777777777777776 - N[(N[(0.041666666666666664 * x), $MachinePrecision] * N[(0.041666666666666664 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 35.9

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

   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. distribute-lft-neg-out N/A

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

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

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

   8. remove-double-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 91.4

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

7. Applied rewrites 91.4%

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

9. Applied rewrites 91.4%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.027777777777777776 - \left(0.041666666666666664 \cdot x\right) \cdot \left(0.041666666666666664 \cdot x\right)}{0.16666666666666666 + 0.041666666666666664 \cdot x} \cdot x - 0.5, x, 1\right) \cdot x} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\frac{1}{36} - \left(\frac{1}{24} \cdot x\right) \cdot \left(\frac{1}{24} \cdot x\right)}{\frac{1}{6}} \cdot x - \frac{1}{2}, x, 1\right) \cdot x} \]
11. Step-by-step derivation

12. Applied rewrites 93.1%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.027777777777777776 - \left(0.041666666666666664 \cdot x\right) \cdot \left(0.041666666666666664 \cdot x\right)}{0.16666666666666666} \cdot x - 0.5, x, 1\right) \cdot x} \]

13. Final simplification 93.1%

    \[\leadsto {\left(\mathsf{fma}\left(\frac{0.027777777777777776 - \left(0.041666666666666664 \cdot x\right) \cdot \left(0.041666666666666664 \cdot x\right)}{0.16666666666666666} \cdot x - 0.5, x, 1\right) \cdot x\right)}^{-1} \]
14. Add Preprocessing

Alternative 7: 91.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(-0.041666666666666664, x, 0.16666666666666666\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6)
   (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 <= -3.6) {
		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 <= -3.6)
		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, -3.6], 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 < -3.60000000000000009

   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. distribute-lft-neg-out N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 74.4

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

   7. Applied rewrites 74.4%

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

   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 74.4%

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

   if -3.60000000000000009 < x

   1. Initial program 6.3%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 6.3

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

      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 99.6%

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

4. Final simplification 91.6%

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

Alternative 8: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{e^{x}}{\left(1 + x\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.8)
   (/ (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 <= -3.8) {
		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 <= -3.8)
		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, -3.8], 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 < -3.7999999999999998

   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 -3.7999999999999998 < x

   1. Initial program 6.3%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 6.3

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

      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 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{e^{x}}{\left(1 + x\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, {x}^{-1} - -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.5% 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 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 35.9

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

   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. distribute-lft-neg-out N/A

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

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

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

   8. remove-double-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 91.4

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

7. Applied rewrites 91.4%

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

8. Final simplification 91.4%

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

Alternative 10: 91.1% 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 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 35.9

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

   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. distribute-lft-neg-out N/A

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

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

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

   8. remove-double-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 91.4

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

7. Applied rewrites 91.4%

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

9. Applied rewrites 91.4%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.027777777777777776 - \left(0.041666666666666664 \cdot x\right) \cdot \left(0.041666666666666664 \cdot x\right)}{0.16666666666666666 + 0.041666666666666664 \cdot x} \cdot x - 0.5, x, 1\right) \cdot x} \]

10. Taylor expanded in x around inf

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

12. Applied rewrites 90.9%

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

13. Final simplification 90.9%

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

Alternative 11: 88.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 35.9

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

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

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

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

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

   5. +-commutative N/A

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

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

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

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

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

   8. remove-double-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-*.f64 89.3

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

7. Applied rewrites 89.3%

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

8. Final simplification 89.3%

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

Alternative 12: 83.0% 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 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 35.9

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

   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 82.7

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

7. Applied rewrites 82.7%

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

8. Final simplification 82.7%

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

Alternative 13: 66.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma 0.08333333333333333 x (- (pow x -1.0) -0.5)))

C

double code(double x) {
	return fma(0.08333333333333333, x, (pow(x, -1.0) - -0.5));
}

Julia

function code(x)
	return fma(0.08333333333333333, x, Float64((x ^ -1.0) - -0.5))
end

Wolfram

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

Derivation

1. Initial program 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 68.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} - -0.5\right)} \]

6. Final simplification 68.7%

   \[\leadsto \mathsf{fma}\left(0.08333333333333333, x, {x}^{-1} - -0.5\right) \]
7. Add Preprocessing

Alternative 14: 66.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-1} - -0.5 \end{array} \]

FPCore

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

C

double code(double x) {
	return 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 = (x ** (-1.0d0)) - (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 68.4

       \[\leadsto \frac{1}{x} - \color{blue}{-0.5} \]

5. Applied rewrites 68.4%

   \[\leadsto \color{blue}{\frac{1}{x} - -0.5} \]

6. Final simplification 68.4%

   \[\leadsto {x}^{-1} - -0.5 \]
7. 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 35.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 68.2

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

5. Applied rewrites 68.2%

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

6. Final simplification 68.2%

   \[\leadsto {x}^{-1} \]
7. Add Preprocessing

Alternative 16: 3.3% 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 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 68.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} - -0.5\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{12} \cdot \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 3.2%

   \[\leadsto 0.08333333333333333 \cdot \color{blue}{x} \]
9. 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 35.9%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 68.4

       \[\leadsto \frac{1}{x} - \color{blue}{-0.5} \]

5. Applied rewrites 68.4%

   \[\leadsto \color{blue}{\frac{1}{x} - -0.5} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{2} \]
7. Step-by-step derivation

8. Applied rewrites 3.0%

   \[\leadsto 0.5 \]
9. 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 2024357 
(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)))