expq2 (section 3.11)

Percentage Accurate: 37.4% → 100.0%

Time: 3.1s

Alternatives: 15

Speedup: 17.9×

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: 37.4% 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, 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 35.6%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 98.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 98.7

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

5. Applied rewrites 98.7%

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

Alternative 3: 99.3% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.7)
   (/ (exp x) x)
   (/
    (fma
     (fma (fma (* x x) -0.001388888888888889 0.08333333333333333) x 0.5)
     x
     1.0)
    x)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   if -3.7000000000000002 < x

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{1}{2}\right), x, 1\right)}{x} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{1}{2}\right), x, 1\right)}{x} \]

      12. lower-*.f64 99.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, 0.5\right), x, 1\right)}{x} \]

   5. Applied rewrites 99.1%

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

Alternative 4: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right) \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \cdot x}\\ \mathbf{elif}\;x \leq -5.2:\\ \;\;\;\;\frac{1}{\frac{x \cdot x - t\_0 \cdot t\_0}{x - t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, 0.5\right), x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          (* (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x)
          x)))
   (if (<= x -2.5e+77)
     (/ 1.0 (* (fma (* (* x x) 0.041666666666666664) x 1.0) x))
     (if (<= x -5.2)
       (/ 1.0 (/ (- (* x x) (* t_0 t_0)) (- x t_0)))
       (/
        (fma
         (fma (fma (* x x) -0.001388888888888889 0.08333333333333333) x 0.5)
         x
         1.0)
        x)))))

C

double code(double x) {
	double t_0 = (fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5) * x) * x;
	double tmp;
	if (x <= -2.5e+77) {
		tmp = 1.0 / (fma(((x * x) * 0.041666666666666664), x, 1.0) * x);
	} else if (x <= -5.2) {
		tmp = 1.0 / (((x * x) - (t_0 * t_0)) / (x - t_0));
	} else {
		tmp = fma(fma(fma((x * x), -0.001388888888888889, 0.08333333333333333), x, 0.5), x, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5) * x) * x)
	tmp = 0.0
	if (x <= -2.5e+77)
		tmp = Float64(1.0 / Float64(fma(Float64(Float64(x * x) * 0.041666666666666664), x, 1.0) * x));
	elseif (x <= -5.2)
		tmp = Float64(1.0 / Float64(Float64(Float64(x * x) - Float64(t_0 * t_0)) / Float64(x - t_0)));
	else
		tmp = Float64(fma(fma(fma(Float64(x * x), -0.001388888888888889, 0.08333333333333333), x, 0.5), x, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.5e+77], N[(1.0 / N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2], N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.08333333333333333), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000002e77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.9%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 100.0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24}, x, 1\right) \cdot x} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24}, x, 1\right) \cdot x} \]

       3. pow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24}, x, 1\right) \cdot x} \]

       4. lift-*.f64 100.0

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

   14. Applied rewrites 100.0%

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

   if -2.50000000000000002e77 < x < -5.20000000000000018

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 5.8

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

   11. Applied rewrites 5.8%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. *-commutative N/A

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

       13. *-rgt-identity N/A

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

       14. flip-+ N/A

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

       15. lower-/.f64 N/A

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

   13. Applied rewrites 66.7%

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

   if -5.20000000000000018 < x

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{1}{2}\right), x, 1\right)}{x} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{1}{2}\right), x, 1\right)}{x} \]

      12. lower-*.f64 99.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, 0.5\right), x, 1\right)}{x} \]

   5. Applied rewrites 99.1%

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

Alternative 5: 94.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x}\\ \mathbf{elif}\;x \leq -5.2:\\ \;\;\;\;\frac{1}{\frac{1 - t\_0 \cdot t\_0}{1 - t\_0} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, 0.5\right), x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (* x (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5))))
   (if (<= x -1e+104)
     (/ 1.0 (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))
     (if (<= x -5.2)
       (/ 1.0 (* (/ (- 1.0 (* t_0 t_0)) (- 1.0 t_0)) x))
       (/
        (fma
         (fma (fma (* x x) -0.001388888888888889 0.08333333333333333) x 0.5)
         x
         1.0)
        x)))))

C

double code(double x) {
	double t_0 = x * fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5);
	double tmp;
	if (x <= -1e+104) {
		tmp = 1.0 / (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x);
	} else if (x <= -5.2) {
		tmp = 1.0 / (((1.0 - (t_0 * t_0)) / (1.0 - t_0)) * x);
	} else {
		tmp = fma(fma(fma((x * x), -0.001388888888888889, 0.08333333333333333), x, 0.5), x, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5))
	tmp = 0.0
	if (x <= -1e+104)
		tmp = Float64(1.0 / Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x));
	elseif (x <= -5.2)
		tmp = Float64(1.0 / Float64(Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(1.0 - t_0)) * x));
	else
		tmp = Float64(fma(fma(fma(Float64(x * x), -0.001388888888888889, 0.08333333333333333), x, 0.5), x, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e+104], N[(1.0 / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2], N[(1.0 / N[(N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.08333333333333333), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e104

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.2%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -1e104 < x < -5.20000000000000018

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.6%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 25.8

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

   11. Applied rewrites 25.8%

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

       1. lift-fma.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. +-commutative N/A

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

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

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

       10. flip-- N/A

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

       11. lower-/.f64 N/A

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

   13. Applied rewrites 50.8%

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

   if -5.20000000000000018 < x

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{1}{2}\right), x, 1\right)}{x} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{1}{2}\right), x, 1\right)}{x} \]

      12. lower-*.f64 99.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, 0.5\right), x, 1\right)}{x} \]

   5. Applied rewrites 99.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x}\\ \mathbf{elif}\;x \leq -5.2:\\ \;\;\;\;\frac{1}{\frac{1 - \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right)\right)}{1 - x \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right)} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, 0.5\right), x, 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, x, 0.5\right), x, 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, 0.5\right), x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.6)
   (/ 1.0 (* (fma (fma (* 0.041666666666666664 x) x 0.5) x 1.0) x))
   (/
    (fma
     (fma (fma (* x x) -0.001388888888888889 0.08333333333333333) x 0.5)
     x
     1.0)
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -4.6) {
		tmp = 1.0 / (fma(fma((0.041666666666666664 * x), x, 0.5), x, 1.0) * x);
	} else {
		tmp = fma(fma(fma((x * x), -0.001388888888888889, 0.08333333333333333), x, 0.5), x, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.6)
		tmp = Float64(1.0 / Float64(fma(fma(Float64(0.041666666666666664 * x), x, 0.5), x, 1.0) * x));
	else
		tmp = Float64(fma(fma(fma(Float64(x * x), -0.001388888888888889, 0.08333333333333333), x, 0.5), x, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.6], N[(1.0 / N[(N[(N[(N[(0.041666666666666664 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.08333333333333333), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.1%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 68.2

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

   11. Applied rewrites 68.2%

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

   12. Taylor expanded in x around inf

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

       1. lower-*.f64 68.2

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

   14. Applied rewrites 68.2%

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

   if -4.5999999999999996 < x

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{1}{2}\right), x, 1\right)}{x} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{1}{2}\right), x, 1\right)}{x} \]

      12. lower-*.f64 99.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, 0.5\right), x, 1\right)}{x} \]

   5. Applied rewrites 99.1%

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

Alternative 7: 91.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, x, 0.5\right), x, 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.5\right), x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -700.0)
   (/ 1.0 (* (fma (fma (* 0.041666666666666664 x) x 0.5) x 1.0) x))
   (/ (fma (fma 0.08333333333333333 x 0.5) x 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -700.0) {
		tmp = 1.0 / (fma(fma((0.041666666666666664 * x), x, 0.5), x, 1.0) * x);
	} else {
		tmp = fma(fma(0.08333333333333333, x, 0.5), x, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(1.0 / Float64(fma(fma(Float64(0.041666666666666664 * x), x, 0.5), x, 1.0) * x));
	else
		tmp = Float64(fma(fma(0.08333333333333333, x, 0.5), x, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -700.0], N[(1.0 / N[(N[(N[(N[(0.041666666666666664 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.08333333333333333 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.1%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 68.2

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

   11. Applied rewrites 68.2%

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

   12. Taylor expanded in x around inf

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

       1. lower-*.f64 68.2

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

   14. Applied rewrites 68.2%

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

   if -700 < x

   1. Initial program 7.9%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 8: 91.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.5\right), x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -700.0)
   (/ 1.0 (* (fma (* (* x x) 0.041666666666666664) x 1.0) x))
   (/ (fma (fma 0.08333333333333333 x 0.5) x 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -700.0) {
		tmp = 1.0 / (fma(((x * x) * 0.041666666666666664), x, 1.0) * x);
	} else {
		tmp = fma(fma(0.08333333333333333, x, 0.5), x, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(1.0 / Float64(fma(Float64(Float64(x * x) * 0.041666666666666664), x, 1.0) * x));
	else
		tmp = Float64(fma(fma(0.08333333333333333, x, 0.5), x, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -700.0], N[(1.0 / N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.08333333333333333 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.1%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 68.2

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

   11. Applied rewrites 68.2%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24}, x, 1\right) \cdot x} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24}, x, 1\right) \cdot x} \]

       3. pow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24}, x, 1\right) \cdot x} \]

       4. lift-*.f64 68.2

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

   14. Applied rewrites 68.2%

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

   if -700 < x

   1. Initial program 7.9%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 9: 89.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.5\right), x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.6)
   (/ 1.0 (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))
   (/ (fma (fma 0.08333333333333333 x 0.5) x 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -4.6) {
		tmp = 1.0 / (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x);
	} else {
		tmp = fma(fma(0.08333333333333333, x, 0.5), x, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.6)
		tmp = Float64(1.0 / Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x));
	else
		tmp = Float64(fma(fma(0.08333333333333333, x, 0.5), x, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.6], N[(1.0 / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.08333333333333333 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.1%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 59.3

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

   11. Applied rewrites 59.3%

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

   if -4.5999999999999996 < x

   1. Initial program 7.9%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 10: 83.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, x, 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.5\right), x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -700.0)
   (/ 1.0 (* (fma 0.5 x 1.0) x))
   (/ (fma (fma 0.08333333333333333 x 0.5) x 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -700.0) {
		tmp = 1.0 / (fma(0.5, x, 1.0) * x);
	} else {
		tmp = fma(fma(0.08333333333333333, x, 0.5), x, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(1.0 / Float64(fma(0.5, x, 1.0) * x));
	else
		tmp = Float64(fma(fma(0.08333333333333333, x, 0.5), x, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -700.0], N[(1.0 / N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.08333333333333333 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.1%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 39.8

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

   11. Applied rewrites 39.8%

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

   if -700 < x

   1. Initial program 7.9%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 11: 82.8% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 97.6%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

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

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

   5. metadata-eval N/A

      \[\leadsto \frac{x - -1}{x} \]

   6. lower--.f64 68.5

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

8. Applied rewrites 68.5%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 80.2

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

11. Applied rewrites 80.2%

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

Alternative 12: 82.1% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 97.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 69.0%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 79.4

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

11. Applied rewrites 79.4%

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

Alternative 13: 67.2% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 69.6

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

5. Applied rewrites 69.6%

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

Alternative 14: 67.1% accurate, 17.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 97.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 69.0%

   \[\leadsto \frac{\color{blue}{1}}{x} \]
9. Add Preprocessing

Alternative 15: 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 35.6%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 69.6

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

5. Applied rewrites 69.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 3.4%

   \[\leadsto 0.5 \]
9. 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 2025051 
(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)))