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

Percentage Accurate: 95.5% → 99.9%

Time: 9.4s

Alternatives: 9

Speedup: 8.5×

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

real(8) function code(x, y, z)
    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: 95.5% 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

real(8) function code(x, y, z)
    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, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ 1.0 (- (* (exp z) (/ 1.1283791670955126 y)) x))))

C

double code(double x, double y, double z) {
	return x + (1.0 / ((exp(z) * (1.1283791670955126 / y)) - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / ((exp(z) * (1.1283791670955126d0 / y)) - x))
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / ((Math.exp(z) * (1.1283791670955126 / y)) - x));
}

Python

def code(x, y, z):
	return x + (1.0 / ((math.exp(z) * (1.1283791670955126 / y)) - x))

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / Float64(Float64(exp(z) * Float64(1.1283791670955126 / y)) - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / ((exp(z) * (1.1283791670955126 / y)) - x));
end

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

4. Final simplification 100.0%

   \[\leadsto x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot 0.8862269254527579}{e^{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (- x (/ 1.0 x))
   (if (<= (exp z) 2.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     (+ x (/ (* y 0.8862269254527579) (exp z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (exp(z) <= 2.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x + ((y * 0.8862269254527579) / exp(z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x - (1.0d0 / x)
    else if (exp(z) <= 2.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * 1.1283791670955126d0)) - (x * y)))
    else
        tmp = x + ((y * 0.8862269254527579d0) / exp(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (Math.exp(z) <= 2.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x + ((y * 0.8862269254527579) / Math.exp(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x - (1.0 / x)
	elif math.exp(z) <= 2.0:
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)))
	else:
		tmp = x + ((y * 0.8862269254527579) / math.exp(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (exp(z) <= 2.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * 1.1283791670955126)) - Float64(x * y))));
	else
		tmp = Float64(x + Float64(Float64(y * 0.8862269254527579) / exp(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x - (1.0 / x);
	elseif (exp(z) <= 2.0)
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	else
		tmp = x + ((y * 0.8862269254527579) / exp(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * 0.8862269254527579), $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 88.1%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.9%

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

   if 2 < (exp.f64 z)

   1. Initial program 95.7%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in y around 0 100.0%

      \[\leadsto x + \color{blue}{0.8862269254527579 \cdot \frac{y}{e^{z}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

      \[\leadsto x + \color{blue}{\frac{y \cdot 0.8862269254527579}{e^{z}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot 0.8862269254527579}{e^{z}}\\ \end{array} \]

Alternative 3: 99.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -340000000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 160:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -340000000000.0)
   (- x (/ 1.0 x))
   (if (<= z 160.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -340000000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 160.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-340000000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 160.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * 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 <= -340000000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 160.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -340000000000.0:
		tmp = x - (1.0 / x)
	elif z <= 160.0:
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -340000000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 160.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * 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 <= -340000000000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 160.0)
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -340000000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 160.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4e11

   1. Initial program 87.5%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in x around inf 100.0%

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

   if -3.4e11 < z < 160

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.2%

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

   if 160 < z

   1. Initial program 95.7%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -340000000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 160:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 73.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+25}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -410000000000:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-131}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6e+25)
   x
   (if (<= z -410000000000.0)
     (/ -1.0 x)
     (if (<= z -6.8e-54)
       x
       (if (<= z 1.9e-131) (+ x (* y 0.8862269254527579)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6e+25) {
		tmp = x;
	} else if (z <= -410000000000.0) {
		tmp = -1.0 / x;
	} else if (z <= -6.8e-54) {
		tmp = x;
	} else if (z <= 1.9e-131) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6d+25)) then
        tmp = x
    else if (z <= (-410000000000.0d0)) then
        tmp = (-1.0d0) / x
    else if (z <= (-6.8d-54)) then
        tmp = x
    else if (z <= 1.9d-131) then
        tmp = x + (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 (z <= -6e+25) {
		tmp = x;
	} else if (z <= -410000000000.0) {
		tmp = -1.0 / x;
	} else if (z <= -6.8e-54) {
		tmp = x;
	} else if (z <= 1.9e-131) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6e+25:
		tmp = x
	elif z <= -410000000000.0:
		tmp = -1.0 / x
	elif z <= -6.8e-54:
		tmp = x
	elif z <= 1.9e-131:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6e+25)
		tmp = x;
	elseif (z <= -410000000000.0)
		tmp = Float64(-1.0 / x);
	elseif (z <= -6.8e-54)
		tmp = x;
	elseif (z <= 1.9e-131)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6e+25)
		tmp = x;
	elseif (z <= -410000000000.0)
		tmp = -1.0 / x;
	elseif (z <= -6.8e-54)
		tmp = x;
	elseif (z <= 1.9e-131)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6e+25], x, If[LessEqual[z, -410000000000.0], N[(-1.0 / x), $MachinePrecision], If[LessEqual[z, -6.8e-54], x, If[LessEqual[z, 1.9e-131], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.00000000000000011e25 or -4.1e11 < z < -6.79999999999999975e-54 or 1.89999999999999997e-131 < z

   1. Initial program 94.2%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in x around inf 83.7%

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

   if -6.00000000000000011e25 < z < -4.1e11

   1. Initial program 80.5%

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

   2. Taylor expanded in z around 0 5.5%

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

   3. Taylor expanded in x around inf 80.5%

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

      1. associate-*r* 80.5%

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

      2. neg-mul-1 80.5%

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

   5. Simplified 80.5%

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

      1. frac-2neg 80.5%

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

      2. distribute-frac-neg 80.5%

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

      3. distribute-lft-neg-in 80.5%

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

      4. add-sqr-sqrt 19.7%

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

      5. sqrt-unprod 20.1%

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

      6. sqr-neg 20.1%

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

      7. sqrt-unprod 0.5%

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

      8. add-sqr-sqrt 0.7%

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

      9. sub-neg 0.7%

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

      10. add-sqr-sqrt 0.6%

          \[\leadsto x - \frac{y}{\color{blue}{\sqrt{\left(-x\right) \cdot y} \cdot \sqrt{\left(-x\right) \cdot y}}} \]

      11. add-sqr-sqrt 0.7%

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

      12. *-commutative 0.7%

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

      13. add-sqr-sqrt 0.3%

          \[\leadsto x - \frac{y}{y \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} \]

      14. sqrt-unprod 23.7%

          \[\leadsto x - \frac{y}{y \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}} \]

      15. sqr-neg 23.7%

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

      16. sqrt-unprod 60.2%

          \[\leadsto x - \frac{y}{y \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

      17. add-sqr-sqrt 80.5%

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

   7. Applied egg-rr 80.5%

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

   8. Taylor expanded in x around 0 100.0%

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

   if -6.79999999999999975e-54 < z < 1.89999999999999997e-131

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in z around 0 99.8%

      \[\leadsto x + \color{blue}{\frac{1}{1.1283791670955126 \cdot \frac{1}{y} - x}} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto x + \frac{1}{\color{blue}{\frac{1.1283791670955126 \cdot 1}{y}} - x} \]

      2. metadata-eval 99.9%

         \[\leadsto x + \frac{1}{\frac{\color{blue}{1.1283791670955126}}{y} - x} \]

   6. Simplified 99.9%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126}{y} - x}} \]

   7. Taylor expanded in y around 0 79.7%

      \[\leadsto x + \color{blue}{0.8862269254527579 \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 79.7%

         \[\leadsto x + \color{blue}{y \cdot 0.8862269254527579} \]

   9. Simplified 79.7%

      \[\leadsto x + \color{blue}{y \cdot 0.8862269254527579} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+25}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -410000000000:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-131}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 73.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -580000000000:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -8.4 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-131}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8e+22)
   x
   (if (<= z -580000000000.0)
     (/ -1.0 x)
     (if (<= z -8.4e-54)
       x
       (if (<= z 3e-131) (+ x (/ y 1.1283791670955126)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e+22) {
		tmp = x;
	} else if (z <= -580000000000.0) {
		tmp = -1.0 / x;
	} else if (z <= -8.4e-54) {
		tmp = x;
	} else if (z <= 3e-131) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-8d+22)) then
        tmp = x
    else if (z <= (-580000000000.0d0)) then
        tmp = (-1.0d0) / x
    else if (z <= (-8.4d-54)) then
        tmp = x
    else if (z <= 3d-131) then
        tmp = x + (y / 1.1283791670955126d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e+22) {
		tmp = x;
	} else if (z <= -580000000000.0) {
		tmp = -1.0 / x;
	} else if (z <= -8.4e-54) {
		tmp = x;
	} else if (z <= 3e-131) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8e+22:
		tmp = x
	elif z <= -580000000000.0:
		tmp = -1.0 / x
	elif z <= -8.4e-54:
		tmp = x
	elif z <= 3e-131:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8e+22)
		tmp = x;
	elseif (z <= -580000000000.0)
		tmp = Float64(-1.0 / x);
	elseif (z <= -8.4e-54)
		tmp = x;
	elseif (z <= 3e-131)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8e+22)
		tmp = x;
	elseif (z <= -580000000000.0)
		tmp = -1.0 / x;
	elseif (z <= -8.4e-54)
		tmp = x;
	elseif (z <= 3e-131)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8e+22], x, If[LessEqual[z, -580000000000.0], N[(-1.0 / x), $MachinePrecision], If[LessEqual[z, -8.4e-54], x, If[LessEqual[z, 3e-131], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8e22 or -5.8e11 < z < -8.4e-54 or 2.99999999999999996e-131 < z

   1. Initial program 94.2%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in x around inf 83.7%

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

   if -8e22 < z < -5.8e11

   1. Initial program 80.5%

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

   2. Taylor expanded in z around 0 5.5%

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

   3. Taylor expanded in x around inf 80.5%

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

      1. associate-*r* 80.5%

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

      2. neg-mul-1 80.5%

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

   5. Simplified 80.5%

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

      1. frac-2neg 80.5%

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

      2. distribute-frac-neg 80.5%

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

      3. distribute-lft-neg-in 80.5%

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

      4. add-sqr-sqrt 19.7%

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

      5. sqrt-unprod 20.1%

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

      6. sqr-neg 20.1%

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

      7. sqrt-unprod 0.5%

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

      8. add-sqr-sqrt 0.7%

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

      9. sub-neg 0.7%

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

      10. add-sqr-sqrt 0.6%

          \[\leadsto x - \frac{y}{\color{blue}{\sqrt{\left(-x\right) \cdot y} \cdot \sqrt{\left(-x\right) \cdot y}}} \]

      11. add-sqr-sqrt 0.7%

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

      12. *-commutative 0.7%

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

      13. add-sqr-sqrt 0.3%

          \[\leadsto x - \frac{y}{y \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} \]

      14. sqrt-unprod 23.7%

          \[\leadsto x - \frac{y}{y \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}} \]

      15. sqr-neg 23.7%

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

      16. sqrt-unprod 60.2%

          \[\leadsto x - \frac{y}{y \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

      17. add-sqr-sqrt 80.5%

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

   7. Applied egg-rr 80.5%

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

   8. Taylor expanded in x around 0 100.0%

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

   if -8.4e-54 < z < 2.99999999999999996e-131

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in z around 0 99.8%

      \[\leadsto x + \color{blue}{\frac{1}{1.1283791670955126 \cdot \frac{1}{y} - x}} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto x + \frac{1}{\color{blue}{\frac{1.1283791670955126 \cdot 1}{y}} - x} \]

      2. metadata-eval 99.9%

         \[\leadsto x + \frac{1}{\frac{\color{blue}{1.1283791670955126}}{y} - x} \]

   6. Simplified 99.9%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126}{y} - x}} \]

   7. Taylor expanded in y around 0 79.7%

      \[\leadsto x + \color{blue}{0.8862269254527579 \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 79.7%

         \[\leadsto x + \color{blue}{y \cdot 0.8862269254527579} \]

   9. Simplified 79.7%

      \[\leadsto x + \color{blue}{y \cdot 0.8862269254527579} \]
   10. Step-by-step derivation

       1. metadata-eval 79.7%

          \[\leadsto x + y \cdot \color{blue}{\frac{1}{1.1283791670955126}} \]

       2. div-inv 79.8%

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

   11. Applied egg-rr 79.8%

       \[\leadsto x + \color{blue}{\frac{y}{1.1283791670955126}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -580000000000:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -8.4 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-131}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 84.8% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{1}{x}\\ t_1 := x + \frac{y}{1.1283791670955126}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (/ 1.0 x))) (t_1 (+ x (/ y 1.1283791670955126))))
   (if (<= z -8.5e-54)
     t_0
     (if (<= z 1.15e-197)
       t_1
       (if (<= z 1.1e-179) t_0 (if (<= z 3.8e-132) t_1 x))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (1.0 / x);
	double t_1 = x + (y / 1.1283791670955126);
	double tmp;
	if (z <= -8.5e-54) {
		tmp = t_0;
	} else if (z <= 1.15e-197) {
		tmp = t_1;
	} else if (z <= 1.1e-179) {
		tmp = t_0;
	} else if (z <= 3.8e-132) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    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)
    if (z <= (-8.5d-54)) then
        tmp = t_0
    else if (z <= 1.15d-197) then
        tmp = t_1
    else if (z <= 1.1d-179) then
        tmp = t_0
    else if (z <= 3.8d-132) then
        tmp = t_1
    else
        tmp = x
    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);
	double tmp;
	if (z <= -8.5e-54) {
		tmp = t_0;
	} else if (z <= 1.15e-197) {
		tmp = t_1;
	} else if (z <= 1.1e-179) {
		tmp = t_0;
	} else if (z <= 3.8e-132) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (1.0 / x)
	t_1 = x + (y / 1.1283791670955126)
	tmp = 0
	if z <= -8.5e-54:
		tmp = t_0
	elif z <= 1.15e-197:
		tmp = t_1
	elif z <= 1.1e-179:
		tmp = t_0
	elif z <= 3.8e-132:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(1.0 / x))
	t_1 = Float64(x + Float64(y / 1.1283791670955126))
	tmp = 0.0
	if (z <= -8.5e-54)
		tmp = t_0;
	elseif (z <= 1.15e-197)
		tmp = t_1;
	elseif (z <= 1.1e-179)
		tmp = t_0;
	elseif (z <= 3.8e-132)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (1.0 / x);
	t_1 = x + (y / 1.1283791670955126);
	tmp = 0.0;
	if (z <= -8.5e-54)
		tmp = t_0;
	elseif (z <= 1.15e-197)
		tmp = t_1;
	elseif (z <= 1.1e-179)
		tmp = t_0;
	elseif (z <= 3.8e-132)
		tmp = t_1;
	else
		tmp = x;
	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 / 1.1283791670955126), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.5e-54], t$95$0, If[LessEqual[z, 1.15e-197], t$95$1, If[LessEqual[z, 1.1e-179], t$95$0, If[LessEqual[z, 3.8e-132], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.5e-54 or 1.15e-197 < z < 1.10000000000000002e-179

   1. Initial program 90.4%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in x around inf 96.4%

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

   if -8.5e-54 < z < 1.15e-197 or 1.10000000000000002e-179 < z < 3.7999999999999997e-132

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in z around 0 99.8%

      \[\leadsto x + \color{blue}{\frac{1}{1.1283791670955126 \cdot \frac{1}{y} - x}} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto x + \frac{1}{\color{blue}{\frac{1.1283791670955126 \cdot 1}{y}} - x} \]

      2. metadata-eval 99.9%

         \[\leadsto x + \frac{1}{\frac{\color{blue}{1.1283791670955126}}{y} - x} \]

   6. Simplified 99.9%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126}{y} - x}} \]

   7. Taylor expanded in y around 0 84.1%

      \[\leadsto x + \color{blue}{0.8862269254527579 \cdot y} \]
   8. Step-by-step derivation

      1. *-commutative 84.1%

         \[\leadsto x + \color{blue}{y \cdot 0.8862269254527579} \]

   9. Simplified 84.1%

      \[\leadsto x + \color{blue}{y \cdot 0.8862269254527579} \]
   10. Step-by-step derivation

       1. metadata-eval 84.1%

          \[\leadsto x + y \cdot \color{blue}{\frac{1}{1.1283791670955126}} \]

       2. div-inv 84.3%

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

   11. Applied egg-rr 84.3%

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

   if 3.7999999999999997e-132 < z

   1. Initial program 96.8%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in x around inf 93.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-54}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-197}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-179}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-132}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 99.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -340000000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 230:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -340000000000.0)
   (- x (/ 1.0 x))
   (if (<= z 230.0) (+ x (/ 1.0 (- (/ 1.1283791670955126 y) x))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -340000000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 230.0) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-340000000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 230.0d0) then
        tmp = x + (1.0d0 / ((1.1283791670955126d0 / y) - 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 <= -340000000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 230.0) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -340000000000.0:
		tmp = x - (1.0 / x)
	elif z <= 230.0:
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -340000000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 230.0)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(1.1283791670955126 / y) - x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -340000000000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 230.0)
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -340000000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 230.0], N[(x + N[(1.0 / N[(N[(1.1283791670955126 / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4e11

   1. Initial program 87.5%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in x around inf 100.0%

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

   if -3.4e11 < z < 230

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in z around 0 99.1%

      \[\leadsto x + \color{blue}{\frac{1}{1.1283791670955126 \cdot \frac{1}{y} - x}} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.1%

         \[\leadsto x + \frac{1}{\color{blue}{\frac{1.1283791670955126 \cdot 1}{y}} - x} \]

      2. metadata-eval 99.1%

         \[\leadsto x + \frac{1}{\frac{\color{blue}{1.1283791670955126}}{y} - x} \]

   6. Simplified 99.1%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126}{y} - x}} \]

   if 230 < z

   1. Initial program 95.7%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -340000000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 230:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 68.6% accurate, 21.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+254}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y -6e+254) (/ -1.0 x) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e+254) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-6d+254)) 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 (y <= -6e+254) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6e+254:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6e+254)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6e+254)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6e+254], N[(-1.0 / x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -6.00000000000000014e254

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0 77.4%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. associate-*r* 99.4%

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

      2. neg-mul-1 99.4%

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

   5. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-lft-neg-in 99.4%

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

      4. add-sqr-sqrt 74.4%

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

      5. sqrt-unprod 60.5%

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

      6. sqr-neg 60.5%

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

      7. sqrt-unprod 0.2%

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

      8. add-sqr-sqrt 13.2%

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

      9. sub-neg 13.2%

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

      10. add-sqr-sqrt 0.2%

          \[\leadsto x - \frac{y}{\color{blue}{\sqrt{\left(-x\right) \cdot y} \cdot \sqrt{\left(-x\right) \cdot y}}} \]

      11. add-sqr-sqrt 13.2%

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

      12. *-commutative 13.2%

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

      13. add-sqr-sqrt 13.0%

          \[\leadsto x - \frac{y}{y \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}} \]

      14. sqrt-unprod 26.0%

          \[\leadsto x - \frac{y}{y \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}} \]

      15. sqr-neg 26.0%

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

      16. sqrt-unprod 24.6%

          \[\leadsto x - \frac{y}{y \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

      17. add-sqr-sqrt 99.4%

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

   7. Applied egg-rr 99.4%

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

   8. Taylor expanded in x around 0 87.7%

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

   if -6.00000000000000014e254 < y

   1. Initial program 95.8%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

   4. Taylor expanded in x around inf 74.3%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+254}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 69.8% accurate, 111.0× 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

real(8) function code(x, y, z)
    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 95.9%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x + \frac{1}{e^{z} \cdot \frac{1.1283791670955126}{y} - x}} \]

4. Taylor expanded in x around inf 72.4%

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

5. Final simplification 72.4%

   \[\leadsto x \]

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x))))

C

double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / (((1.1283791670955126d0 / y) * exp(z)) - x))
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * Math.exp(z)) - x));
}

Python

def code(x, y, z):
	return x + (1.0 / (((1.1283791670955126 / y) * math.exp(z)) - x))

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / Float64(Float64(Float64(1.1283791670955126 / y) * exp(z)) - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
end

Wolfram

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))