expq2 (section 3.11)

Percentage Accurate: 37.5% → 100.0%

Time: 2.2s

Alternatives: 10

Speedup: 6.2×

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: 37.5% 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, 1.3× 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 37.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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: 88.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, 0.5\right) + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ 1.0 (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))
   (+
    (fma (fma (* x x) -0.001388888888888889 0.08333333333333333) x 0.5)
    (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = 1.0 / (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x);
	} else {
		tmp = fma(fma((x * x), -0.001388888888888889, 0.08333333333333333), x, 0.5) + (1.0 / x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[(1.0 / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.08333333333333333), $MachinePrecision] * x + 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 37.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 66.5

         \[\leadsto \frac{1 + \color{blue}{x}}{\mathsf{expm1}\left(x\right)} \]

   6. Applied rewrites 66.5%

      \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 87.9

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

   9. Applied rewrites 87.9%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 87.3%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 37.5%

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

   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 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\mathsf{fma}\left(0.5, x, 1\right)}{x} \]
   8. 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. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 66.9

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

   9. Applied rewrites 66.9%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. pow2 N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lower-*.f64 66.9

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

   12. Applied rewrites 66.9%

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

Alternative 3: 88.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ 1.0 (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))
   (+ (fma 0.08333333333333333 x 0.5) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = 1.0 / (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x);
	} else {
		tmp = fma(0.08333333333333333, x, 0.5) + (1.0 / x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[(1.0 / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.08333333333333333 * x + 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 37.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 66.5

         \[\leadsto \frac{1 + \color{blue}{x}}{\mathsf{expm1}\left(x\right)} \]

   6. Applied rewrites 66.5%

      \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 87.9

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

   9. Applied rewrites 87.9%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 87.3%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 37.5%

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

   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 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\mathsf{fma}\left(0.5, x, 1\right)}{x} \]
   8. 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. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 66.9

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

   9. Applied rewrites 66.9%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 67.0

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

   12. Applied rewrites 67.0%

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

Alternative 4: 88.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{1 + x}{\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, 0.5\right) + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -700.0)
   (/ (+ 1.0 x) (* (* (* 0.16666666666666666 x) x) x))
   (+ (fma 0.08333333333333333 x 0.5) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -700.0) {
		tmp = (1.0 + x) / (((0.16666666666666666 * x) * x) * x);
	} else {
		tmp = fma(0.08333333333333333, x, 0.5) + (1.0 / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(Float64(1.0 + x) / Float64(Float64(Float64(0.16666666666666666 * x) * x) * x));
	else
		tmp = Float64(fma(0.08333333333333333, x, 0.5) + Float64(1.0 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -700.0], N[(N[(1.0 + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.08333333333333333 * x + 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 37.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 66.5

         \[\leadsto \frac{1 + \color{blue}{x}}{\mathsf{expm1}\left(x\right)} \]

   6. Applied rewrites 66.5%

      \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 87.9

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

   9. Applied rewrites 87.9%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 26.3

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

   12. Applied rewrites 26.3%

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

   if -700 < x

   1. Initial program 37.5%

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

   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 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\mathsf{fma}\left(0.5, x, 1\right)}{x} \]
   8. 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. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 66.9

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

   9. Applied rewrites 66.9%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 67.0

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

   12. Applied rewrites 67.0%

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

Alternative 5: 87.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]
5. Step-by-step derivation

   1. lower-+.f64 66.5

      \[\leadsto \frac{1 + \color{blue}{x}}{\mathsf{expm1}\left(x\right)} \]

6. Applied rewrites 66.5%

   \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 87.9

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

9. Applied rewrites 87.9%

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

Alternative 6: 83.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -700.0)
   (/ 1.0 (* (fma 0.5 x 1.0) x))
   (+ (fma 0.08333333333333333 x 0.5) (/ 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(0.08333333333333333, x, 0.5) + (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(0.08333333333333333, x, 0.5) + Float64(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[(0.08333333333333333 * x + 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 37.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 66.5

         \[\leadsto \frac{1 + \color{blue}{x}}{\mathsf{expm1}\left(x\right)} \]

   6. Applied rewrites 66.5%

      \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 82.8

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

   9. Applied rewrites 82.8%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 82.2%

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

   if -700 < x

   1. Initial program 37.5%

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

   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 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\mathsf{fma}\left(0.5, x, 1\right)}{x} \]
   8. 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. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 66.9

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

   9. Applied rewrites 66.9%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 67.0

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

   12. Applied rewrites 67.0%

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

Alternative 7: 82.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]
5. Step-by-step derivation

   1. lower-+.f64 66.5

      \[\leadsto \frac{1 + \color{blue}{x}}{\mathsf{expm1}\left(x\right)} \]

6. Applied rewrites 66.5%

   \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 82.8

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

9. Applied rewrites 82.8%

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

Alternative 8: 67.0% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (+ (fma 0.08333333333333333 x 0.5) (/ 1.0 x)))

C

double code(double x) {
	return fma(0.08333333333333333, x, 0.5) + (1.0 / x);
}

Julia

function code(x)
	return Float64(fma(0.08333333333333333, x, 0.5) + Float64(1.0 / x))
end

Wolfram

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

Derivation

1. Initial program 37.5%

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

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 66.9

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

4. Applied rewrites 66.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 66.9%

   \[\leadsto \frac{\mathsf{fma}\left(0.5, x, 1\right)}{x} \]
8. 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. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-/.f64 66.9

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

9. Applied rewrites 66.9%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 67.0

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

12. Applied rewrites 67.0%

    \[\leadsto \mathsf{fma}\left(0.08333333333333333, x, 0.5\right) + \frac{\color{blue}{1}}{x} \]
13. Add Preprocessing

Alternative 9: 66.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (1.0 / x) - -0.5;
}

Python

def code(x):
	return (1.0 / x) - -0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]
5. Step-by-step derivation

   1. lower-+.f64 66.5

      \[\leadsto \frac{1 + \color{blue}{x}}{\mathsf{expm1}\left(x\right)} \]

6. Applied rewrites 66.5%

   \[\leadsto \frac{\color{blue}{1 + x}}{\mathsf{expm1}\left(x\right)} \]

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 87.9

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

9. Applied rewrites 87.9%

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

10. Taylor expanded in x around 0

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

    1. div-add N/A

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

    2. associate-/l* N/A

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

    3. *-inverses N/A

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

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

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

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

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

    6. metadata-eval N/A

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

    7. lower--.f64 N/A

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

    8. lift-/.f64 N/A

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

    9. metadata-eval 66.9

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

12. Applied rewrites 66.9%

    \[\leadsto \color{blue}{\frac{1}{x} - -0.5} \]
13. Add Preprocessing

Alternative 10: 66.9% accurate, 6.2× 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 37.5%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 66.9

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

4. Applied rewrites 66.9%

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

Developer Target 1: 100.0% accurate, 2.1× 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 2025130 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :alt
  (! :herbie-platform c (/ (- 1) (expm1 (- x))))

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