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

Percentage Accurate: 95.7% → 99.9%

Time: 3.0s

Alternatives: 10

Speedup: 0.8×

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: 95.7% 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: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13500000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-x, y, e^{z} \cdot 1.1283791670955126\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13500000000.0)
   (+ x (/ -1.0 x))
   (+ (/ y (fma (- x) y (* (exp z) 1.1283791670955126))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13500000000.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = (y / fma(-x, y, (exp(z) * 1.1283791670955126))) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13500000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(Float64(y / fma(Float64(-x), y, Float64(exp(z) * 1.1283791670955126))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13500000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[((-x) * y + N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e10

   1. Initial program 89.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -1.35e10 < z

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

Alternative 2: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13500000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13500000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 1.4e-26)
     (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13500000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.4e-26) {
		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 <= (-13500000000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.4d-26) 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 <= -13500000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.4e-26) {
		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 <= -13500000000.0:
		tmp = x + (-1.0 / x)
	elif z <= 1.4e-26:
		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 <= -13500000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.4e-26)
		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 <= -13500000000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.4e-26)
		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, -13500000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.4e-26], 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 < -1.35e10

   1. Initial program 89.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -1.35e10 < z < 1.4000000000000001e-26

   1. Initial program 99.8%

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

   if 1.4000000000000001e-26 < z

   1. Initial program 93.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 97.0%

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

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13500000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13500000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 1.4e-26) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13500000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.4e-26) {
		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 <= (-13500000000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.4d-26) 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 <= -13500000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.4e-26) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -13500000000.0:
		tmp = x + (-1.0 / x)
	elif z <= 1.4e-26:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13500000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.4e-26)
		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 <= -13500000000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.4e-26)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13500000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.4e-26], 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 < -1.35e10

   1. Initial program 89.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -1.35e10 < z < 1.4000000000000001e-26

   1. Initial program 99.8%

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

   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 99.1%

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

   if 1.4000000000000001e-26 < z

   1. Initial program 93.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 97.0%

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

Alternative 4: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -4000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (<= t_1 -4000.0)
     t_0
     (if (<= t_1 4.0) (+ x (/ y (* (exp z) 1.1283791670955126))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = t_0;
	} else if (t_1 <= 4.0) {
		tmp = x + (y / (exp(z) * 1.1283791670955126));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    if (t_1 <= (-4000.0d0)) then
        tmp = t_0
    else if (t_1 <= 4.0d0) then
        tmp = x + (y / (exp(z) * 1.1283791670955126d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = t_0;
	} else if (t_1 <= 4.0) {
		tmp = x + (y / (Math.exp(z) * 1.1283791670955126));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	tmp = 0
	if t_1 <= -4000.0:
		tmp = t_0
	elif t_1 <= 4.0:
		tmp = x + (y / (math.exp(z) * 1.1283791670955126))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -4000.0)
		tmp = t_0;
	elseif (t_1 <= 4.0)
		tmp = Float64(x + Float64(y / Float64(exp(z) * 1.1283791670955126)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	tmp = 0.0;
	if (t_1 <= -4000.0)
		tmp = t_0;
	elseif (t_1 <= 4.0)
		tmp = x + (y / (exp(z) * 1.1283791670955126));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4000.0], t$95$0, If[LessEqual[t$95$1, 4.0], N[(x + N[(y / N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -4e3 or 4 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 91.2

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

   4. Applied rewrites 91.2%

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

   if -4e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 4

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-exp.f64 98.4

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

   4. Applied rewrites 98.4%

      \[\leadsto x + \frac{y}{\color{blue}{e^{z} \cdot 1.1283791670955126}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -4000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{e^{z}}, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (<= t_1 -4000.0)
     t_0
     (if (<= t_1 4.0) (fma (/ y (exp z)) 0.8862269254527579 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = t_0;
	} else if (t_1 <= 4.0) {
		tmp = fma((y / exp(z)), 0.8862269254527579, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -4000.0)
		tmp = t_0;
	elseif (t_1 <= 4.0)
		tmp = fma(Float64(y / exp(z)), 0.8862269254527579, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4000.0], t$95$0, If[LessEqual[t$95$1, 4.0], N[(N[(y / N[Exp[z], $MachinePrecision]), $MachinePrecision] * 0.8862269254527579 + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -4e3 or 4 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 91.2

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

   4. Applied rewrites 91.2%

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

   if -4e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 4

   1. Initial program 99.9%

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

   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 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -600000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-68}:\\ \;\;\;\;\frac{y}{1.1283791670955126} + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -600000.0)
   (+ x (/ -1.0 x))
   (if (<= z 1.12e-68) (+ (/ y 1.1283791670955126) x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -600000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.12e-68) {
		tmp = (y / 1.1283791670955126) + 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 (z <= (-600000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.12d-68) then
        tmp = (y / 1.1283791670955126d0) + x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -600000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.12e-68) {
		tmp = (y / 1.1283791670955126) + x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -600000.0:
		tmp = x + (-1.0 / x)
	elif z <= 1.12e-68:
		tmp = (y / 1.1283791670955126) + x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -600000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.12e-68)
		tmp = Float64(Float64(y / 1.1283791670955126) + x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -600000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.12e-68)
		tmp = (y / 1.1283791670955126) + x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -600000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.12e-68], N[(N[(y / 1.1283791670955126), $MachinePrecision] + x), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -6e5

   1. Initial program 89.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -6e5 < z < 1.11999999999999992e-68

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

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

         \[\leadsto \frac{y}{\frac{5641895835477563}{5000000000000000} + -1 \cdot \left(x \cdot y\right)} + x \]

      2. mul-1-neg N/A

         \[\leadsto \frac{y}{\frac{5641895835477563}{5000000000000000} + -1 \cdot \left(x \cdot y\right)} + x \]

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto \frac{y}{\frac{5641895835477563}{5000000000000000} + -1 \cdot \left(x \cdot y\right)} + x \]

      5. mul-1-neg N/A

         \[\leadsto \frac{y}{\frac{5641895835477563}{5000000000000000} + -1 \cdot \left(x \cdot y\right)} + x \]

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

         \[\leadsto \frac{y}{\frac{5641895835477563}{5000000000000000} + -1 \cdot \left(x \cdot y\right)} + x \]

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

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

         \[\leadsto \frac{y}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \frac{5641895835477563}{5000000000000000}} + x \]

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

          \[\leadsto \frac{y}{\left(\mathsf{neg}\left(x\right)\right) \cdot y + \frac{5641895835477563}{5000000000000000}} + x \]

      11. lower-fma.f64 N/A

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

      12. lift-neg.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{y}{\frac{5641895835477563}{5000000000000000}} + x \]
   8. Step-by-step derivation

   9. Applied rewrites 72.9%

      \[\leadsto \frac{y}{1.1283791670955126} + x \]

   if 1.11999999999999992e-68 < z

   1. Initial program 94.6%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 92.8%

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

Alternative 7: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -600000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.96 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -600000.0)
   (+ x (/ -1.0 x))
   (if (<= z 1.96e-28) (fma y 0.8862269254527579 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -600000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.96e-28) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -600000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.96e-28)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -600000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.96e-28], N[(y * 0.8862269254527579 + x), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -6e5

   1. Initial program 89.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -6e5 < z < 1.9599999999999999e-28

   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 73.2

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

   4. Applied rewrites 73.2%

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

   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 73.0%

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

   if 1.9599999999999999e-28 < z

   1. Initial program 93.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 96.9%

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

Alternative 8: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -600000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.96 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -600000.0) x (if (<= z 1.96e-28) (fma y 0.8862269254527579 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -600000.0) {
		tmp = x;
	} else if (z <= 1.96e-28) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -600000.0)
		tmp = x;
	elseif (z <= 1.96e-28)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -600000.0], x, If[LessEqual[z, 1.96e-28], N[(y * 0.8862269254527579 + x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -6e5 or 1.9599999999999999e-28 < z

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 75.8%

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

   if -6e5 < z < 1.9599999999999999e-28

   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 73.2

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

   4. Applied rewrites 73.2%

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

   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 73.0%

      \[\leadsto \mathsf{fma}\left(y, 0.8862269254527579, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-159}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.05e-179) x (if (<= x 4.1e-159) (* y 0.8862269254527579) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.05e-179) {
		tmp = x;
	} else if (x <= 4.1e-159) {
		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 <= (-2.05d-179)) then
        tmp = x
    else if (x <= 4.1d-159) 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 <= -2.05e-179) {
		tmp = x;
	} else if (x <= 4.1e-159) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.05e-179:
		tmp = x
	elif x <= 4.1e-159:
		tmp = y * 0.8862269254527579
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.05e-179)
		tmp = x;
	elseif (x <= 4.1e-159)
		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 <= -2.05e-179)
		tmp = x;
	elseif (x <= 4.1e-159)
		tmp = y * 0.8862269254527579;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.05e-179], x, If[LessEqual[x, 4.1e-159], N[(y * 0.8862269254527579), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05e-179 or 4.10000000000000014e-159 < x

   1. Initial program 96.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 80.4%

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

   if -2.05e-179 < x < 4.10000000000000014e-159

   1. Initial program 91.7%

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

   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 42.9

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

   4. Applied rewrites 42.9%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 40.8%

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

Alternative 10: 69.3% accurate, 12.0× 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 95.7%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 69.3%

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

Reproduce

herbie shell --seed 2025101 
(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)))))