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

Percentage Accurate: 95.8% → 99.9%

Time: 9.4s

Alternatives: 12

Speedup: 5.8×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+ x (/ y (* x (- (/ (* (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 / (x * (((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 / (x * (((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 / (x * (((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 / (x * (((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(x * Float64(Float64(Float64(exp(z) * 1.1283791670955126) / 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 / (x * (((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[(x * N[(N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] / x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x} - y\right)}\right)\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{5641895835477563}{5000000000000000} \cdot \frac{e^{z}}{x}\right), \color{blue}{y}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{\frac{5641895835477563}{5000000000000000} \cdot e^{z}}{x}\right), y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{5641895835477563}{5000000000000000} \cdot e^{z}\right), x\right), y\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(e^{z} \cdot \frac{5641895835477563}{5000000000000000}\right), x\right), y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(e^{z}\right), \frac{5641895835477563}{5000000000000000}\right), x\right), y\right)\right)\right)\right) \]

      7. exp-lowering-exp.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(z\right), \frac{5641895835477563}{5000000000000000}\right), x\right), y\right)\right)\right)\right) \]

   5. Simplified 99.9%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot \left(0.5641895835477563 + z \cdot 0.18806319451591877\right)\right)\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot \frac{y}{e^{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.0)
     (+
      x
      (/
       y
       (-
        (+
         1.1283791670955126
         (*
          z
          (+
           1.1283791670955126
           (* z (+ 0.5641895835477563 (* z 0.18806319451591877))))))
        (* x y))))
     (+ x (* 0.8862269254527579 (/ y (exp z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * (0.5641895835477563 + (z * 0.18806319451591877)))))) - (x * y)));
	} else {
		tmp = x + (0.8862269254527579 * (y / exp(z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * (1.1283791670955126d0 + (z * (0.5641895835477563d0 + (z * 0.18806319451591877d0)))))) - (x * y)))
    else
        tmp = x + (0.8862269254527579d0 * (y / exp(z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 1.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * (0.5641895835477563 + (z * 0.18806319451591877)))))) - (x * y)));
	} else {
		tmp = x + (0.8862269254527579 * (y / Math.exp(z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.0:
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * (0.5641895835477563 + (z * 0.18806319451591877)))))) - (x * y)))
	else:
		tmp = x + (0.8862269254527579 * (y / math.exp(z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * Float64(1.1283791670955126 + Float64(z * Float64(0.5641895835477563 + Float64(z * 0.18806319451591877)))))) - Float64(x * y))));
	else
		tmp = Float64(x + Float64(0.8862269254527579 * Float64(y / exp(z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 1.0)
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * (0.5641895835477563 + (z * 0.18806319451591877)))))) - (x * y)));
	else
		tmp = x + (0.8862269254527579 * (y / exp(z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * N[(1.1283791670955126 + N[(z * N[(0.5641895835477563 + N[(z * 0.18806319451591877), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(0.8862269254527579 * N[(y / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{10000000000000000}, \left(\frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{10000000000000000}, \left(z \cdot \frac{5641895835477563}{30000000000000000}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{10000000000000000}, \mathsf{*.f64}\left(z, \frac{5641895835477563}{30000000000000000}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

   5. Simplified 99.6%

      \[\leadsto x + \frac{y}{\color{blue}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot \left(0.5641895835477563 + z \cdot 0.18806319451591877\right)\right)\right)} - x \cdot y} \]

   if 1 < (exp.f64 z)

   1. Initial program 96.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}} + \color{blue}{x} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}\right), \color{blue}{x}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{5000000000000000}{5641895835477563}, \left(\frac{y}{e^{z}}\right)\right), x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{5000000000000000}{5641895835477563}, \mathsf{/.f64}\left(y, \left(e^{z}\right)\right)\right), x\right) \]

      5. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{5000000000000000}{5641895835477563}, \mathsf{/.f64}\left(y, \mathsf{exp.f64}\left(z\right)\right)\right), x\right) \]

   5. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot \left(0.5641895835477563 + z \cdot 0.18806319451591877\right)\right)\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot \frac{y}{e^{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot \left(0.5641895835477563 + z \cdot 0.18806319451591877\right)\right)\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 2.0)
     (+
      x
      (/
       y
       (-
        (+
         1.1283791670955126
         (*
          z
          (+
           1.1283791670955126
           (* z (+ 0.5641895835477563 (* z 0.18806319451591877))))))
        (* 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) <= 2.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * (0.5641895835477563 + (z * 0.18806319451591877)))))) - (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) <= 2.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * (1.1283791670955126d0 + (z * (0.5641895835477563d0 + (z * 0.18806319451591877d0)))))) - (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) <= 2.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * (0.5641895835477563 + (z * 0.18806319451591877)))))) - (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) <= 2.0:
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * (0.5641895835477563 + (z * 0.18806319451591877)))))) - (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) <= 2.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * Float64(1.1283791670955126 + Float64(z * Float64(0.5641895835477563 + Float64(z * 0.18806319451591877)))))) - 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) <= 2.0)
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * (0.5641895835477563 + (z * 0.18806319451591877)))))) - (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], 2.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * N[(1.1283791670955126 + N[(z * N[(0.5641895835477563 + N[(z * 0.18806319451591877), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 84.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{10000000000000000}, \left(\frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{10000000000000000}, \left(z \cdot \frac{5641895835477563}{30000000000000000}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{10000000000000000}, \mathsf{*.f64}\left(z, \frac{5641895835477563}{30000000000000000}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

   5. Simplified 99.6%

      \[\leadsto x + \frac{y}{\color{blue}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot \left(0.5641895835477563 + z \cdot 0.18806319451591877\right)\right)\right)} - x \cdot y} \]

   if 2 < (exp.f64 z)

   1. Initial program 96.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

Alternative 4: 98.2% 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 84.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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: 99.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -280:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 160:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot 0.5641895835477563\right)\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -280.0)
   (+ x (/ -1.0 x))
   (if (<= z 160.0)
     (+
      x
      (/
       y
       (-
        (+
         1.1283791670955126
         (* z (+ 1.1283791670955126 (* z 0.5641895835477563))))
        (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -280.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 160.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * 0.5641895835477563)))) - (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 <= (-280.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 160.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * (1.1283791670955126d0 + (z * 0.5641895835477563d0)))) - (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 <= -280.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 160.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * 0.5641895835477563)))) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -280.0:
		tmp = x + (-1.0 / x)
	elif z <= 160.0:
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * 0.5641895835477563)))) - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -280.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 160.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * Float64(1.1283791670955126 + Float64(z * 0.5641895835477563)))) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -280.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 160.0)
		tmp = x + (y / ((1.1283791670955126 + (z * (1.1283791670955126 + (z * 0.5641895835477563)))) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -280.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 160.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * N[(1.1283791670955126 + N[(z * 0.5641895835477563), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -280

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if -280 < z < 160

   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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(\frac{5641895835477563}{10000000000000000} \cdot z\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(z \cdot \frac{5641895835477563}{10000000000000000}\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      5. *-lowering-*.f64 99.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \frac{5641895835477563}{10000000000000000}\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

   5. Simplified 99.4%

      \[\leadsto x + \frac{y}{\color{blue}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot 0.5641895835477563\right)\right)} - x \cdot y} \]

   if 160 < z

   1. Initial program 96.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

Alternative 6: 99.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -190:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 82:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -190.0)
   (+ x (/ -1.0 x))
   (if (<= z 82.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -190.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 82.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-190.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 82.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * 1.1283791670955126d0)) - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -190.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 82.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -190.0:
		tmp = x + (-1.0 / x)
	elif z <= 82.0:
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -190.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 82.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * 1.1283791670955126)) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -190.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 82.0)
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -190.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 82.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -190

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if -190 < z < 82

   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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(\frac{5641895835477563}{5000000000000000} \cdot z\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(z \cdot \frac{5641895835477563}{5000000000000000}\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(z, \frac{5641895835477563}{5000000000000000}\right)\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

   5. Simplified 99.2%

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

   if 82 < z

   1. Initial program 96.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

Alternative 7: 99.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -145:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 122:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -145.0)
   (+ x (/ -1.0 x))
   (if (<= z 122.0) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -145.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 122.0) {
		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 <= (-145.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 122.0d0) 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 <= -145.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 122.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -145.0:
		tmp = x + (-1.0 / x)
	elif z <= 122.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -145.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 122.0)
		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 <= -145.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 122.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -145.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 122.0], 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 < -145

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if -145 < z < 122

   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

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   5. Simplified 98.7%

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

   if 122 < z

   1. Initial program 96.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -145:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 122:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.00115:\\ \;\;\;\;x + y \cdot \left(0.8862269254527579 + z \cdot -0.8862269254527579\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.15e-11)
   (+ x (/ -1.0 x))
   (if (<= z 0.00115)
     (+ x (* y (+ 0.8862269254527579 (* z -0.8862269254527579))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e-11) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.00115) {
		tmp = x + (y * (0.8862269254527579 + (z * -0.8862269254527579)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.15d-11)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.00115d0) then
        tmp = x + (y * (0.8862269254527579d0 + (z * (-0.8862269254527579d0))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e-11) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.00115) {
		tmp = x + (y * (0.8862269254527579 + (z * -0.8862269254527579)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.15e-11:
		tmp = x + (-1.0 / x)
	elif z <= 0.00115:
		tmp = x + (y * (0.8862269254527579 + (z * -0.8862269254527579)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15e-11)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.00115)
		tmp = Float64(x + Float64(y * Float64(0.8862269254527579 + Float64(z * -0.8862269254527579))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.15e-11)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.00115)
		tmp = x + (y * (0.8862269254527579 + (z * -0.8862269254527579)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15e-11], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.00115], N[(x + N[(y * N[(0.8862269254527579 + N[(z * -0.8862269254527579), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15000000000000007e-11

   1. Initial program 85.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 98.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 98.5%

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

   if -1.15000000000000007e-11 < z < 0.00115

   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

      \[\leadsto \color{blue}{x + \left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y} + \color{blue}{\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) + \color{blue}{\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right), \color{blue}{\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)\right), \left(\color{blue}{\frac{-5641895835477563}{5000000000000000}} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)\right)\right), \left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(x \cdot y\right)\right)\right)\right), \left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot x\right)\right)\right)\right), \left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \left(\frac{\frac{-5641895835477563}{5000000000000000} \cdot \left(y \cdot z\right)}{\color{blue}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{-5641895835477563}{5000000000000000} \cdot \left(y \cdot z\right)\right), \color{blue}{\left({\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-5641895835477563}{5000000000000000}, \left(y \cdot z\right)\right), \left({\color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, z\right)\right), \left({\left(\frac{5641895835477563}{5000000000000000} - \color{blue}{x \cdot y}\right)}^{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, z\right)\right), \left(\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right) \cdot \color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right), \color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}\right)\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(x \cdot y\right)\right), \left(\color{blue}{\frac{5641895835477563}{5000000000000000}} - x \cdot y\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot x\right)\right), \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right), \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)\right)\right)\right) \]

      18. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right), \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right), \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right), \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right)\right) \]

   5. Simplified 99.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\frac{5000000000000000}{5641895835477563} + \frac{-5000000000000000}{5641895835477563} \cdot z\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot \left(\frac{5000000000000000}{5641895835477563} + \frac{-5000000000000000}{5641895835477563} \cdot z\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{5000000000000000}{5641895835477563} + \frac{-5000000000000000}{5641895835477563} \cdot z\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{5000000000000000}{5641895835477563}, \color{blue}{\left(\frac{-5000000000000000}{5641895835477563} \cdot z\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{5000000000000000}{5641895835477563}, \left(z \cdot \color{blue}{\frac{-5000000000000000}{5641895835477563}}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 77.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{5000000000000000}{5641895835477563}, \mathsf{*.f64}\left(z, \color{blue}{\frac{-5000000000000000}{5641895835477563}}\right)\right)\right)\right) \]

   8. Simplified 77.9%

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

   if 0.00115 < z

   1. Initial program 96.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

Alternative 9: 86.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.00068:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.7e-12)
   (+ x (/ -1.0 x))
   (if (<= z 0.00068) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e-12) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.00068) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.7d-12)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.00068d0) then
        tmp = x + (y / 1.1283791670955126d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e-12) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.00068) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.7e-12:
		tmp = x + (-1.0 / x)
	elif z <= 0.00068:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.7e-12)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.00068)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.7e-12)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.00068)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.7e-12], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.00068], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6999999999999998e-12

   1. Initial program 85.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 98.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 98.5%

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

   if -2.6999999999999998e-12 < z < 6.8e-4

   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

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   5. Simplified 98.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\frac{5641895835477563}{5000000000000000}}\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 77.4%

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

   if 6.8e-4 < z

   1. Initial program 96.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

Alternative 10: 70.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-303}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.4e-117) x (if (<= x 2.8e-303) (* y 0.8862269254527579) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.4e-117) {
		tmp = x;
	} else if (x <= 2.8e-303) {
		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 <= (-5.4d-117)) then
        tmp = x
    else if (x <= 2.8d-303) 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 <= -5.4e-117) {
		tmp = x;
	} else if (x <= 2.8e-303) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.4e-117:
		tmp = x
	elif x <= 2.8e-303:
		tmp = y * 0.8862269254527579
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.4e-117)
		tmp = x;
	elseif (x <= 2.8e-303)
		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 <= -5.4e-117)
		tmp = x;
	elseif (x <= 2.8e-303)
		tmp = y * 0.8862269254527579;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.4e-117], x, If[LessEqual[x, 2.8e-303], N[(y * 0.8862269254527579), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.40000000000000005e-117 or 2.8e-303 < x

   1. Initial program 95.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 79.5%

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

   if -5.40000000000000005e-117 < x < 2.8e-303

   1. Initial program 95.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 67.6%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   5. Simplified 67.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 56.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{5000000000000000}{5641895835477563}, \color{blue}{y}\right) \]

   8. Simplified 56.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-303}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.6% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -43:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z -43.0) (+ x (/ -1.0 x)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -43.0) {
		tmp = x + (-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 (z <= (-43.0d0)) then
        tmp = x + ((-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 (z <= -43.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -43.0:
		tmp = x + (-1.0 / x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -43.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -43.0)
		tmp = x + (-1.0 / x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -43.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -43

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if -43 < z

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 74.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.9% 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 95.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 69.6%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 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 2024138 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ 1 (- (* (/ 5641895835477563/5000000000000000 y) (exp z)) x))))

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