Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A

Percentage Accurate: 95.0% → 99.7%

Time: 4.5s

Alternatives: 10

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(-x, y, e^{z} \cdot 1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.15e+21)
   (- x (/ 1.0 x))
   (+ x (/ y (fma (- x) y (* (exp z) 1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e+21) {
		tmp = x - (1.0 / x);
	} else {
		tmp = x + (y / fma(-x, y, (exp(z) * 1.1283791670955126)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15e+21)
		tmp = Float64(x - Float64(1.0 / x));
	else
		tmp = Float64(x + Float64(y / fma(Float64(-x), y, Float64(exp(z) * 1.1283791670955126))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15e+21], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[((-x) * y + N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15e21

   1. Initial program 87.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -1.15e21 < z

   1. Initial program 97.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - \color{blue}{x \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z}} - x \cdot y} \]

      4. lift-exp.f64 N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{e^{z}} - x \cdot y} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + \left(\mathsf{neg}\left(x\right)\right) \cdot y}} \]

      6. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + \color{blue}{\left(-1 \cdot x\right)} \cdot y} \]

      7. associate-*r* N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + \color{blue}{-1 \cdot \left(x \cdot y\right)}} \]

      8. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{-1 \cdot \left(x \cdot y\right) + \frac{5641895835477563}{5000000000000000} \cdot e^{z}}} \]

      9. associate-*r* N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(-1 \cdot x\right) \cdot y} + \frac{5641895835477563}{5000000000000000} \cdot e^{z}} \]

      10. mul-1-neg N/A

          \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot y + \frac{5641895835477563}{5000000000000000} \cdot e^{z}} \]

      11. lower-fma.f64 N/A

          \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), y, \frac{5641895835477563}{5000000000000000} \cdot e^{z}\right)}} \]

      12. lower-neg.f64 N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{-x}, y, \frac{5641895835477563}{5000000000000000} \cdot e^{z}\right)} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{e^{z} \cdot \frac{5641895835477563}{5000000000000000}}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{e^{z} \cdot \frac{5641895835477563}{5000000000000000}}\right)} \]

      15. lift-exp.f64 99.6

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{e^{z}} \cdot 1.1283791670955126\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(-x, y, e^{z} \cdot 1.1283791670955126\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.000135:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.15e+21)
   (- x (/ 1.0 x))
   (if (<= z 0.000135)
     (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e+21) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.000135) {
		tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	} else {
		tmp = 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.15d+21)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 0.000135d0) then
        tmp = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e+21) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.000135) {
		tmp = x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.15e+21:
		tmp = x - (1.0 / x)
	elif z <= 0.000135:
		tmp = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15e+21)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 0.000135)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.15e+21)
		tmp = x - (1.0 / x);
	elseif (z <= 0.000135)
		tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15e+21], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.000135], N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15e21

   1. Initial program 87.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -1.15e21 < z < 1.35000000000000002e-4

   1. Initial program 99.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   if 1.35000000000000002e-4 < z

   1. Initial program 93.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.4%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(-x, y, 1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{e^{z}}, 0.8862269254527579, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.15e+21)
   (- x (/ 1.0 x))
   (if (<= z 2.1e-23)
     (+ x (/ y (fma (- x) y 1.1283791670955126)))
     (fma (/ y (exp z)) 0.8862269254527579 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e+21) {
		tmp = x - (1.0 / x);
	} else if (z <= 2.1e-23) {
		tmp = x + (y / fma(-x, y, 1.1283791670955126));
	} else {
		tmp = fma((y / exp(z)), 0.8862269254527579, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15e+21)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 2.1e-23)
		tmp = Float64(x + Float64(y / fma(Float64(-x), y, 1.1283791670955126)));
	else
		tmp = fma(Float64(y / exp(z)), 0.8862269254527579, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15e+21], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e-23], N[(x + N[(y / N[((-x) * y + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[Exp[z], $MachinePrecision]), $MachinePrecision] * 0.8862269254527579 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15e21

   1. Initial program 87.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -1.15e21 < z < 2.1000000000000001e-23

   1. Initial program 99.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - \color{blue}{x \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z}} - x \cdot y} \]

      4. lift-exp.f64 N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{e^{z}} - x \cdot y} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + \left(\mathsf{neg}\left(x\right)\right) \cdot y}} \]

      6. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + \color{blue}{\left(-1 \cdot x\right)} \cdot y} \]

      7. associate-*r* N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + \color{blue}{-1 \cdot \left(x \cdot y\right)}} \]

      8. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{-1 \cdot \left(x \cdot y\right) + \frac{5641895835477563}{5000000000000000} \cdot e^{z}}} \]

      9. associate-*r* N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(-1 \cdot x\right) \cdot y} + \frac{5641895835477563}{5000000000000000} \cdot e^{z}} \]

      10. mul-1-neg N/A

          \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot y + \frac{5641895835477563}{5000000000000000} \cdot e^{z}} \]

      11. lower-fma.f64 N/A

          \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), y, \frac{5641895835477563}{5000000000000000} \cdot e^{z}\right)}} \]

      12. lower-neg.f64 N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{-x}, y, \frac{5641895835477563}{5000000000000000} \cdot e^{z}\right)} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{e^{z} \cdot \frac{5641895835477563}{5000000000000000}}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{e^{z} \cdot \frac{5641895835477563}{5000000000000000}}\right)} \]

      15. lift-exp.f64 99.4

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{e^{z}} \cdot 1.1283791670955126\right)} \]

   3. Applied rewrites 99.4%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(-x, y, e^{z} \cdot 1.1283791670955126\right)}} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{1} \cdot \frac{5641895835477563}{5000000000000000}\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 98.3%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{1} \cdot 1.1283791670955126\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{\frac{5641895835477563}{5000000000000000}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{1.1283791670955126}\right)} \]

   if 2.1000000000000001e-23 < z

   1. Initial program 93.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}} + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{e^{z}}, \color{blue}{\frac{5000000000000000}{5641895835477563}}, x\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{e^{z}}, \frac{5000000000000000}{5641895835477563}, x\right) \]

      5. lift-exp.f64 98.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{e^{z}}, 0.8862269254527579, x\right) \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{e^{z}}, 0.8862269254527579, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.000135:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(-x, y, 1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.15e+21)
   (- x (/ 1.0 x))
   (if (<= z 0.000135) (+ x (/ y (fma (- x) y 1.1283791670955126))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e+21) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.000135) {
		tmp = x + (y / fma(-x, y, 1.1283791670955126));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15e+21)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 0.000135)
		tmp = Float64(x + Float64(y / fma(Float64(-x), y, 1.1283791670955126)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15e+21], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.000135], N[(x + N[(y / N[((-x) * y + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15e21

   1. Initial program 87.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -1.15e21 < z < 1.35000000000000002e-4

   1. Initial program 99.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - \color{blue}{x \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z}} - x \cdot y} \]

      4. lift-exp.f64 N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{e^{z}} - x \cdot y} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + \left(\mathsf{neg}\left(x\right)\right) \cdot y}} \]

      6. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + \color{blue}{\left(-1 \cdot x\right)} \cdot y} \]

      7. associate-*r* N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + \color{blue}{-1 \cdot \left(x \cdot y\right)}} \]

      8. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{-1 \cdot \left(x \cdot y\right) + \frac{5641895835477563}{5000000000000000} \cdot e^{z}}} \]

      9. associate-*r* N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(-1 \cdot x\right) \cdot y} + \frac{5641895835477563}{5000000000000000} \cdot e^{z}} \]

      10. mul-1-neg N/A

          \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot y + \frac{5641895835477563}{5000000000000000} \cdot e^{z}} \]

      11. lower-fma.f64 N/A

          \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), y, \frac{5641895835477563}{5000000000000000} \cdot e^{z}\right)}} \]

      12. lower-neg.f64 N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{-x}, y, \frac{5641895835477563}{5000000000000000} \cdot e^{z}\right)} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{e^{z} \cdot \frac{5641895835477563}{5000000000000000}}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{e^{z} \cdot \frac{5641895835477563}{5000000000000000}}\right)} \]

      15. lift-exp.f64 99.4

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{e^{z}} \cdot 1.1283791670955126\right)} \]

   3. Applied rewrites 99.4%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(-x, y, e^{z} \cdot 1.1283791670955126\right)}} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{1} \cdot \frac{5641895835477563}{5000000000000000}\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 98.2%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{1} \cdot 1.1283791670955126\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{\frac{5641895835477563}{5000000000000000}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.2%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(-x, y, \color{blue}{1.1283791670955126}\right)} \]

   if 1.35000000000000002e-4 < z

   1. Initial program 93.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.4%

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

Alternative 5: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.000135:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.15e+21)
   (- x (/ 1.0 x))
   (if (<= z 0.000135) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e+21) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.000135) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.15d+21)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 0.000135d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e+21) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.000135) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.15e+21:
		tmp = x - (1.0 / x)
	elif z <= 0.000135:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15e+21)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 0.000135)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.15e+21)
		tmp = x - (1.0 / x);
	elseif (z <= 0.000135)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15e+21], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.000135], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15e21

   1. Initial program 87.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -1.15e21 < z < 1.35000000000000002e-4

   1. Initial program 99.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000}} - x \cdot y} \]
   3. Step-by-step derivation

   4. Applied rewrites 98.2%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]

   if 1.35000000000000002e-4 < z

   1. Initial program 93.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.4%

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

Alternative 6: 85.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-24}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.7e-24)
   (- x (/ 1.0 x))
   (if (<= z 7.8e-222) (fma y 0.8862269254527579 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.7e-24) {
		tmp = x - (1.0 / x);
	} else if (z <= 7.8e-222) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.7e-24)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 7.8e-222)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.7e-24], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.8e-222], N[(y * 0.8862269254527579 + x), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.69999999999999996e-24

   1. Initial program 88.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

   if -1.69999999999999996e-24 < z < 7.8000000000000002e-222

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}} + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{e^{z}}, \color{blue}{\frac{5000000000000000}{5641895835477563}}, x\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{e^{z}}, \frac{5000000000000000}{5641895835477563}, x\right) \]

      5. lift-exp.f64 74.4

         \[\leadsto \mathsf{fma}\left(\frac{y}{e^{z}}, 0.8862269254527579, x\right) \]

   4. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{e^{z}}, 0.8862269254527579, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{5000000000000000}{5641895835477563}, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 74.4%

      \[\leadsto \mathsf{fma}\left(y, 0.8862269254527579, x\right) \]

   if 7.8000000000000002e-222 < z

   1. Initial program 95.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 84.9%

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

Alternative 7: 72.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+23}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.08e+23)
   (/ -1.0 x)
   (if (<= z 7.8e-222) (fma y 0.8862269254527579 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.08e+23) {
		tmp = -1.0 / x;
	} else if (z <= 7.8e-222) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.08e+23)
		tmp = Float64(-1.0 / x);
	elseif (z <= 7.8e-222)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.08e+23], N[(-1.0 / x), $MachinePrecision], If[LessEqual[z, 7.8e-222], N[(y * 0.8862269254527579 + x), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0800000000000001e23

   1. Initial program 87.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 50.4

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

   7. Applied rewrites 50.4%

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

   if -1.0800000000000001e23 < z < 7.8000000000000002e-222

   1. Initial program 99.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}} + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{e^{z}}, \color{blue}{\frac{5000000000000000}{5641895835477563}}, x\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{e^{z}}, \frac{5000000000000000}{5641895835477563}, x\right) \]

      5. lift-exp.f64 71.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{e^{z}}, 0.8862269254527579, x\right) \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{e^{z}}, 0.8862269254527579, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{5000000000000000}{5641895835477563}, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 71.9%

      \[\leadsto \mathsf{fma}\left(y, 0.8862269254527579, x\right) \]

   if 7.8000000000000002e-222 < z

   1. Initial program 95.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 84.9%

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

Alternative 8: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0005:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-101}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-219}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-170}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 0.012:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0005)
   x
   (if (<= x -3.7e-101)
     (/ -1.0 x)
     (if (<= x -9e-219)
       x
       (if (<= x 5.3e-170)
         (* y 0.8862269254527579)
         (if (<= x 0.012) (/ -1.0 x) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0005) {
		tmp = x;
	} else if (x <= -3.7e-101) {
		tmp = -1.0 / x;
	} else if (x <= -9e-219) {
		tmp = x;
	} else if (x <= 5.3e-170) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 0.012) {
		tmp = -1.0 / x;
	} else {
		tmp = 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-0.0005d0)) then
        tmp = x
    else if (x <= (-3.7d-101)) then
        tmp = (-1.0d0) / x
    else if (x <= (-9d-219)) then
        tmp = x
    else if (x <= 5.3d-170) then
        tmp = y * 0.8862269254527579d0
    else if (x <= 0.012d0) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0005) {
		tmp = x;
	} else if (x <= -3.7e-101) {
		tmp = -1.0 / x;
	} else if (x <= -9e-219) {
		tmp = x;
	} else if (x <= 5.3e-170) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 0.012) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.0005:
		tmp = x
	elif x <= -3.7e-101:
		tmp = -1.0 / x
	elif x <= -9e-219:
		tmp = x
	elif x <= 5.3e-170:
		tmp = y * 0.8862269254527579
	elif x <= 0.012:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.0005)
		tmp = x;
	elseif (x <= -3.7e-101)
		tmp = Float64(-1.0 / x);
	elseif (x <= -9e-219)
		tmp = x;
	elseif (x <= 5.3e-170)
		tmp = Float64(y * 0.8862269254527579);
	elseif (x <= 0.012)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.0005)
		tmp = x;
	elseif (x <= -3.7e-101)
		tmp = -1.0 / x;
	elseif (x <= -9e-219)
		tmp = x;
	elseif (x <= 5.3e-170)
		tmp = y * 0.8862269254527579;
	elseif (x <= 0.012)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.0005], x, If[LessEqual[x, -3.7e-101], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, -9e-219], x, If[LessEqual[x, 5.3e-170], N[(y * 0.8862269254527579), $MachinePrecision], If[LessEqual[x, 0.012], N[(-1.0 / x), $MachinePrecision], x]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.0000000000000001e-4 or -3.70000000000000005e-101 < x < -9.00000000000000029e-219 or 0.012 < x

   1. Initial program 95.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 89.2%

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

   if -5.0000000000000001e-4 < x < -3.70000000000000005e-101 or 5.3e-170 < x < 0.012

   1. Initial program 96.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 44.9

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

   4. Applied rewrites 44.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 44.2

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

   7. Applied rewrites 44.2%

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

   if -9.00000000000000029e-219 < x < 5.3e-170

   1. Initial program 95.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y}{e^{z}} \cdot \color{blue}{\frac{5000000000000000}{5641895835477563}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{e^{z}} \cdot \color{blue}{\frac{5000000000000000}{5641895835477563}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563} \]

      4. lift-exp.f64 7.1

         \[\leadsto \frac{y}{e^{z}} \cdot 0.8862269254527579 \]

   4. Applied rewrites 7.1%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   5. Taylor expanded in z around 0

      \[\leadsto y \cdot \frac{5000000000000000}{5641895835477563} \]
   6. Step-by-step derivation

   7. Applied rewrites 7.3%

      \[\leadsto y \cdot 0.8862269254527579 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-219}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-56}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9e-219) x (if (<= x 2.7e-56) (* y 0.8862269254527579) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e-219) {
		tmp = x;
	} else if (x <= 2.7e-56) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-9d-219)) then
        tmp = x
    else if (x <= 2.7d-56) then
        tmp = y * 0.8862269254527579d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e-219) {
		tmp = x;
	} else if (x <= 2.7e-56) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9e-219:
		tmp = x
	elif x <= 2.7e-56:
		tmp = y * 0.8862269254527579
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9e-219)
		tmp = x;
	elseif (x <= 2.7e-56)
		tmp = Float64(y * 0.8862269254527579);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9e-219)
		tmp = x;
	elseif (x <= 2.7e-56)
		tmp = y * 0.8862269254527579;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9e-219], x, If[LessEqual[x, 2.7e-56], N[(y * 0.8862269254527579), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000029e-219 or 2.69999999999999995e-56 < x

   1. Initial program 96.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 81.9%

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

   if -9.00000000000000029e-219 < x < 2.69999999999999995e-56

   1. Initial program 91.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y}{e^{z}} \cdot \color{blue}{\frac{5000000000000000}{5641895835477563}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{e^{z}} \cdot \color{blue}{\frac{5000000000000000}{5641895835477563}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563} \]

      4. lift-exp.f64 34.3

         \[\leadsto \frac{y}{e^{z}} \cdot 0.8862269254527579 \]

   4. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   5. Taylor expanded in z around 0

      \[\leadsto y \cdot \frac{5000000000000000}{5641895835477563} \]
   6. Step-by-step derivation

   7. Applied rewrites 32.5%

      \[\leadsto y \cdot 0.8862269254527579 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.8% accurate, 25.8× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return 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, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 95.0%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

2. Taylor expanded in x around inf

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

4. Applied rewrites 68.8%

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

Reproduce

herbie shell --seed 2025120 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64
  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))