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

Percentage Accurate: 95.6% → 99.9%

Time: 9.3s

Alternatives: 12

Speedup: 5.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 95.6% 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, 0.4× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (- x (/ y (fma 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 {
		tmp = x - (y / fma(x, y, (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));
	else
		tmp = Float64(x - Float64(y / fma(x, y, Float64(exp(z) * -1.1283791670955126))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.7%

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

      1. remove-double-neg 97.7%

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

      2. distribute-frac-neg 97.7%

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

      3. unsub-neg 97.7%

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

      4. distribute-frac-neg 97.7%

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

      5. distribute-neg-frac2 97.7%

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

      6. neg-sub0 97.7%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 97.7%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 97.7%

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

      9. +-commutative 97.7%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 5 \cdot 10^{+58}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 5e+58)
     (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))
     x)))

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) <= 5e+58) {
		tmp = x + (y / ((exp(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) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 5d+58) then
        tmp = x + (y / ((exp(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) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 5e+58) {
		tmp = x + (y / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	} else {
		tmp = x;
	}
	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) <= 5e+58:
		tmp = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	else:
		tmp = x
	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) <= 5e+58)
		tmp = Float64(x + Float64(y / Float64(Float64(exp(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) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 5e+58)
		tmp = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	else
		tmp = x;
	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], 5e+58], N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 4.99999999999999986e58

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   if 4.99999999999999986e58 < (exp.f64 z)

   1. Initial program 91.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 5 \cdot 10^{+58}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   if 2 < (exp.f64 z)

   1. Initial program 91.5%

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

      1. remove-double-neg 91.5%

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

      2. distribute-frac-neg 91.5%

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

      3. unsub-neg 91.5%

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

      4. distribute-frac-neg 91.5%

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

      5. distribute-neg-frac2 91.5%

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

      6. neg-sub0 91.5%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 91.5%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 91.5%

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

      9. +-commutative 91.5%

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

      10. fma-define 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 99.7%

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

Alternative 4: 70.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-123}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.1 \cdot 10^{-232}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-145}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 10^{-28}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.00105)
   x
   (if (<= x -4.7e-85)
     (/ -1.0 x)
     (if (<= x -5.5e-123)
       x
       (if (<= x -6.1e-232)
         (/ -1.0 x)
         (if (<= x 1.75e-145)
           (* y 0.8862269254527579)
           (if (<= x 1e-28) (/ -1.0 x) x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.00105) {
		tmp = x;
	} else if (x <= -4.7e-85) {
		tmp = -1.0 / x;
	} else if (x <= -5.5e-123) {
		tmp = x;
	} else if (x <= -6.1e-232) {
		tmp = -1.0 / x;
	} else if (x <= 1.75e-145) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 1e-28) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-0.00105d0)) then
        tmp = x
    else if (x <= (-4.7d-85)) then
        tmp = (-1.0d0) / x
    else if (x <= (-5.5d-123)) then
        tmp = x
    else if (x <= (-6.1d-232)) then
        tmp = (-1.0d0) / x
    else if (x <= 1.75d-145) then
        tmp = y * 0.8862269254527579d0
    else if (x <= 1d-28) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.00105) {
		tmp = x;
	} else if (x <= -4.7e-85) {
		tmp = -1.0 / x;
	} else if (x <= -5.5e-123) {
		tmp = x;
	} else if (x <= -6.1e-232) {
		tmp = -1.0 / x;
	} else if (x <= 1.75e-145) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 1e-28) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.00105:
		tmp = x
	elif x <= -4.7e-85:
		tmp = -1.0 / x
	elif x <= -5.5e-123:
		tmp = x
	elif x <= -6.1e-232:
		tmp = -1.0 / x
	elif x <= 1.75e-145:
		tmp = y * 0.8862269254527579
	elif x <= 1e-28:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.00105)
		tmp = x;
	elseif (x <= -4.7e-85)
		tmp = Float64(-1.0 / x);
	elseif (x <= -5.5e-123)
		tmp = x;
	elseif (x <= -6.1e-232)
		tmp = Float64(-1.0 / x);
	elseif (x <= 1.75e-145)
		tmp = Float64(y * 0.8862269254527579);
	elseif (x <= 1e-28)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.00105)
		tmp = x;
	elseif (x <= -4.7e-85)
		tmp = -1.0 / x;
	elseif (x <= -5.5e-123)
		tmp = x;
	elseif (x <= -6.1e-232)
		tmp = -1.0 / x;
	elseif (x <= 1.75e-145)
		tmp = y * 0.8862269254527579;
	elseif (x <= 1e-28)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.00105], x, If[LessEqual[x, -4.7e-85], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, -5.5e-123], x, If[LessEqual[x, -6.1e-232], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 1.75e-145], N[(y * 0.8862269254527579), $MachinePrecision], If[LessEqual[x, 1e-28], N[(-1.0 / x), $MachinePrecision], x]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00104999999999999994 or -4.70000000000000009e-85 < x < -5.5e-123 or 9.99999999999999971e-29 < x

   1. Initial program 96.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 94.5%

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

   if -0.00104999999999999994 < x < -4.70000000000000009e-85 or -5.5e-123 < x < -6.1000000000000001e-232 or 1.74999999999999998e-145 < x < 9.99999999999999971e-29

   1. Initial program 96.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 62.4%

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

   4. Taylor expanded in x around 0 61.4%

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

   if -6.1000000000000001e-232 < x < 1.74999999999999998e-145

   1. Initial program 92.1%

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

      1. remove-double-neg 92.1%

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

      2. distribute-frac-neg 92.1%

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

      3. unsub-neg 92.1%

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

      4. distribute-frac-neg 92.1%

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

      5. distribute-neg-frac2 92.1%

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

      6. neg-sub0 91.9%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 91.9%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 92.2%

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

      9. +-commutative 92.2%

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

      10. fma-define 92.3%

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

      11. *-commutative 92.3%

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

      12. distribute-rgt-neg-in 92.3%

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

      13. metadata-eval 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in z around 0 68.4%

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

      1. *-commutative 68.4%

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

      2. fma-neg 68.4%

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

      3. metadata-eval 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in x around 0 56.6%

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

      1. *-commutative 56.6%

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

   10. Simplified 56.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-123}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.1 \cdot 10^{-232}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-145}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 10^{-28}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+73}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2600:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq 195:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.9e+73)
   (/ -1.0 x)
   (if (<= z -1.9e+20)
     x
     (if (<= z -2600.0)
       (/ -1.0 x)
       (if (<= z 195.0) (- x (* y -0.8862269254527579)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.9e+73) {
		tmp = -1.0 / x;
	} else if (z <= -1.9e+20) {
		tmp = x;
	} else if (z <= -2600.0) {
		tmp = -1.0 / x;
	} else if (z <= 195.0) {
		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 <= (-3.9d+73)) then
        tmp = (-1.0d0) / x
    else if (z <= (-1.9d+20)) then
        tmp = x
    else if (z <= (-2600.0d0)) then
        tmp = (-1.0d0) / x
    else if (z <= 195.0d0) 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 <= -3.9e+73) {
		tmp = -1.0 / x;
	} else if (z <= -1.9e+20) {
		tmp = x;
	} else if (z <= -2600.0) {
		tmp = -1.0 / x;
	} else if (z <= 195.0) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.9e+73:
		tmp = -1.0 / x
	elif z <= -1.9e+20:
		tmp = x
	elif z <= -2600.0:
		tmp = -1.0 / x
	elif z <= 195.0:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.9e+73)
		tmp = Float64(-1.0 / x);
	elseif (z <= -1.9e+20)
		tmp = x;
	elseif (z <= -2600.0)
		tmp = Float64(-1.0 / x);
	elseif (z <= 195.0)
		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 <= -3.9e+73)
		tmp = -1.0 / x;
	elseif (z <= -1.9e+20)
		tmp = x;
	elseif (z <= -2600.0)
		tmp = -1.0 / x;
	elseif (z <= 195.0)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.9e+73], N[(-1.0 / x), $MachinePrecision], If[LessEqual[z, -1.9e+20], x, If[LessEqual[z, -2600.0], N[(-1.0 / x), $MachinePrecision], If[LessEqual[z, 195.0], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.9000000000000001e73 or -1.9e20 < z < -2600

   1. Initial program 90.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   4. Taylor expanded in x around 0 60.5%

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

   if -3.9000000000000001e73 < z < -1.9e20 or 195 < z

   1. Initial program 91.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.2%

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

   if -2600 < z < 195

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 98.8%

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

      1. *-commutative 98.8%

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

      2. fma-neg 98.8%

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

      3. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in y around 0 78.6%

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

      1. *-commutative 78.6%

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

   10. Simplified 78.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+73}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2600:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq 195:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1700000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1700000.0)
   (+ x (/ -1.0 x))
   (if (<= z 190.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1700000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 190.0) {
		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 <= (-1700000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 190.0d0) 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 <= -1700000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 190.0) {
		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 <= -1700000.0:
		tmp = x + (-1.0 / x)
	elif z <= 190.0:
		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 <= -1700000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 190.0)
		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 <= -1700000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 190.0)
		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, -1700000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 190.0], 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 < -1.7e6

   1. Initial program 89.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   if -1.7e6 < z < 190

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 99.2%

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

      1. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 190 < z

   1. Initial program 91.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1700000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 140:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.32e-30)
   (+ x (/ -1.0 x))
   (if (<= z 140.0)
     (+ x (/ y (+ 1.1283791670955126 (* z 1.1283791670955126))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.32e-30) {
		tmp = x + (-1.0 / x);
	} else if (z <= 140.0) {
		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 <= (-1.32d-30)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 140.0d0) 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 <= -1.32e-30) {
		tmp = x + (-1.0 / x);
	} else if (z <= 140.0) {
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.32e-30:
		tmp = x + (-1.0 / x)
	elif z <= 140.0:
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.32e-30)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 140.0)
		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 <= -1.32e-30)
		tmp = x + (-1.0 / x);
	elseif (z <= 140.0)
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.32e-30], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 140.0], 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 < -1.32e-30

   1. Initial program 90.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   if -1.32e-30 < z < 140

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 99.2%

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

      1. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around 0 79.6%

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

   if 140 < z

   1. Initial program 91.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 140:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1700000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 145:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1700000.0)
   (+ 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 <= -1700000.0) {
		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 <= (-1700000.0d0)) 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 <= -1700000.0) {
		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 <= -1700000.0:
		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 <= -1700000.0)
		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 <= -1700000.0)
		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, -1700000.0], 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 < -1.7e6

   1. Initial program 89.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   if -1.7e6 < z < 145

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 98.8%

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

   if 145 < z

   1. Initial program 91.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1700000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 145:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 140:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.2e-30)
   (+ x (/ -1.0 x))
   (if (<= z 140.0) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.2e-30) {
		tmp = x + (-1.0 / x);
	} else if (z <= 140.0) {
		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 <= (-4.2d-30)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 140.0d0) 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 <= -4.2e-30) {
		tmp = x + (-1.0 / x);
	} else if (z <= 140.0) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.2e-30:
		tmp = x + (-1.0 / x)
	elif z <= 140.0:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.2e-30)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 140.0)
		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 <= -4.2e-30)
		tmp = x + (-1.0 / x);
	elseif (z <= 140.0)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.2e-30], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 140.0], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2000000000000004e-30

   1. Initial program 90.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   if -4.2000000000000004e-30 < z < 140

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 98.8%

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

      1. *-commutative 98.8%

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

      2. fma-neg 98.8%

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

      3. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in y around 0 79.2%

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

      1. *-commutative 79.2%

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

   10. Simplified 79.2%

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

   if 140 < z

   1. Initial program 91.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 140:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 140:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.7e-30)
   (+ x (/ -1.0 x))
   (if (<= z 140.0) (- x (/ y -1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.7e-30) {
		tmp = x + (-1.0 / x);
	} else if (z <= 140.0) {
		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 <= (-4.7d-30)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 140.0d0) 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 <= -4.7e-30) {
		tmp = x + (-1.0 / x);
	} else if (z <= 140.0) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.7e-30:
		tmp = x + (-1.0 / x)
	elif z <= 140.0:
		tmp = x - (y / -1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.7e-30)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 140.0)
		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 <= -4.7e-30)
		tmp = x + (-1.0 / x);
	elseif (z <= 140.0)
		tmp = x - (y / -1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.7e-30], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 140.0], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -4.69999999999999969e-30

   1. Initial program 90.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

   if -4.69999999999999969e-30 < z < 140

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 99.8%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 98.8%

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

   6. Taylor expanded in x around 0 79.2%

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

   if 140 < z

   1. Initial program 91.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 140:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-161}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-74}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.2e-161) x (if (<= x 6.8e-74) (* y 0.8862269254527579) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.2e-161) {
		tmp = x;
	} else if (x <= 6.8e-74) {
		tmp = 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 (x <= (-7.2d-161)) then
        tmp = x
    else if (x <= 6.8d-74) then
        tmp = y * 0.8862269254527579d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.2e-161) {
		tmp = x;
	} else if (x <= 6.8e-74) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.2e-161:
		tmp = x
	elif x <= 6.8e-74:
		tmp = y * 0.8862269254527579
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.2e-161)
		tmp = x;
	elseif (x <= 6.8e-74)
		tmp = Float64(y * 0.8862269254527579);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.2e-161)
		tmp = x;
	elseif (x <= 6.8e-74)
		tmp = y * 0.8862269254527579;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.2e-161], x, If[LessEqual[x, 6.8e-74], N[(y * 0.8862269254527579), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000036e-161 or 6.8000000000000001e-74 < x

   1. Initial program 97.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 79.9%

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

   if -7.20000000000000036e-161 < x < 6.8000000000000001e-74

   1. Initial program 92.5%

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

      1. remove-double-neg 92.5%

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

      2. distribute-frac-neg 92.5%

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

      3. unsub-neg 92.5%

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

      4. distribute-frac-neg 92.5%

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

      5. distribute-neg-frac2 92.5%

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

      6. neg-sub0 92.2%

         \[\leadsto x - \frac{y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      7. associate--r- 92.2%

         \[\leadsto x - \frac{y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      8. neg-sub0 92.6%

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

      9. +-commutative 92.6%

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

      10. fma-define 92.6%

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

      11. *-commutative 92.6%

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

      12. distribute-rgt-neg-in 92.6%

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

      13. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in z around 0 69.3%

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

      1. *-commutative 69.3%

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

      2. fma-neg 69.3%

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

      3. metadata-eval 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in x around 0 49.2%

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

      1. *-commutative 49.2%

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

   10. Simplified 49.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-161}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-74}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 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 95.5%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 58.5%

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

4. Final simplification 58.5%

   \[\leadsto x \]
5. Add Preprocessing

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 2024055 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

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

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