expq2 (section 3.11)

Percentage Accurate: 36.8% → 100.0%

Time: 2.7s

Alternatives: 14

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: 36.8% 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 36.8%

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

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

3. Applied rewrites 100.0%

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

Alternative 2: 98.9% 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 36.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \frac{e^{x}}{\color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right)}} \]
3. 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 98.9

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

4. Applied rewrites 98.9%

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

Alternative 3: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 98.2%

   \[\leadsto \frac{e^{x}}{\color{blue}{x}} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right)\\ t_1 := \left(-x\right) \cdot t\_0\\ \mathbf{if}\;x \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{x - -1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -1}{\frac{1 - t\_1 \cdot t\_1}{1 - t\_0 \cdot x} \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5))
        (t_1 (* (- x) t_0)))
   (if (<= x -1e+103)
     (/ (- x -1.0) (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))
     (/ (- x -1.0) (* (/ (- 1.0 (* t_1 t_1)) (- 1.0 (* t_0 x))) x)))))

C

double code(double x) {
	double t_0 = fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5);
	double t_1 = -x * t_0;
	double tmp;
	if (x <= -1e+103) {
		tmp = (x - -1.0) / (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x);
	} else {
		tmp = (x - -1.0) / (((1.0 - (t_1 * t_1)) / (1.0 - (t_0 * x))) * x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5)
	t_1 = Float64(Float64(-x) * t_0)
	tmp = 0.0
	if (x <= -1e+103)
		tmp = Float64(Float64(x - -1.0) / Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x));
	else
		tmp = Float64(Float64(x - -1.0) / Float64(Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(1.0 - Float64(t_0 * x))) * x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[((-x) * t$95$0), $MachinePrecision]}, If[LessEqual[x, -1e+103], N[(N[(x - -1.0), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x - -1.0), $MachinePrecision] / N[(N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e103

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + x}}{x} \]
   6. 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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 3.1

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -1e103 < x

   1. Initial program 19.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + x}}{x} \]
   6. 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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 84.8

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 89.0

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

   10. Applied rewrites 89.0%

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

       1. lift-fma.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. +-commutative N/A

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

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

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

       10. flip-- N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. +-commutative N/A

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

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

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

   12. Applied rewrites 92.0%

       \[\leadsto \frac{x - -1}{\frac{1 - \left(\left(-x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right)\right) \cdot \left(\left(-x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right)\right)}{1 - \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right) \cdot x} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1 + x}}{x} \]
6. 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. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 66.8

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

7. Applied rewrites 66.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 91.4

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

10. Applied rewrites 91.4%

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

Alternative 6: 90.8% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1 + x}}{x} \]
6. 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. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 66.8

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

7. Applied rewrites 66.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 91.4

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

10. Applied rewrites 91.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 90.8%

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

Alternative 7: 84.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.08333333333333333, x, 0.5\right) \cdot x - -1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6.0)
   (/ x (* x x))
   (/ (- (* (fma 0.08333333333333333 x 0.5) x) -1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -6.0) {
		tmp = x / (x * x);
	} else {
		tmp = ((fma(0.08333333333333333, x, 0.5) * x) - -1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -6.0)
		tmp = Float64(x / Float64(x * x));
	else
		tmp = Float64(Float64(Float64(fma(0.08333333333333333, x, 0.5) * x) - -1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -6.0], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.08333333333333333 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   3. 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 3.1

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

   4. Applied rewrites 3.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 1.6

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

   6. Applied rewrites 1.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 52.6%

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

   if -6 < x

   1. Initial program 6.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   3. 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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.5\right), x, 1\right)}{x} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 99.3

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

   9. Applied rewrites 99.3%

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

Alternative 8: 88.9% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1 + x}}{x} \]
6. 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. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 66.8

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

7. Applied rewrites 66.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 88.9

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

10. Applied rewrites 88.9%

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

Alternative 9: 84.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, 0.5\right), x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6.0)
   (/ x (* x x))
   (/ (fma (fma 0.08333333333333333 x 0.5) x 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -6.0) {
		tmp = x / (x * 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 <= -6.0)
		tmp = Float64(x / Float64(x * 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, -6.0], N[(x / N[(x * 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 < -6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   3. 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 3.1

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

   4. Applied rewrites 3.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 1.6

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

   6. Applied rewrites 1.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 52.6%

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

   if -6 < x

   1. Initial program 6.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   3. 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 99.3

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

   4. Applied rewrites 99.3%

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

Alternative 10: 83.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x}, -1, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.7) (/ x (* x x)) (fma (/ -1.0 x) -1.0 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= -1.7) {
		tmp = x / (x * x);
	} else {
		tmp = fma((-1.0 / x), -1.0, 0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.7)
		tmp = Float64(x / Float64(x * x));
	else
		tmp = fma(Float64(-1.0 / x), -1.0, 0.5);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.7], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] * -1.0 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   3. 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 3.1

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

   4. Applied rewrites 3.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 1.6

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

   6. Applied rewrites 1.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 52.6%

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

   if -1.69999999999999996 < x

   1. Initial program 6.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   3. 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 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 54.1

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. frac-add N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. *-rgt-identity N/A

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

      13. times-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 98.7

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

   8. Applied rewrites 98.7%

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

Alternative 11: 83.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, x, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.7) (/ x (* x x)) (/ (fma 0.5 x 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.7) {
		tmp = x / (x * x);
	} else {
		tmp = fma(0.5, x, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.7)
		tmp = Float64(x / Float64(x * x));
	else
		tmp = Float64(fma(0.5, x, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.7], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * x + 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   3. 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 3.1

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

   4. Applied rewrites 3.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 1.6

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

   6. Applied rewrites 1.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 52.6%

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

   if -1.69999999999999996 < x

   1. Initial program 6.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   3. 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 98.7

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

   4. Applied rewrites 98.7%

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

Alternative 12: 83.0% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-15) (/ x (* x x)) (/ (- x -1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-15) {
		tmp = x / (x * x);
	} else {
		tmp = (x - -1.0) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-15)) then
        tmp = x / (x * x)
    else
        tmp = (x - (-1.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e-15) {
		tmp = x / (x * x);
	} else {
		tmp = (x - -1.0) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-15:
		tmp = x / (x * x)
	else:
		tmp = (x - -1.0) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-15)
		tmp = Float64(x / Float64(x * x));
	else
		tmp = Float64(Float64(x - -1.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-15)
		tmp = x / (x * x);
	else
		tmp = (x - -1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-15], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999999e-15

   1. Initial program 98.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
   3. 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 6.5

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

   4. Applied rewrites 6.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 5.0

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

   6. Applied rewrites 5.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 52.6%

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

   if -4.99999999999999999e-15 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + x}}{x} \]
   6. 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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 98.8

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

   7. Applied rewrites 98.8%

      \[\leadsto \frac{\color{blue}{x - -1}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 67.6% 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 36.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 67.6%

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

Alternative 14: 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 36.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}} \]
3. 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 67.5

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

4. Applied rewrites 67.5%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 3.2%

   \[\leadsto 0.5 \]
8. 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 2025106 
(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)))