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

Percentage Accurate: 96.0% → 99.4%

Time: 10.0s

Alternatives: 11

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: 96.0% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1.1283791670955126 - x \cdot y\\ \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.00000000002:\\ \;\;\;\;\left(x + \frac{y}{t\_0}\right) + \frac{-1.1283791670955126 \cdot \left(z \cdot y\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.1283791670955126 (* x y))))
   (if (<= (exp z) 0.0)
     (+ x (/ -1.0 x))
     (if (<= (exp z) 1.00000000002)
       (+ (+ x (/ y t_0)) (/ (* -1.1283791670955126 (* z y)) (* t_0 t_0)))
       x))))

C

double code(double x, double y, double z) {
	double t_0 = 1.1283791670955126 - (x * y);
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.00000000002) {
		tmp = (x + (y / t_0)) + ((-1.1283791670955126 * (z * y)) / (t_0 * t_0));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.1283791670955126d0 - (x * y)
    if (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.00000000002d0) then
        tmp = (x + (y / t_0)) + (((-1.1283791670955126d0) * (z * y)) / (t_0 * t_0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.1283791670955126 - (x * y);
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 1.00000000002) {
		tmp = (x + (y / t_0)) + ((-1.1283791670955126 * (z * y)) / (t_0 * t_0));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.1283791670955126 - (x * y)
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.00000000002:
		tmp = (x + (y / t_0)) + ((-1.1283791670955126 * (z * y)) / (t_0 * t_0))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.1283791670955126 - Float64(x * y))
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.00000000002)
		tmp = Float64(Float64(x + Float64(y / t_0)) + Float64(Float64(-1.1283791670955126 * Float64(z * y)) / Float64(t_0 * t_0)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.1283791670955126 - (x * y);
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 1.00000000002)
		tmp = (x + (y / t_0)) + ((-1.1283791670955126 * (z * y)) / (t_0 * t_0));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.00000000002], N[(N[(x + N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.1283791670955126 * N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 88.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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) < 1.00000000002

   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 99.8%

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

   5. Simplified 99.8%

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

   if 1.00000000002 < (exp.f64 z)

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.00000000002:\\ \;\;\;\;\left(x + \frac{y}{1.1283791670955126 - x \cdot y}\right) + \frac{-1.1283791670955126 \cdot \left(z \cdot y\right)}{\left(1.1283791670955126 - x \cdot y\right) \cdot \left(1.1283791670955126 - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+ x (/ 1.0 (/ (- (* (exp z) 1.1283791670955126) (* x y)) 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 + (1.0 / (((exp(z) * 1.1283791670955126) - (x * y)) / 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 + (1.0d0 / (((exp(z) * 1.1283791670955126d0) - (x * y)) / 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 + (1.0 / (((Math.exp(z) * 1.1283791670955126) - (x * y)) / 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 + (1.0 / (((math.exp(z) * 1.1283791670955126) - (x * y)) / 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(1.0 / Float64(Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y)) / 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 + (1.0 / (((exp(z) * 1.1283791670955126) - (x * y)) / 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[(1.0 / N[(N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 88.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}{y}}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y}{y}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{5641895835477563}{5000000000000000} \cdot e^{z} - x \cdot y\right), \color{blue}{y}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{5641895835477563}{5000000000000000} \cdot e^{z}\right), \left(x \cdot y\right)\right), y\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(e^{z}\right)\right), \left(x \cdot y\right)\right), y\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{exp.f64}\left(z\right)\right), \left(x \cdot y\right)\right), y\right)\right)\right) \]

      7. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{exp.f64}\left(z\right)\right), \mathsf{*.f64}\left(x, y\right)\right), y\right)\right)\right) \]

   4. Applied egg-rr 98.2%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126 \cdot e^{z} - x \cdot y}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

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

Alternative 3: 98.5% 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 88.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 98.2%

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

4. Final simplification 98.7%

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

Alternative 4: 70.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-148}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-293}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-175}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-61}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.9e-148)
   x
   (if (<= x -9.5e-293)
     (/ -1.0 x)
     (if (<= x 1.15e-175)
       (/ y 1.1283791670955126)
       (if (<= x 1.85e-61) (/ -1.0 x) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e-148) {
		tmp = x;
	} else if (x <= -9.5e-293) {
		tmp = -1.0 / x;
	} else if (x <= 1.15e-175) {
		tmp = y / 1.1283791670955126;
	} else if (x <= 1.85e-61) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.9d-148)) then
        tmp = x
    else if (x <= (-9.5d-293)) then
        tmp = (-1.0d0) / x
    else if (x <= 1.15d-175) then
        tmp = y / 1.1283791670955126d0
    else if (x <= 1.85d-61) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e-148) {
		tmp = x;
	} else if (x <= -9.5e-293) {
		tmp = -1.0 / x;
	} else if (x <= 1.15e-175) {
		tmp = y / 1.1283791670955126;
	} else if (x <= 1.85e-61) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.9e-148:
		tmp = x
	elif x <= -9.5e-293:
		tmp = -1.0 / x
	elif x <= 1.15e-175:
		tmp = y / 1.1283791670955126
	elif x <= 1.85e-61:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.9e-148)
		tmp = x;
	elseif (x <= -9.5e-293)
		tmp = Float64(-1.0 / x);
	elseif (x <= 1.15e-175)
		tmp = Float64(y / 1.1283791670955126);
	elseif (x <= 1.85e-61)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.9e-148)
		tmp = x;
	elseif (x <= -9.5e-293)
		tmp = -1.0 / x;
	elseif (x <= 1.15e-175)
		tmp = y / 1.1283791670955126;
	elseif (x <= 1.85e-61)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.9e-148], x, If[LessEqual[x, -9.5e-293], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 1.15e-175], N[(y / 1.1283791670955126), $MachinePrecision], If[LessEqual[x, 1.85e-61], N[(-1.0 / x), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.90000000000000007e-148 or 1.85e-61 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 87.0%

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

   if -1.90000000000000007e-148 < x < -9.50000000000000049e-293 or 1.15e-175 < x < 1.85e-61

   1. Initial program 90.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 31.9%

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

   5. Simplified 31.9%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 61.6%

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

   8. Simplified 61.6%

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

   if -9.50000000000000049e-293 < x < 1.15e-175

   1. Initial program 89.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. *-lowering-*.f64 51.6%

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

   8. Simplified 51.6%

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

   9. Taylor expanded in z around 0

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

   11. Simplified 51.2%

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

Alternative 5: 70.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-148}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-290}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-176}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.05e-148)
   x
   (if (<= x -1.2e-290)
     (/ -1.0 x)
     (if (<= x 6.6e-176)
       (* y 0.8862269254527579)
       (if (<= x 4.5e-61) (/ -1.0 x) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e-148) {
		tmp = x;
	} else if (x <= -1.2e-290) {
		tmp = -1.0 / x;
	} else if (x <= 6.6e-176) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 4.5e-61) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.05d-148)) then
        tmp = x
    else if (x <= (-1.2d-290)) then
        tmp = (-1.0d0) / x
    else if (x <= 6.6d-176) then
        tmp = y * 0.8862269254527579d0
    else if (x <= 4.5d-61) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.05e-148) {
		tmp = x;
	} else if (x <= -1.2e-290) {
		tmp = -1.0 / x;
	} else if (x <= 6.6e-176) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 4.5e-61) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.05e-148:
		tmp = x
	elif x <= -1.2e-290:
		tmp = -1.0 / x
	elif x <= 6.6e-176:
		tmp = y * 0.8862269254527579
	elif x <= 4.5e-61:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.05e-148)
		tmp = x;
	elseif (x <= -1.2e-290)
		tmp = Float64(-1.0 / x);
	elseif (x <= 6.6e-176)
		tmp = Float64(y * 0.8862269254527579);
	elseif (x <= 4.5e-61)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.05e-148)
		tmp = x;
	elseif (x <= -1.2e-290)
		tmp = -1.0 / x;
	elseif (x <= 6.6e-176)
		tmp = y * 0.8862269254527579;
	elseif (x <= 4.5e-61)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.05e-148], x, If[LessEqual[x, -1.2e-290], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 6.6e-176], N[(y * 0.8862269254527579), $MachinePrecision], If[LessEqual[x, 4.5e-61], N[(-1.0 / x), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e-148 or 4.5e-61 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 87.0%

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

   if -1.05e-148 < x < -1.2e-290 or 6.60000000000000025e-176 < x < 4.5e-61

   1. Initial program 90.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 31.9%

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

   5. Simplified 31.9%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 61.6%

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

   8. Simplified 61.6%

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

   if -1.2e-290 < x < 6.60000000000000025e-176

   1. Initial program 89.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 52.6%

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

   5. Simplified 52.6%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\frac{5000000000000000}{5641895835477563}} \]

      2. *-lowering-*.f64 51.1%

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

   8. Simplified 51.1%

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

Alternative 6: 99.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -300:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -300.0)
   (+ x (/ -1.0 x))
   (if (<= z 3.8e-11)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -300.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.8e-11) {
		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 <= (-300.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 3.8d-11) 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 <= -300.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.8e-11) {
		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 <= -300.0:
		tmp = x + (-1.0 / x)
	elif z <= 3.8e-11:
		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 <= -300.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 3.8e-11)
		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 <= -300.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 3.8e-11)
		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, -300.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e-11], 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 < -300

   1. Initial program 88.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 -300 < z < 3.7999999999999998e-11

   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 99.8%

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

   5. Simplified 99.8%

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

   if 3.7999999999999998e-11 < z

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 7: 72.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+261}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-101}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.8e+261)
   x
   (if (<= z -2.2e+30)
     (/ -1.0 x)
     (if (<= z 4.1e-101) (+ x (* y 0.8862269254527579)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e+261) {
		tmp = x;
	} else if (z <= -2.2e+30) {
		tmp = -1.0 / x;
	} else if (z <= 4.1e-101) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.8d+261)) then
        tmp = x
    else if (z <= (-2.2d+30)) then
        tmp = (-1.0d0) / x
    else if (z <= 4.1d-101) then
        tmp = x + (y * 0.8862269254527579d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e+261) {
		tmp = x;
	} else if (z <= -2.2e+30) {
		tmp = -1.0 / x;
	} else if (z <= 4.1e-101) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.8e+261:
		tmp = x
	elif z <= -2.2e+30:
		tmp = -1.0 / x
	elif z <= 4.1e-101:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.8e+261)
		tmp = x;
	elseif (z <= -2.2e+30)
		tmp = Float64(-1.0 / x);
	elseif (z <= 4.1e-101)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.8e+261)
		tmp = x;
	elseif (z <= -2.2e+30)
		tmp = -1.0 / x;
	elseif (z <= 4.1e-101)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.8e+261], x, If[LessEqual[z, -2.2e+30], N[(-1.0 / x), $MachinePrecision], If[LessEqual[z, 4.1e-101], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7999999999999998e261 or 4.10000000000000026e-101 < z

   1. Initial program 96.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 92.1%

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

   if -2.7999999999999998e261 < z < -2.2e30

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   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 71.4%

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

   5. Simplified 71.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 63.3%

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

   8. Simplified 63.3%

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

   if -2.2e30 < z < 4.10000000000000026e-101

   1. Initial program 98.9%

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

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 76.7%

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

   8. Simplified 76.7%

      \[\leadsto \color{blue}{x + y \cdot 0.8862269254527579} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -63:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -63.0)
   (+ x (/ -1.0 x))
   (if (<= z 3.8e-11) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -63.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.8e-11) {
		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 <= (-63.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 3.8d-11) 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 <= -63.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.8e-11) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -63.0:
		tmp = x + (-1.0 / x)
	elif z <= 3.8e-11:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -63.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 3.8e-11)
		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 <= -63.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 3.8e-11)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -63.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e-11], 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 < -63

   1. Initial program 88.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 -63 < z < 3.7999999999999998e-11

   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 99.8%

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

   5. Simplified 99.8%

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

   if 3.7999999999999998e-11 < z

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -63:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.65 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{-100}:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.65e-9)
   (+ x (/ -1.0 x))
   (if (<= z 3.15e-100) (+ x (/ 1.0 (/ 1.1283791670955126 y))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.65e-9) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.15e-100) {
		tmp = x + (1.0 / (1.1283791670955126 / 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 <= (-3.65d-9)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 3.15d-100) then
        tmp = x + (1.0d0 / (1.1283791670955126d0 / 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 <= -3.65e-9) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.15e-100) {
		tmp = x + (1.0 / (1.1283791670955126 / y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.65e-9:
		tmp = x + (-1.0 / x)
	elif z <= 3.15e-100:
		tmp = x + (1.0 / (1.1283791670955126 / y))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.65e-9)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 3.15e-100)
		tmp = Float64(x + Float64(1.0 / Float64(1.1283791670955126 / y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.65e-9)
		tmp = x + (-1.0 / x);
	elseif (z <= 3.15e-100)
		tmp = x + (1.0 / (1.1283791670955126 / y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.65e-9], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.15e-100], N[(x + N[(1.0 / N[(1.1283791670955126 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -3.65000000000000001e-9

   1. Initial program 89.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 -3.65000000000000001e-9 < z < 3.1499999999999998e-100

   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 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 79.6%

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

   8. Simplified 79.6%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \frac{1}{\color{blue}{\frac{5641895835477563}{5000000000000000}}}\right)\right) \]

      2. div-inv N/A

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

      3. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{\frac{5641895835477563}{5000000000000000}}{y}}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{5641895835477563}{5000000000000000}}{y}\right)}\right)\right) \]

      5. /-lowering-/.f64 79.7%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{5641895835477563}{5000000000000000}, \color{blue}{y}\right)\right)\right) \]

   10. Applied egg-rr 79.7%

       \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126}{y}}} \]

   if 3.1499999999999998e-100 < z

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 94.9%

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

Alternative 10: 85.2% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-99}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.8e-9)
   (+ x (/ -1.0 x))
   (if (<= z 3.5e-99) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.8e-9) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.5e-99) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6.8d-9)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 3.5d-99) then
        tmp = x + (y * 0.8862269254527579d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.8e-9) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.5e-99) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.8e-9:
		tmp = x + (-1.0 / x)
	elif z <= 3.5e-99:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.8e-9)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 3.5e-99)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.8e-9)
		tmp = x + (-1.0 / x);
	elseif (z <= 3.5e-99)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.8e-9], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e-99], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -6.7999999999999997e-9

   1. Initial program 89.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 -6.7999999999999997e-9 < z < 3.4999999999999999e-99

   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 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 79.6%

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

   8. Simplified 79.6%

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

   if 3.4999999999999999e-99 < z

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 94.9%

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

Alternative 11: 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 95.5%

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

3. Taylor expanded in x around inf

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

5. Simplified 68.4%

   \[\leadsto \color{blue}{x} \]
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 2024161 
(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)))))