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

Percentage Accurate: 95.8% → 99.9%

Time: 8.9s

Alternatives: 10

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: 95.8% 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 96.6%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t_0 \leq 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_0 1e+179) t_0 (+ x (/ -1.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 1e+179) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / 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) :: tmp
    t_0 = x + (y / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    if (t_0 <= 1d+179) then
        tmp = t_0
    else
        tmp = x + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 1e+179) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	tmp = 0
	if t_0 <= 1e+179:
		tmp = t_0
	else:
		tmp = x + (-1.0 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_0 <= 1e+179)
		tmp = t_0;
	else
		tmp = Float64(x + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	tmp = 0.0;
	if (t_0 <= 1e+179)
		tmp = t_0;
	else
		tmp = x + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+179], t$95$0, N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))) < 9.9999999999999998e178

   1. Initial program 99.0%

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

   if 9.9999999999999998e178 < (+.f64 x (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))

   1. Initial program 83.6%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 10^{+179}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -250:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot -0.8862269254527579}{e^{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -250.0)
   (+ x (/ -1.0 x))
   (if (<= z 4.1e-48)
     (+ 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 (z <= -250.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 4.1e-48) {
		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 (z <= (-250.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 4.1d-48) 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 (z <= -250.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 4.1e-48) {
		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 z <= -250.0:
		tmp = x + (-1.0 / x)
	elif z <= 4.1e-48:
		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 (z <= -250.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 4.1e-48)
		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 (z <= -250.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 4.1e-48)
		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[z, -250.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.1e-48], 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 z < -250

   1. Initial program 94.2%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -250 < z < 4.10000000000000014e-48

   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%

      \[\leadsto x + \frac{y}{\color{blue}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right)} - x \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   if 4.10000000000000014e-48 < z

   1. Initial program 93.2%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -250:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot -0.8862269254527579}{e^{z}}\\ \end{array} \]

Alternative 4: 99.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -114:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 7.1:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -114.0)
   (+ x (/ -1.0 x))
   (if (<= z 7.1)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -114.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 7.1) {
		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 <= (-114.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 7.1d0) 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 <= -114.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 7.1) {
		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 <= -114.0:
		tmp = x + (-1.0 / x)
	elif z <= 7.1:
		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 <= -114.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 7.1)
		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 <= -114.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 7.1)
		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, -114.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.1], 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 < -114

   1. Initial program 94.2%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -114 < z < 7.0999999999999996

   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%

      \[\leadsto x + \frac{y}{\color{blue}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right)} - x \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   if 7.0999999999999996 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -114:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 7.1:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 86.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-56}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.0016:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8e-56)
   (+ x (/ -1.0 x))
   (if (<= z 0.0016)
     (+ x (/ y (+ 1.1283791670955126 (* z 1.1283791670955126))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e-56) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.0016) {
		tmp = x + (y / (1.1283791670955126 + (z * 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-56)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.0016d0) then
        tmp = x + (y / (1.1283791670955126d0 + (z * 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-56) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.0016) {
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8e-56:
		tmp = x + (-1.0 / x)
	elif z <= 0.0016:
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8e-56)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.0016)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 + Float64(z * 1.1283791670955126))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8e-56)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.0016)
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8e-56], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.0016], N[(x + N[(y / N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -8.0000000000000003e-56

   1. Initial program 95.1%

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

   2. Taylor expanded in x around inf 95.9%

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

   if -8.0000000000000003e-56 < z < 0.00160000000000000008

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right)} - x \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 78.6%

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

   if 0.00160000000000000008 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-56}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.0016:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 99.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -215:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 7.1:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -215.0)
   (+ x (/ -1.0 x))
   (if (<= z 7.1) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -215.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 7.1) {
		tmp = x + (y / (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 <= (-215.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 7.1d0) 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 <= -215.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 7.1) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -215.0:
		tmp = x + (-1.0 / x)
	elif z <= 7.1:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -215.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 7.1)
		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 <= -215.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 7.1)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -215.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.1], 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 < -215

   1. Initial program 94.2%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -215 < z < 7.0999999999999996

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.6%

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

      1. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   if 7.0999999999999996 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -215:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 7.1:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 99.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -350:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 7.1:\\ \;\;\;\;x - \frac{-1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -350.0)
   (+ x (/ -1.0 x))
   (if (<= z 7.1) (- x (/ -1.0 (- (/ 1.1283791670955126 y) x))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -350.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 7.1) {
		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 <= (-350.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 7.1d0) 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 <= -350.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 7.1) {
		tmp = x - (-1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -350.0:
		tmp = x + (-1.0 / x)
	elif z <= 7.1:
		tmp = x - (-1.0 / ((1.1283791670955126 / y) - x))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -350.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 7.1)
		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 <= -350.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 7.1)
		tmp = x - (-1.0 / ((1.1283791670955126 / y) - x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -350.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.1], 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 < -350

   1. Initial program 94.2%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -350 < z < 7.0999999999999996

   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 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   if 7.0999999999999996 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -350:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 7.1:\\ \;\;\;\;x - \frac{-1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 73.6% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-185}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-8}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.05e-185)
   x
   (if (<= z 1.45e-8) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.05e-185) {
		tmp = x;
	} else if (z <= 1.45e-8) {
		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 <= (-2.05d-185)) then
        tmp = x
    else if (z <= 1.45d-8) 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 <= -2.05e-185) {
		tmp = x;
	} else if (z <= 1.45e-8) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.05e-185:
		tmp = x
	elif z <= 1.45e-8:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.05e-185)
		tmp = x;
	elseif (z <= 1.45e-8)
		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 <= -2.05e-185)
		tmp = x;
	elseif (z <= 1.45e-8)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.05e-185], x, If[LessEqual[z, 1.45e-8], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -2.05e-185 or 1.4500000000000001e-8 < z

   1. Initial program 95.0%

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

   2. Taylor expanded in x around inf 75.9%

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

   if -2.05e-185 < z < 1.4500000000000001e-8

   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 99.5%

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

   5. Taylor expanded in y around 0 81.0%

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

      1. *-commutative 81.0%

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

   7. Simplified 81.0%

      \[\leadsto x - \color{blue}{y \cdot -0.8862269254527579} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-185}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-8}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 85.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-55}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.0132:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.3e-55)
   (+ x (/ -1.0 x))
   (if (<= z 0.0132) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e-55) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.0132) {
		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 <= (-1.3d-55)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.0132d0) 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 <= -1.3e-55) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.0132) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.3e-55:
		tmp = x + (-1.0 / x)
	elif z <= 0.0132:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.3e-55)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.0132)
		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 <= -1.3e-55)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.0132)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.3e-55], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.0132], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2999999999999999e-55

   1. Initial program 95.1%

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

   2. Taylor expanded in x around inf 95.9%

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

   if -1.2999999999999999e-55 < z < 0.0132

   1. Initial program 99.8%

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

   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 99.6%

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

   5. Taylor expanded in y around 0 78.3%

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

      1. *-commutative 78.3%

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

   7. Simplified 78.3%

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

   if 0.0132 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-55}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.0132:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 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 96.6%

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

2. Taylor expanded in x around inf 70.0%

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

3. Final simplification 70.0%

   \[\leadsto x \]

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 2023336 
(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)))))