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

Percentage Accurate: 95.8% → 99.7%

Time: 9.4s

Alternatives: 9

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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot \left(1.1283791670955126 + z \cdot \left(0.5641895835477563 + z \cdot 0.18806319451591877\right)\right) - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 5e-11)
   (+ 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) <= 5e-11) {
		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) <= 5d-11) 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) <= 5e-11) {
		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) <= 5e-11:
		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) <= 5e-11)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 2.0)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 + Float64(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) <= 5e-11)
		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], 5e-11], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(x + N[(y / N[(1.1283791670955126 + N[(N[(z * N[(1.1283791670955126 + N[(z * N[(0.5641895835477563 + N[(z * 0.18806319451591877), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 5.00000000000000018e-11

   1. Initial program 96.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 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 5.00000000000000018e-11 < (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 99.1%

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

   5. Simplified 99.1%

      \[\leadsto x + \frac{y}{\color{blue}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot \left(0.5641895835477563 + z \cdot 0.18806319451591877\right)\right)\right)} - x \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + z \cdot \frac{5641895835477563}{30000000000000000}\right)\right)\right) - x \cdot y} + \color{blue}{x} \]

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

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

   7. Applied egg-rr 99.1%

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

   if 2 < (exp.f64 z)

   1. Initial program 94.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot \left(1.1283791670955126 + z \cdot \left(0.5641895835477563 + z \cdot 0.18806319451591877\right)\right) - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+163}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (<= t_0 4e+163) t_0 (+ x (/ -1.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	double tmp;
	if (t_0 <= 4e+163) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / 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 + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    if (t_0 <= 4d+163) then
        tmp = t_0
    else
        tmp = x + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
	double tmp;
	if (t_0 <= 4e+163) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	tmp = 0
	if t_0 <= 4e+163:
		tmp = t_0
	else:
		tmp = x + (-1.0 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
	tmp = 0.0
	if (t_0 <= 4e+163)
		tmp = t_0;
	else
		tmp = Float64(x + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	tmp = 0.0;
	if (t_0 <= 4e+163)
		tmp = t_0;
	else
		tmp = x + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e+163], t$95$0, N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 3.9999999999999998e163

   1. Initial program 99.0%

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

   if 3.9999999999999998e163 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 86.5%

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

      \[\leadsto \color{blue}{x + \frac{-1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -23:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 340:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot \left(0.5641895835477563 + z \cdot 0.18806319451591877\right)\right)\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -23.0)
   (+ x (/ -1.0 x))
   (if (<= z 340.0)
     (+
      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 (z <= -23.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 340.0) {
		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 (z <= (-23.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 340.0d0) 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 (z <= -23.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 340.0) {
		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 z <= -23.0:
		tmp = x + (-1.0 / x)
	elif z <= 340.0:
		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 (z <= -23.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 340.0)
		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 (z <= -23.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 340.0)
		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[z, -23.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 340.0], 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 z < -23

   1. Initial program 96.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 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 -23 < z < 340

   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 99.1%

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

   5. Simplified 99.1%

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

   if 340 < z

   1. Initial program 94.0%

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

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -23:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 185:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot \left(1.1283791670955126 + z \cdot 0.5641895835477563\right)\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -23.0)
   (+ x (/ -1.0 x))
   (if (<= z 185.0)
     (+
      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 <= -23.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 185.0) {
		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 <= (-23.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 185.0d0) 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 <= -23.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 185.0) {
		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 <= -23.0:
		tmp = x + (-1.0 / x)
	elif z <= 185.0:
		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 <= -23.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 185.0)
		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 <= -23.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 185.0)
		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, -23.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 185.0], 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 < -23

   1. Initial program 96.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 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 -23 < z < 185

   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 98.9%

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

   5. Simplified 98.9%

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

   if 185 < z

   1. Initial program 94.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 5: 99.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -23:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 110:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -23.0)
   (+ x (/ -1.0 x))
   (if (<= z 110.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* 1.1283791670955126 z)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -23.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 110.0) {
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (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 <= (-23.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 110.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (1.1283791670955126d0 * z)) - (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 <= -23.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 110.0) {
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -23.0:
		tmp = x + (-1.0 / x)
	elif z <= 110.0:
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -23.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 110.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(1.1283791670955126 * z)) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -23.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 110.0)
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -23.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 110.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(1.1283791670955126 * z), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -23

   1. Initial program 96.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 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 -23 < z < 110

   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 98.7%

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

   5. Simplified 98.7%

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

   if 110 < z

   1. Initial program 94.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -23:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 110:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -23:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 380:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -23.0)
   (+ x (/ -1.0 x))
   (if (<= z 380.0) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -23.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 380.0) {
		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 <= (-23.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 380.0d0) 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 <= -23.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 380.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -23.0:
		tmp = x + (-1.0 / x)
	elif z <= 380.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -23.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 380.0)
		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 <= -23.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 380.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -23.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 380.0], 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 < -23

   1. Initial program 96.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 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 -23 < z < 380

   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 98.2%

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

   5. Simplified 98.2%

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

   if 380 < z

   1. Initial program 94.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -23:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 380:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e-9) x (if (<= x 1.45e-52) (/ -1.0 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-9) {
		tmp = x;
	} else if (x <= 1.45e-52) {
		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 <= (-2d-9)) then
        tmp = x
    else if (x <= 1.45d-52) 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 <= -2e-9) {
		tmp = x;
	} else if (x <= 1.45e-52) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2e-9:
		tmp = x
	elif x <= 1.45e-52:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e-9)
		tmp = x;
	elseif (x <= 1.45e-52)
		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 <= -2e-9)
		tmp = x;
	elseif (x <= 1.45e-52)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e-9], x, If[LessEqual[x, 1.45e-52], N[(-1.0 / x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000012e-9 or 1.4500000000000001e-52 < x

   1. Initial program 97.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.4%

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

   if -2.00000000000000012e-9 < x < 1.4500000000000001e-52

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + \color{blue}{1}\right) \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\left(0 - \frac{1}{{x}^{2}}\right) + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \left(0 - \color{blue}{\left(\frac{1}{{x}^{2}} - 1\right)}\right) \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)}\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{\left(\frac{1}{{x}^{2}} - 1\right)}\right)\right) \]

      8. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(0 - \frac{1}{{x}^{2}}\right) + \color{blue}{1}\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{{x}^{2}}}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{{\color{blue}{x}}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

      15. unpow2 N/A

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

      16. *-lowering-*.f64 26.4%

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

   5. Simplified 26.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 43.0%

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

   8. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{-1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.4% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-295}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z -3.6e-295) (+ x (/ -1.0 x)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.6e-295) {
		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 <= (-3.6d-295)) 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 <= -3.6e-295) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.6e-295:
		tmp = x + (-1.0 / x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.6e-295)
		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 <= -3.6e-295)
		tmp = x + (-1.0 / x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.6e-295], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -3.6000000000000001e-295

   1. Initial program 97.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 91.0%

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

   5. Simplified 91.0%

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

   if -3.6000000000000001e-295 < z

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 80.8%

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

Alternative 9: 68.9% 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 97.2%

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

3. Taylor expanded in x around inf

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

5. Simplified 71.8%

   \[\leadsto \color{blue}{x} \]
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 2024160 
(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)))))