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

Percentage Accurate: 95.6% → 99.7%

Time: 9.8s

Alternatives: 13

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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.01:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{e^{z}} \cdot -0.8862269254527579\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.01)
     (+ 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) <= 1.01) {
		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) <= 1.01d0) 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) <= 1.01) {
		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) <= 1.01:
		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) <= 1.01)
		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) <= 1.01)
		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], 1.01], 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 83.9%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1.01000000000000001

   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 1.01000000000000001 < (exp.f64 z)

   1. Initial program 96.8%

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

      1. remove-double-neg 96.8%

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

      2. neg-sub0 96.8%

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

      3. associate-+l- 96.8%

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

      4. add0 96.8%

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

      5. associate--l+ 96.8%

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

      6. associate--r+ 96.8%

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

      7. neg-sub0 96.8%

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

      8. remove-double-neg 96.8%

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

      9. neg-sub0 96.8%

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

      10. distribute-neg-frac2 96.8%

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

      11. neg-sub0 96.8%

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

      12. associate--r- 96.8%

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

      13. neg-sub0 96.8%

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

      14. +-commutative 96.8%

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

      15. sub-neg 96.8%

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

      16. fma-neg 100.0%

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

      17. *-commutative 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. 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 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.01:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{e^{z}} \cdot -0.8862269254527579\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 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 83.9%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.8%

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

      1. remove-double-neg 98.8%

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

      2. neg-sub0 98.8%

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

      3. associate-+l- 98.8%

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

      4. add0 98.8%

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

      5. associate--l+ 98.8%

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

      6. associate--r+ 98.8%

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

      7. neg-sub0 98.8%

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

      8. remove-double-neg 98.8%

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

      9. neg-sub0 98.8%

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

      10. distribute-neg-frac2 98.8%

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

      11. neg-sub0 98.8%

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

      12. associate--r- 98.8%

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

      13. neg-sub0 98.8%

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

      14. +-commutative 98.8%

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

      15. sub-neg 98.8%

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

      16. fma-neg 99.9%

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

      17. *-commutative 99.9%

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

      18. distribute-rgt-neg-in 99.9%

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

      19. 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 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 10000:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 10000.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* 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) <= 10000.0) {
		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 (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 10000.0d0) 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 (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 10000.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 10000.0:
		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 (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 10000.0)
		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 (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 10000.0)
		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[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 10000.0], 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 (exp.f64 z) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1e4

   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 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

   if 1e4 < (exp.f64 z)

   1. Initial program 96.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 10000:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

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 / ((exp(z) * 1.1283791670955126) - (x * y)));
	}
	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
        tmp = x + (y / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    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 {
		tmp = x + (y / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	else:
		tmp = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	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 / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))));
	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);
	else
		tmp = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	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], N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

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

Alternative 5: 71.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-62}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-187}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-300}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-305}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{-214}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.25e-5)
   x
   (if (<= x -4e-62)
     (/ -1.0 x)
     (if (<= x -2.9e-187)
       x
       (if (<= x -8.6e-300)
         (* y 0.8862269254527579)
         (if (<= x 3.35e-305)
           (/ -1.0 x)
           (if (<= x 1.76e-214)
             (* y 0.8862269254527579)
             (if (<= x 1.35e-28) (/ -1.0 x) x))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e-5) {
		tmp = x;
	} else if (x <= -4e-62) {
		tmp = -1.0 / x;
	} else if (x <= -2.9e-187) {
		tmp = x;
	} else if (x <= -8.6e-300) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 3.35e-305) {
		tmp = -1.0 / x;
	} else if (x <= 1.76e-214) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 1.35e-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 <= (-1.25d-5)) then
        tmp = x
    else if (x <= (-4d-62)) then
        tmp = (-1.0d0) / x
    else if (x <= (-2.9d-187)) then
        tmp = x
    else if (x <= (-8.6d-300)) then
        tmp = y * 0.8862269254527579d0
    else if (x <= 3.35d-305) then
        tmp = (-1.0d0) / x
    else if (x <= 1.76d-214) then
        tmp = y * 0.8862269254527579d0
    else if (x <= 1.35d-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 <= -1.25e-5) {
		tmp = x;
	} else if (x <= -4e-62) {
		tmp = -1.0 / x;
	} else if (x <= -2.9e-187) {
		tmp = x;
	} else if (x <= -8.6e-300) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 3.35e-305) {
		tmp = -1.0 / x;
	} else if (x <= 1.76e-214) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 1.35e-28) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.25e-5:
		tmp = x
	elif x <= -4e-62:
		tmp = -1.0 / x
	elif x <= -2.9e-187:
		tmp = x
	elif x <= -8.6e-300:
		tmp = y * 0.8862269254527579
	elif x <= 3.35e-305:
		tmp = -1.0 / x
	elif x <= 1.76e-214:
		tmp = y * 0.8862269254527579
	elif x <= 1.35e-28:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.25e-5)
		tmp = x;
	elseif (x <= -4e-62)
		tmp = Float64(-1.0 / x);
	elseif (x <= -2.9e-187)
		tmp = x;
	elseif (x <= -8.6e-300)
		tmp = Float64(y * 0.8862269254527579);
	elseif (x <= 3.35e-305)
		tmp = Float64(-1.0 / x);
	elseif (x <= 1.76e-214)
		tmp = Float64(y * 0.8862269254527579);
	elseif (x <= 1.35e-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 <= -1.25e-5)
		tmp = x;
	elseif (x <= -4e-62)
		tmp = -1.0 / x;
	elseif (x <= -2.9e-187)
		tmp = x;
	elseif (x <= -8.6e-300)
		tmp = y * 0.8862269254527579;
	elseif (x <= 3.35e-305)
		tmp = -1.0 / x;
	elseif (x <= 1.76e-214)
		tmp = y * 0.8862269254527579;
	elseif (x <= 1.35e-28)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.25e-5], x, If[LessEqual[x, -4e-62], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, -2.9e-187], x, If[LessEqual[x, -8.6e-300], N[(y * 0.8862269254527579), $MachinePrecision], If[LessEqual[x, 3.35e-305], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 1.76e-214], N[(y * 0.8862269254527579), $MachinePrecision], If[LessEqual[x, 1.35e-28], N[(-1.0 / x), $MachinePrecision], x]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25000000000000006e-5 or -4.0000000000000002e-62 < x < -2.89999999999999988e-187 or 1.3499999999999999e-28 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf 90.3%

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

   if -1.25000000000000006e-5 < x < -4.0000000000000002e-62 or -8.6000000000000001e-300 < x < 3.3500000000000002e-305 or 1.75999999999999997e-214 < x < 1.3499999999999999e-28

   1. Initial program 92.6%

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

   3. Taylor expanded in x around inf 56.2%

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

   4. Taylor expanded in x around 0 55.8%

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

   if -2.89999999999999988e-187 < x < -8.6000000000000001e-300 or 3.3500000000000002e-305 < x < 1.75999999999999997e-214

   1. Initial program 91.6%

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

      1. remove-double-neg 91.6%

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

      2. neg-sub0 91.6%

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

      3. associate-+l- 91.6%

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

      4. add0 91.6%

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

      5. associate--l+ 91.6%

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

      6. associate--r+ 91.6%

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

      7. neg-sub0 91.6%

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

      8. remove-double-neg 91.6%

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

      9. neg-sub0 91.6%

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

      10. distribute-neg-frac2 91.6%

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

      11. neg-sub0 91.3%

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

      12. associate--r- 91.3%

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

      13. neg-sub0 91.8%

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

      14. +-commutative 91.8%

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

      15. sub-neg 91.8%

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

      16. fma-neg 91.8%

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

      17. *-commutative 91.8%

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

      18. distribute-rgt-neg-in 91.8%

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

      19. metadata-eval 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in z around 0 64.5%

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

   6. Taylor expanded in x around 0 55.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-62}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-187}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-300}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-305}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{-214}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-63}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-186}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-300}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-214}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 10^{-27}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.5e-5)
   x
   (if (<= x -7e-63)
     (/ -1.0 x)
     (if (<= x -1.7e-186)
       x
       (if (<= x -7.6e-300)
         (* y 0.8862269254527579)
         (if (<= x 2.4e-308)
           (/ -1.0 x)
           (if (<= x 3.1e-214)
             (/ y 1.1283791670955126)
             (if (<= x 1e-27) (/ -1.0 x) x))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e-5) {
		tmp = x;
	} else if (x <= -7e-63) {
		tmp = -1.0 / x;
	} else if (x <= -1.7e-186) {
		tmp = x;
	} else if (x <= -7.6e-300) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 2.4e-308) {
		tmp = -1.0 / x;
	} else if (x <= 3.1e-214) {
		tmp = y / 1.1283791670955126;
	} else if (x <= 1e-27) {
		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 <= (-2.5d-5)) then
        tmp = x
    else if (x <= (-7d-63)) then
        tmp = (-1.0d0) / x
    else if (x <= (-1.7d-186)) then
        tmp = x
    else if (x <= (-7.6d-300)) then
        tmp = y * 0.8862269254527579d0
    else if (x <= 2.4d-308) then
        tmp = (-1.0d0) / x
    else if (x <= 3.1d-214) then
        tmp = y / 1.1283791670955126d0
    else if (x <= 1d-27) 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 <= -2.5e-5) {
		tmp = x;
	} else if (x <= -7e-63) {
		tmp = -1.0 / x;
	} else if (x <= -1.7e-186) {
		tmp = x;
	} else if (x <= -7.6e-300) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 2.4e-308) {
		tmp = -1.0 / x;
	} else if (x <= 3.1e-214) {
		tmp = y / 1.1283791670955126;
	} else if (x <= 1e-27) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.5e-5:
		tmp = x
	elif x <= -7e-63:
		tmp = -1.0 / x
	elif x <= -1.7e-186:
		tmp = x
	elif x <= -7.6e-300:
		tmp = y * 0.8862269254527579
	elif x <= 2.4e-308:
		tmp = -1.0 / x
	elif x <= 3.1e-214:
		tmp = y / 1.1283791670955126
	elif x <= 1e-27:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.5e-5)
		tmp = x;
	elseif (x <= -7e-63)
		tmp = Float64(-1.0 / x);
	elseif (x <= -1.7e-186)
		tmp = x;
	elseif (x <= -7.6e-300)
		tmp = Float64(y * 0.8862269254527579);
	elseif (x <= 2.4e-308)
		tmp = Float64(-1.0 / x);
	elseif (x <= 3.1e-214)
		tmp = Float64(y / 1.1283791670955126);
	elseif (x <= 1e-27)
		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 <= -2.5e-5)
		tmp = x;
	elseif (x <= -7e-63)
		tmp = -1.0 / x;
	elseif (x <= -1.7e-186)
		tmp = x;
	elseif (x <= -7.6e-300)
		tmp = y * 0.8862269254527579;
	elseif (x <= 2.4e-308)
		tmp = -1.0 / x;
	elseif (x <= 3.1e-214)
		tmp = y / 1.1283791670955126;
	elseif (x <= 1e-27)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.5e-5], x, If[LessEqual[x, -7e-63], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, -1.7e-186], x, If[LessEqual[x, -7.6e-300], N[(y * 0.8862269254527579), $MachinePrecision], If[LessEqual[x, 2.4e-308], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 3.1e-214], N[(y / 1.1283791670955126), $MachinePrecision], If[LessEqual[x, 1e-27], N[(-1.0 / x), $MachinePrecision], x]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.50000000000000012e-5 or -7.00000000000000006e-63 < x < -1.7e-186 or 1e-27 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf 90.3%

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

   if -2.50000000000000012e-5 < x < -7.00000000000000006e-63 or -7.60000000000000026e-300 < x < 2.40000000000000008e-308 or 3.10000000000000004e-214 < x < 1e-27

   1. Initial program 92.6%

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

   3. Taylor expanded in x around inf 56.2%

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

   4. Taylor expanded in x around 0 55.8%

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

   if -1.7e-186 < x < -7.60000000000000026e-300

   1. Initial program 91.7%

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

      1. remove-double-neg 91.7%

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

      2. neg-sub0 91.7%

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

      3. associate-+l- 91.7%

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

      4. add0 91.7%

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

      5. associate--l+ 91.7%

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

      6. associate--r+ 91.7%

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

      7. neg-sub0 91.7%

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

      8. remove-double-neg 91.7%

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

      9. neg-sub0 91.7%

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

      10. distribute-neg-frac2 91.7%

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

      11. neg-sub0 91.7%

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

      12. associate--r- 91.7%

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

      13. neg-sub0 92.0%

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

      14. +-commutative 92.0%

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

      15. sub-neg 92.0%

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

      16. fma-neg 92.0%

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

      17. *-commutative 92.0%

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

      18. distribute-rgt-neg-in 92.0%

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

      19. metadata-eval 92.0%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in z around 0 55.5%

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

   6. Taylor expanded in x around 0 48.5%

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

   if 2.40000000000000008e-308 < x < 3.10000000000000004e-214

   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%

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

      2. neg-sub0 91.5%

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

      3. associate-+l- 91.5%

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

      4. add0 91.5%

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

      5. associate--l+ 91.5%

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

      6. associate--r+ 91.5%

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

      7. neg-sub0 91.5%

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

      8. remove-double-neg 91.5%

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

      9. neg-sub0 91.5%

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

      10. distribute-neg-frac2 91.5%

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

      11. neg-sub0 90.8%

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

      12. associate--r- 90.8%

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

      13. neg-sub0 91.5%

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

      14. +-commutative 91.5%

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

      15. sub-neg 91.5%

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

      16. fma-neg 91.5%

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

      17. *-commutative 91.5%

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

      18. distribute-rgt-neg-in 91.5%

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

      19. metadata-eval 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in z around 0 74.7%

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

   6. Taylor expanded in x around 0 64.2%

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

      1. *-commutative 64.2%

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

      2. metadata-eval 64.2%

         \[\leadsto y \cdot \color{blue}{\left(--0.8862269254527579\right)} \]

      3. distribute-rgt-neg-in 64.2%

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

      4. metadata-eval 64.6%

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

      5. div-inv 64.7%

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

      6. distribute-neg-frac2 64.7%

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

      7. metadata-eval 64.7%

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

   8. Applied egg-rr 64.7%

      \[\leadsto \color{blue}{\frac{y}{1.1283791670955126}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-63}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-186}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-300}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-214}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 10^{-27}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x - \frac{y}{-1.1283791670955126}\\ \mathbf{if}\;z \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-246}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))) (t_1 (- x (/ y -1.1283791670955126))))
   (if (<= z -5e-10)
     t_0
     (if (<= z -1.6e-203)
       t_1
       (if (<= z -9e-246)
         t_0
         (if (<= z 6e-135) t_1 (if (<= z 8e-32) t_0 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x - (y / -1.1283791670955126);
	double tmp;
	if (z <= -5e-10) {
		tmp = t_0;
	} else if (z <= -1.6e-203) {
		tmp = t_1;
	} else if (z <= -9e-246) {
		tmp = t_0;
	} else if (z <= 6e-135) {
		tmp = t_1;
	} else if (z <= 8e-32) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x - (y / (-1.1283791670955126d0))
    if (z <= (-5d-10)) then
        tmp = t_0
    else if (z <= (-1.6d-203)) then
        tmp = t_1
    else if (z <= (-9d-246)) then
        tmp = t_0
    else if (z <= 6d-135) then
        tmp = t_1
    else if (z <= 8d-32) then
        tmp = t_0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x - (y / -1.1283791670955126);
	double tmp;
	if (z <= -5e-10) {
		tmp = t_0;
	} else if (z <= -1.6e-203) {
		tmp = t_1;
	} else if (z <= -9e-246) {
		tmp = t_0;
	} else if (z <= 6e-135) {
		tmp = t_1;
	} else if (z <= 8e-32) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x - (y / -1.1283791670955126)
	tmp = 0
	if z <= -5e-10:
		tmp = t_0
	elif z <= -1.6e-203:
		tmp = t_1
	elif z <= -9e-246:
		tmp = t_0
	elif z <= 6e-135:
		tmp = t_1
	elif z <= 8e-32:
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x - Float64(y / -1.1283791670955126))
	tmp = 0.0
	if (z <= -5e-10)
		tmp = t_0;
	elseif (z <= -1.6e-203)
		tmp = t_1;
	elseif (z <= -9e-246)
		tmp = t_0;
	elseif (z <= 6e-135)
		tmp = t_1;
	elseif (z <= 8e-32)
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x - (y / -1.1283791670955126);
	tmp = 0.0;
	if (z <= -5e-10)
		tmp = t_0;
	elseif (z <= -1.6e-203)
		tmp = t_1;
	elseif (z <= -9e-246)
		tmp = t_0;
	elseif (z <= 6e-135)
		tmp = t_1;
	elseif (z <= 8e-32)
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5e-10], t$95$0, If[LessEqual[z, -1.6e-203], t$95$1, If[LessEqual[z, -9e-246], t$95$0, If[LessEqual[z, 6e-135], t$95$1, If[LessEqual[z, 8e-32], t$95$0, x]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.00000000000000031e-10 or -1.6e-203 < z < -8.99999999999999998e-246 or 6.00000000000000024e-135 < z < 8.00000000000000045e-32

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf 93.7%

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

   if -5.00000000000000031e-10 < z < -1.6e-203 or -8.99999999999999998e-246 < z < 6.00000000000000024e-135

   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%

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

      2. neg-sub0 99.8%

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

      3. associate-+l- 99.8%

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

      4. add0 99.8%

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

      5. associate--l+ 99.8%

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

      6. associate--r+ 99.8%

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

      7. neg-sub0 99.8%

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

      8. remove-double-neg 99.8%

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

      9. neg-sub0 99.8%

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

      10. distribute-neg-frac2 99.8%

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

      11. neg-sub0 99.8%

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

      12. associate--r- 99.8%

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

      13. neg-sub0 99.8%

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

      14. +-commutative 99.8%

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

      15. sub-neg 99.8%

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

      16. fma-neg 99.8%

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

      17. *-commutative 99.8%

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

      18. distribute-rgt-neg-in 99.8%

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

      19. 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 99.5%

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

   6. Taylor expanded in x around 0 76.4%

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

   if 8.00000000000000045e-32 < z

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf 94.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-203}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-246}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-203}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-246}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-134}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z -5.8e-12)
     t_0
     (if (<= z -1.95e-203)
       (+ x (/ y (+ 1.1283791670955126 (* z 1.1283791670955126))))
       (if (<= z -3.7e-246)
         t_0
         (if (<= z 6.5e-134)
           (- x (/ y -1.1283791670955126))
           (if (<= z 8.8e-32) t_0 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -5.8e-12) {
		tmp = t_0;
	} else if (z <= -1.95e-203) {
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	} else if (z <= -3.7e-246) {
		tmp = t_0;
	} else if (z <= 6.5e-134) {
		tmp = x - (y / -1.1283791670955126);
	} else if (z <= 8.8e-32) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    if (z <= (-5.8d-12)) then
        tmp = t_0
    else if (z <= (-1.95d-203)) then
        tmp = x + (y / (1.1283791670955126d0 + (z * 1.1283791670955126d0)))
    else if (z <= (-3.7d-246)) then
        tmp = t_0
    else if (z <= 6.5d-134) then
        tmp = x - (y / (-1.1283791670955126d0))
    else if (z <= 8.8d-32) then
        tmp = t_0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -5.8e-12) {
		tmp = t_0;
	} else if (z <= -1.95e-203) {
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	} else if (z <= -3.7e-246) {
		tmp = t_0;
	} else if (z <= 6.5e-134) {
		tmp = x - (y / -1.1283791670955126);
	} else if (z <= 8.8e-32) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if z <= -5.8e-12:
		tmp = t_0
	elif z <= -1.95e-203:
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)))
	elif z <= -3.7e-246:
		tmp = t_0
	elif z <= 6.5e-134:
		tmp = x - (y / -1.1283791670955126)
	elif z <= 8.8e-32:
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if (z <= -5.8e-12)
		tmp = t_0;
	elseif (z <= -1.95e-203)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 + Float64(z * 1.1283791670955126))));
	elseif (z <= -3.7e-246)
		tmp = t_0;
	elseif (z <= 6.5e-134)
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	elseif (z <= 8.8e-32)
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	tmp = 0.0;
	if (z <= -5.8e-12)
		tmp = t_0;
	elseif (z <= -1.95e-203)
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	elseif (z <= -3.7e-246)
		tmp = t_0;
	elseif (z <= 6.5e-134)
		tmp = x - (y / -1.1283791670955126);
	elseif (z <= 8.8e-32)
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.8e-12], t$95$0, If[LessEqual[z, -1.95e-203], N[(x + N[(y / N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.7e-246], t$95$0, If[LessEqual[z, 6.5e-134], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.8e-32], t$95$0, x]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.8000000000000003e-12 or -1.95e-203 < z < -3.7e-246 or 6.4999999999999998e-134 < z < 8.7999999999999999e-32

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf 93.7%

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

   if -5.8000000000000003e-12 < z < -1.95e-203

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 99.7%

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

      1. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 81.2%

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

   if -3.7e-246 < z < 6.4999999999999998e-134

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. neg-sub0 99.9%

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

      3. associate-+l- 99.9%

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

      4. add0 99.9%

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

      5. associate--l+ 99.9%

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

      6. associate--r+ 99.9%

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

      7. neg-sub0 99.9%

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

      8. remove-double-neg 99.9%

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

      9. neg-sub0 99.9%

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

      10. distribute-neg-frac2 99.9%

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

      11. neg-sub0 99.9%

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

      12. associate--r- 99.9%

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

      13. neg-sub0 99.9%

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

      14. +-commutative 99.9%

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

      15. sub-neg 99.9%

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

      16. fma-neg 99.9%

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

      17. *-commutative 99.9%

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

      18. distribute-rgt-neg-in 99.9%

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

      19. 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 99.9%

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

   6. Taylor expanded in x around 0 73.7%

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

   if 8.7999999999999999e-32 < z

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf 94.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-203}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-246}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-134}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -2.85 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-204}:\\ \;\;\;\;x + -0.8862269254527579 \cdot \left(z \cdot y - y\right)\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-247}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-133}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z -2.85e-7)
     t_0
     (if (<= z -6e-204)
       (+ x (* -0.8862269254527579 (- (* z y) y)))
       (if (<= z -1.8e-247)
         t_0
         (if (<= z 2.4e-133)
           (- x (/ y -1.1283791670955126))
           (if (<= z 2.35e-31) t_0 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -2.85e-7) {
		tmp = t_0;
	} else if (z <= -6e-204) {
		tmp = x + (-0.8862269254527579 * ((z * y) - y));
	} else if (z <= -1.8e-247) {
		tmp = t_0;
	} else if (z <= 2.4e-133) {
		tmp = x - (y / -1.1283791670955126);
	} else if (z <= 2.35e-31) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    if (z <= (-2.85d-7)) then
        tmp = t_0
    else if (z <= (-6d-204)) then
        tmp = x + ((-0.8862269254527579d0) * ((z * y) - y))
    else if (z <= (-1.8d-247)) then
        tmp = t_0
    else if (z <= 2.4d-133) then
        tmp = x - (y / (-1.1283791670955126d0))
    else if (z <= 2.35d-31) then
        tmp = t_0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -2.85e-7) {
		tmp = t_0;
	} else if (z <= -6e-204) {
		tmp = x + (-0.8862269254527579 * ((z * y) - y));
	} else if (z <= -1.8e-247) {
		tmp = t_0;
	} else if (z <= 2.4e-133) {
		tmp = x - (y / -1.1283791670955126);
	} else if (z <= 2.35e-31) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if z <= -2.85e-7:
		tmp = t_0
	elif z <= -6e-204:
		tmp = x + (-0.8862269254527579 * ((z * y) - y))
	elif z <= -1.8e-247:
		tmp = t_0
	elif z <= 2.4e-133:
		tmp = x - (y / -1.1283791670955126)
	elif z <= 2.35e-31:
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if (z <= -2.85e-7)
		tmp = t_0;
	elseif (z <= -6e-204)
		tmp = Float64(x + Float64(-0.8862269254527579 * Float64(Float64(z * y) - y)));
	elseif (z <= -1.8e-247)
		tmp = t_0;
	elseif (z <= 2.4e-133)
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	elseif (z <= 2.35e-31)
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	tmp = 0.0;
	if (z <= -2.85e-7)
		tmp = t_0;
	elseif (z <= -6e-204)
		tmp = x + (-0.8862269254527579 * ((z * y) - y));
	elseif (z <= -1.8e-247)
		tmp = t_0;
	elseif (z <= 2.4e-133)
		tmp = x - (y / -1.1283791670955126);
	elseif (z <= 2.35e-31)
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.85e-7], t$95$0, If[LessEqual[z, -6e-204], N[(x + N[(-0.8862269254527579 * N[(N[(z * y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.8e-247], t$95$0, If[LessEqual[z, 2.4e-133], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.35e-31], t$95$0, x]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.8500000000000002e-7 or -5.9999999999999997e-204 < z < -1.7999999999999998e-247 or 2.4e-133 < z < 2.34999999999999993e-31

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf 93.7%

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

   if -2.8500000000000002e-7 < z < -5.9999999999999997e-204

   1. Initial program 99.7%

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

      1. remove-double-neg 99.7%

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

      2. neg-sub0 99.7%

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

      3. associate-+l- 99.7%

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

      4. add0 99.7%

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

      5. associate--l+ 99.7%

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

      6. associate--r+ 99.7%

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

      7. neg-sub0 99.7%

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

      8. remove-double-neg 99.7%

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

      9. neg-sub0 99.7%

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

      10. distribute-neg-frac2 99.7%

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

      11. neg-sub0 99.7%

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

      12. associate--r- 99.7%

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

      13. neg-sub0 99.7%

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

      14. +-commutative 99.7%

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

      15. sub-neg 99.7%

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

      16. fma-neg 99.7%

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

      17. *-commutative 99.7%

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

      18. distribute-rgt-neg-in 99.7%

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

      19. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 81.4%

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

      1. *-commutative 81.4%

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

   7. Simplified 81.4%

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

   8. Taylor expanded in z around 0 81.4%

      \[\leadsto x - \color{blue}{\left(y + -1 \cdot \left(y \cdot z\right)\right)} \cdot -0.8862269254527579 \]
   9. Step-by-step derivation

      1. mul-1-neg 81.4%

         \[\leadsto x - \left(y + \color{blue}{\left(-y \cdot z\right)}\right) \cdot -0.8862269254527579 \]

      2. unsub-neg 81.4%

         \[\leadsto x - \color{blue}{\left(y - y \cdot z\right)} \cdot -0.8862269254527579 \]

   10. Simplified 81.4%

       \[\leadsto x - \color{blue}{\left(y - y \cdot z\right)} \cdot -0.8862269254527579 \]

   if -1.7999999999999998e-247 < z < 2.4e-133

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. neg-sub0 99.9%

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

      3. associate-+l- 99.9%

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

      4. add0 99.9%

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

      5. associate--l+ 99.9%

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

      6. associate--r+ 99.9%

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

      7. neg-sub0 99.9%

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

      8. remove-double-neg 99.9%

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

      9. neg-sub0 99.9%

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

      10. distribute-neg-frac2 99.9%

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

      11. neg-sub0 99.9%

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

      12. associate--r- 99.9%

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

      13. neg-sub0 99.9%

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

      14. +-commutative 99.9%

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

      15. sub-neg 99.9%

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

      16. fma-neg 99.9%

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

      17. *-commutative 99.9%

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

      18. distribute-rgt-neg-in 99.9%

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

      19. 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 99.9%

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

   6. Taylor expanded in x around 0 73.7%

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

   if 2.34999999999999993e-31 < z

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf 94.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-204}:\\ \;\;\;\;x + -0.8862269254527579 \cdot \left(z \cdot y - y\right)\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-247}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-133}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-31}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq 6.2 \cdot 10^{-279}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-256}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z 6.2e-279)
     t_0
     (if (<= z 3e-256) (* y 0.8862269254527579) (if (<= z 2.75e-30) t_0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= 6.2e-279) {
		tmp = t_0;
	} else if (z <= 3e-256) {
		tmp = y * 0.8862269254527579;
	} else if (z <= 2.75e-30) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    if (z <= 6.2d-279) then
        tmp = t_0
    else if (z <= 3d-256) then
        tmp = y * 0.8862269254527579d0
    else if (z <= 2.75d-30) then
        tmp = t_0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= 6.2e-279) {
		tmp = t_0;
	} else if (z <= 3e-256) {
		tmp = y * 0.8862269254527579;
	} else if (z <= 2.75e-30) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if z <= 6.2e-279:
		tmp = t_0
	elif z <= 3e-256:
		tmp = y * 0.8862269254527579
	elif z <= 2.75e-30:
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if (z <= 6.2e-279)
		tmp = t_0;
	elseif (z <= 3e-256)
		tmp = Float64(y * 0.8862269254527579);
	elseif (z <= 2.75e-30)
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	tmp = 0.0;
	if (z <= 6.2e-279)
		tmp = t_0;
	elseif (z <= 3e-256)
		tmp = y * 0.8862269254527579;
	elseif (z <= 2.75e-30)
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 6.2e-279], t$95$0, If[LessEqual[z, 3e-256], N[(y * 0.8862269254527579), $MachinePrecision], If[LessEqual[z, 2.75e-30], t$95$0, x]]]]

Derivation

1. Split input into 3 regimes

2. if z < 6.1999999999999998e-279 or 2.9999999999999998e-256 < z < 2.74999999999999988e-30

   1. Initial program 94.8%

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

   3. Taylor expanded in x around inf 73.5%

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

   if 6.1999999999999998e-279 < z < 2.9999999999999998e-256

   1. Initial program 99.7%

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

      1. remove-double-neg 99.7%

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

      2. neg-sub0 99.7%

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

      3. associate-+l- 99.7%

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

      4. add0 99.7%

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

      5. associate--l+ 99.7%

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

      6. associate--r+ 99.7%

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

      7. neg-sub0 99.7%

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

      8. remove-double-neg 99.7%

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

      9. neg-sub0 99.7%

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

      10. distribute-neg-frac2 99.7%

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

      11. neg-sub0 99.7%

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

      12. associate--r- 99.7%

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

      13. neg-sub0 99.7%

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

      14. +-commutative 99.7%

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

      15. sub-neg 99.7%

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

      16. fma-neg 99.7%

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

      17. *-commutative 99.7%

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

      18. distribute-rgt-neg-in 99.7%

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

      19. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

   if 2.74999999999999988e-30 < z

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf 94.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{-279}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-256}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -150:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 215:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -150.0)
   (+ x (/ -1.0 x))
   (if (<= z 215.0) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -150.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 215.0) {
		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 <= (-150.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 215.0d0) 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 <= -150.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 215.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -150.0:
		tmp = x + (-1.0 / x)
	elif z <= 215.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -150.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 215.0)
		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 <= -150.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 215.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -150.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 215.0], 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 < -150

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf 100.0%

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

   if -150 < z < 215

   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%

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

      2. neg-sub0 99.8%

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

      3. associate-+l- 99.8%

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

      4. add0 99.8%

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

      5. associate--l+ 99.8%

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

      6. associate--r+ 99.8%

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

      7. neg-sub0 99.8%

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

      8. remove-double-neg 99.8%

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

      9. neg-sub0 99.8%

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

      10. distribute-neg-frac2 99.8%

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

      11. neg-sub0 99.8%

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

      12. associate--r- 99.8%

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

      13. neg-sub0 99.8%

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

      14. +-commutative 99.8%

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

      15. sub-neg 99.8%

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

      16. fma-neg 99.8%

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

      17. *-commutative 99.8%

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

      18. distribute-rgt-neg-in 99.8%

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

      19. 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 98.2%

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

   if 215 < z

   1. Initial program 96.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -150:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 215:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-187}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-213}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.6e-187) x (if (<= x 2.45e-213) (* y 0.8862269254527579) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.6e-187) {
		tmp = x;
	} else if (x <= 2.45e-213) {
		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 <= (-4.6d-187)) then
        tmp = x
    else if (x <= 2.45d-213) 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 <= -4.6e-187) {
		tmp = x;
	} else if (x <= 2.45e-213) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.6e-187:
		tmp = x
	elif x <= 2.45e-213:
		tmp = y * 0.8862269254527579
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.6e-187)
		tmp = x;
	elseif (x <= 2.45e-213)
		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 <= -4.6e-187)
		tmp = x;
	elseif (x <= 2.45e-213)
		tmp = y * 0.8862269254527579;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.6e-187], x, If[LessEqual[x, 2.45e-213], N[(y * 0.8862269254527579), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.59999999999999996e-187 or 2.4499999999999999e-213 < x

   1. Initial program 97.6%

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

   3. Taylor expanded in x around inf 71.8%

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

   if -4.59999999999999996e-187 < x < 2.4499999999999999e-213

   1. Initial program 88.2%

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

      1. remove-double-neg 88.2%

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

      2. neg-sub0 88.2%

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

      3. associate-+l- 88.2%

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

      4. add0 88.2%

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

      5. associate--l+ 88.2%

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

      6. associate--r+ 88.2%

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

      7. neg-sub0 88.2%

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

      8. remove-double-neg 88.2%

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

      9. neg-sub0 88.2%

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

      10. distribute-neg-frac2 88.2%

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

      11. neg-sub0 87.5%

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

      12. associate--r- 87.5%

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

      13. neg-sub0 88.5%

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

      14. +-commutative 88.5%

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

      15. sub-neg 88.5%

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

      16. fma-neg 88.5%

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

      17. *-commutative 88.5%

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

      18. distribute-rgt-neg-in 88.5%

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

      19. metadata-eval 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in z around 0 58.0%

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

   6. Taylor expanded in x around 0 49.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-187}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-213}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.4% 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 60.4%

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

4. Final simplification 60.4%

   \[\leadsto x \]
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 2024046 
(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)))))