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

Percentage Accurate: 95.7% → 99.5%

Time: 7.2s

Alternatives: 8

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: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{z} \cdot 1.1283791670955126\\ \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{y}{t_0 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (exp z) 1.1283791670955126)))
   (if (<= (exp z) 0.0)
     (- x (/ 1.0 x))
     (if (<= (exp z) 1.0) (+ x (/ y (- t_0 (* x y)))) (+ x (/ y t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = exp(z) * 1.1283791670955126;
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (exp(z) <= 1.0) {
		tmp = x + (y / (t_0 - (x * y)));
	} else {
		tmp = x + (y / t_0);
	}
	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 = exp(z) * 1.1283791670955126d0
    if (exp(z) <= 0.0d0) then
        tmp = x - (1.0d0 / x)
    else if (exp(z) <= 1.0d0) then
        tmp = x + (y / (t_0 - (x * y)))
    else
        tmp = x + (y / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.exp(z) * 1.1283791670955126;
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (Math.exp(z) <= 1.0) {
		tmp = x + (y / (t_0 - (x * y)));
	} else {
		tmp = x + (y / t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.exp(z) * 1.1283791670955126
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x - (1.0 / x)
	elif math.exp(z) <= 1.0:
		tmp = x + (y / (t_0 - (x * y)))
	else:
		tmp = x + (y / t_0)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]}, If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0], N[(x + N[(y / N[(t$95$0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 82.7%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.9%

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

   if 1 < (exp.f64 z)

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

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) <= 1.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x + (y / (exp(z) * 1.1283791670955126));
	}
	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) <= 1.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * 1.1283791670955126d0)) - (x * y)))
    else
        tmp = x + (y / (exp(z) * 1.1283791670955126d0))
    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) <= 1.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x + (y / (Math.exp(z) * 1.1283791670955126));
	}
	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) <= 1.0:
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)))
	else:
		tmp = x + (y / (math.exp(z) * 1.1283791670955126))
	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) <= 1.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * 1.1283791670955126)) - Float64(x * y))));
	else
		tmp = Float64(x + Float64(y / Float64(exp(z) * 1.1283791670955126)));
	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) <= 1.0)
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	else
		tmp = x + (y / (exp(z) * 1.1283791670955126));
	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], 1.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 82.7%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.1%

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

      1. *-commutative 99.1%

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

   4. Simplified 99.1%

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

   if 1 < (exp.f64 z)

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.6%

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

Alternative 3: 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 91.7%

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

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 4: 99.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -250000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 145:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -250000.0)
   (- x (/ 1.0 x))
   (if (<= z 145.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -250000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 145.0) {
		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 <= (-250000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 145.0d0) 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 <= -250000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 145.0) {
		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 <= -250000.0:
		tmp = x - (1.0 / x)
	elif z <= 145.0:
		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 <= -250000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 145.0)
		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 <= -250000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 145.0)
		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, -250000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 145.0], 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 < -2.5e5

   1. Initial program 82.3%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   if -2.5e5 < z < 145

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 98.9%

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

      1. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   if 145 < z

   1. Initial program 89.5%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 78.1%

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

   5. Taylor expanded in z around inf 73.1%

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

      1. associate-*r/ 73.1%

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

      2. *-commutative 73.1%

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

      3. associate-/l* 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -250000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 145:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 99.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -250000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 145:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -250000.0)
   (- x (/ 1.0 x))
   (if (<= z 145.0) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -250000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 145.0) {
		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 <= (-250000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 145.0d0) 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 <= -250000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 145.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -250000.0:
		tmp = x - (1.0 / x)
	elif z <= 145.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -250000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 145.0)
		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 <= -250000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 145.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -250000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 145.0], 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 < -2.5e5

   1. Initial program 82.3%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 100.0%

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

   if -2.5e5 < z < 145

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 98.3%

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

      1. *-commutative 98.3%

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

   4. Simplified 98.3%

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

   if 145 < z

   1. Initial program 89.5%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 78.1%

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

   5. Taylor expanded in z around inf 73.1%

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

      1. associate-*r/ 73.1%

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

      2. *-commutative 73.1%

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

      3. associate-/l* 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -250000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 145:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 71.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+80}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.8e+47) x (if (<= y 2.2e+80) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.8e+47) {
		tmp = x;
	} else if (y <= 2.2e+80) {
		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 (y <= (-5.8d+47)) then
        tmp = x
    else if (y <= 2.2d+80) 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 (y <= -5.8e+47) {
		tmp = x;
	} else if (y <= 2.2e+80) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.8e+47:
		tmp = x
	elif y <= 2.2e+80:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.8e+47)
		tmp = x;
	elseif (y <= 2.2e+80)
		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 (y <= -5.8e+47)
		tmp = x;
	elseif (y <= 2.2e+80)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.8e+47], x, If[LessEqual[y, 2.2e+80], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -5.79999999999999961e47 or 2.20000000000000003e80 < y

   1. Initial program 94.7%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 79.7%

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

   5. Taylor expanded in z around inf 30.2%

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

      1. associate-*r/ 30.2%

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

      2. *-commutative 30.2%

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

      3. associate-/l* 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in x around inf 70.8%

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

   if -5.79999999999999961e47 < y < 2.20000000000000003e80

   1. Initial program 89.3%

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

   2. Taylor expanded in z around 0 72.0%

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

      1. *-commutative 72.0%

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

   4. Simplified 72.0%

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

   5. Taylor expanded in y around 0 72.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+80}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 84.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-196}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-64}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.1e-196)
   (- x (/ 1.0 x))
   (if (<= z 4e-64) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.1e-196) {
		tmp = x - (1.0 / x);
	} else if (z <= 4e-64) {
		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 <= (-1.1d-196)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 4d-64) 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 <= -1.1e-196) {
		tmp = x - (1.0 / x);
	} else if (z <= 4e-64) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.1e-196:
		tmp = x - (1.0 / x)
	elif z <= 4e-64:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.1e-196)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 4e-64)
		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 <= -1.1e-196)
		tmp = x - (1.0 / x);
	elseif (z <= 4e-64)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.1e-196], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e-64], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.10000000000000007e-196

   1. Initial program 87.3%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 95.9%

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

   if -1.10000000000000007e-196 < z < 3.99999999999999986e-64

   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. Step-by-step derivation

      1. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around 0 76.1%

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

   if 3.99999999999999986e-64 < z

   1. Initial program 91.3%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 81.5%

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

   5. Taylor expanded in z around inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. *-commutative 68.1%

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

      3. associate-/l* 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in x around inf 94.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-196}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-64}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 68.6% 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 91.7%

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

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 76.6%

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

5. Taylor expanded in z around inf 42.4%

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

   1. associate-*r/ 42.4%

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

   2. *-commutative 42.4%

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

   3. associate-/l* 42.4%

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

7. Simplified 42.4%

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

8. Taylor expanded in x around inf 66.7%

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

9. Final simplification 66.7%

   \[\leadsto x \]

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 2023333 
(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)))))