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

Percentage Accurate: 95.3% → 99.7%

Time: 6.0s

Alternatives: 7

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: 95.3% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 2 \cdot 10^{-106}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.01:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 2e-106)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.01)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 2e-106) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.01) {
		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 (exp(z) <= 2d-106) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.01d0) 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 (Math.exp(z) <= 2e-106) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 1.01) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 2e-106:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.01:
		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 (exp(z) <= 2e-106)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.01)
		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 (exp(z) <= 2e-106)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 1.01)
		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[N[Exp[z], $MachinePrecision], 2e-106], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.01], 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 (exp.f64 z) < 1.99999999999999988e-106

   1. Initial program 85.6%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if 1.99999999999999988e-106 < (exp.f64 z) < 1.01000000000000001

   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 1.01000000000000001 < (exp.f64 z)

   1. Initial program 95.5%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 2 \cdot 10^{-106}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.01:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (1.0 / ((1.1283791670955126 * (exp(z) / 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 / ((1.1283791670955126d0 * (exp(z) / y)) - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. *-lft-identity 95.7%

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

   2. associate-/l* 95.7%

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

   3. div-sub 95.8%

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

   4. associate-*r/ 95.8%

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

   5. /-rgt-identity 95.8%

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

   6. metadata-eval 95.8%

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

   7. associate-/l* 95.8%

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

   8. *-commutative 95.8%

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

   9. neg-mul-1 95.8%

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

   10. associate-/l* 95.8%

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

   11. associate-*r* 95.8%

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

   12. *-commutative 95.8%

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

   13. neg-mul-1 95.8%

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

   14. associate-/l* 99.9%

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

   15. *-inverses 99.9%

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

   16. /-rgt-identity 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 84.2% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{1.1283791670955126}\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))) (t_1 (+ x (/ y 1.1283791670955126))))
   (if (<= z -2.3e-124)
     t_0
     (if (<= z 2.8e-302)
       t_1
       (if (<= z 2e-183) t_0 (if (<= z 1.05e-44) 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 <= -2.3e-124) {
		tmp = t_0;
	} else if (z <= 2.8e-302) {
		tmp = t_1;
	} else if (z <= 2e-183) {
		tmp = t_0;
	} else if (z <= 1.05e-44) {
		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 <= (-2.3d-124)) then
        tmp = t_0
    else if (z <= 2.8d-302) then
        tmp = t_1
    else if (z <= 2d-183) then
        tmp = t_0
    else if (z <= 1.05d-44) 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 <= -2.3e-124) {
		tmp = t_0;
	} else if (z <= 2.8e-302) {
		tmp = t_1;
	} else if (z <= 2e-183) {
		tmp = t_0;
	} else if (z <= 1.05e-44) {
		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 <= -2.3e-124:
		tmp = t_0
	elif z <= 2.8e-302:
		tmp = t_1
	elif z <= 2e-183:
		tmp = t_0
	elif z <= 1.05e-44:
		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 <= -2.3e-124)
		tmp = t_0;
	elseif (z <= 2.8e-302)
		tmp = t_1;
	elseif (z <= 2e-183)
		tmp = t_0;
	elseif (z <= 1.05e-44)
		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 <= -2.3e-124)
		tmp = t_0;
	elseif (z <= 2.8e-302)
		tmp = t_1;
	elseif (z <= 2e-183)
		tmp = t_0;
	elseif (z <= 1.05e-44)
		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, -2.3e-124], t$95$0, If[LessEqual[z, 2.8e-302], t$95$1, If[LessEqual[z, 2e-183], t$95$0, If[LessEqual[z, 1.05e-44], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.30000000000000012e-124 or 2.8e-302 < z < 2.00000000000000001e-183

   1. Initial program 92.7%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 90.6%

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

   if -2.30000000000000012e-124 < z < 2.8e-302 or 2.00000000000000001e-183 < z < 1.05000000000000001e-44

   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%

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

   3. Taylor expanded in x around 0 83.1%

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

   if 1.05000000000000001e-44 < z

   1. Initial program 95.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-124}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-302}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-183}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-44}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 99.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -175:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.0165:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -175.0)
   (+ x (/ -1.0 x))
   (if (<= z 0.0165) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -175.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.0165) {
		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 <= (-175.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.0165d0) 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 <= -175.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.0165) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -175.0:
		tmp = x + (-1.0 / x)
	elif z <= 0.0165:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -175.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.0165)
		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 <= -175.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.0165)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -175.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.0165], 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 < -175

   1. Initial program 85.6%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -175 < z < 0.016500000000000001

   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%

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

   if 0.016500000000000001 < z

   1. Initial program 95.5%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -175:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.0165:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 73.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-125}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.2e-125) x (if (<= z 4e-49) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.2e-125) {
		tmp = x;
	} else if (z <= 4e-49) {
		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 <= (-8.2d-125)) then
        tmp = x
    else if (z <= 4d-49) 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 <= -8.2e-125) {
		tmp = x;
	} else if (z <= 4e-49) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.2e-125:
		tmp = x
	elif z <= 4e-49:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.2e-125)
		tmp = x;
	elseif (z <= 4e-49)
		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 <= -8.2e-125)
		tmp = x;
	elseif (z <= 4e-49)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.2e-125], x, If[LessEqual[z, 4e-49], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1999999999999995e-125 or 3.99999999999999975e-49 < z

   1. Initial program 93.1%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 79.4%

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

   if -8.1999999999999995e-125 < z < 3.99999999999999975e-49

   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%

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

   3. Taylor expanded in x around 0 77.2%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-125}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 68.5% 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 95.7%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 73.4%

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

5. Final simplification 73.4%

   \[\leadsto x \]

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 2023322 
(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)))))