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

Percentage Accurate: 95.8% → 99.9%

Time: 6.9s

Alternatives: 6

Speedup: 8.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ -1.0 (fma (exp z) (/ -1.1283791670955126 y) x))))

C

double code(double x, double y, double z) {
	return x + (-1.0 / fma(exp(z), (-1.1283791670955126 / y), x));
}

Julia

function code(x, y, z)
	return Float64(x + Float64(-1.0 / fma(exp(z), Float64(-1.1283791670955126 / y), x)))
end

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. *-lft-identity 95.6%

      \[\leadsto x + \color{blue}{1 \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

   2. metadata-eval 95.6%

      \[\leadsto x + \color{blue}{\frac{-1}{-1}} \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

   3. times-frac 95.6%

      \[\leadsto x + \color{blue}{\frac{-1 \cdot y}{-1 \cdot \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

   4. neg-mul-1 95.6%

      \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{-\left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

   5. sub0-neg 95.7%

      \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

   6. associate-+l- 95.7%

      \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

   7. neg-sub0 95.7%

      \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(-1.1283791670955126 \cdot e^{z}\right)} + x \cdot y} \]

   8. +-commutative 95.7%

      \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y + \left(-1.1283791670955126 \cdot e^{z}\right)}} \]

   9. sub-neg 95.7%

      \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]

   10. associate-/l* 95.8%

       \[\leadsto x + \color{blue}{\frac{-1}{\frac{x \cdot y - 1.1283791670955126 \cdot e^{z}}{y}}} \]

   11. div-sub 95.7%

       \[\leadsto x + \frac{-1}{\color{blue}{\frac{x \cdot y}{y} - \frac{1.1283791670955126 \cdot e^{z}}{y}}} \]

   12. associate-*r/ 100.0%

       \[\leadsto x + \frac{-1}{\color{blue}{x \cdot \frac{y}{y}} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

   13. *-inverses 100.0%

       \[\leadsto x + \frac{-1}{x \cdot \color{blue}{1} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

   14. *-rgt-identity 100.0%

       \[\leadsto x + \frac{-1}{\color{blue}{x} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

   15. associate-*l/ 100.0%

       \[\leadsto x + \frac{-1}{x - \color{blue}{\frac{1.1283791670955126}{y} \cdot e^{z}}} \]

   16. cancel-sign-sub-inv 100.0%

       \[\leadsto x + \frac{-1}{\color{blue}{x + \left(-\frac{1.1283791670955126}{y}\right) \cdot e^{z}}} \]

   17. distribute-lft-neg-in 100.0%

       \[\leadsto x + \frac{-1}{x + \color{blue}{\left(-\frac{1.1283791670955126}{y} \cdot e^{z}\right)}} \]

   18. distribute-rgt-neg-in 100.0%

       \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126}{y} \cdot \left(-e^{z}\right)}} \]

   19. associate-*l/ 100.0%

       \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126 \cdot \left(-e^{z}\right)}{y}}} \]

   20. distribute-rgt-neg-in 100.0%

       \[\leadsto x + \frac{-1}{x + \frac{\color{blue}{-1.1283791670955126 \cdot e^{z}}}{y}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}} \]

4. Final simplification 100.0%

   \[\leadsto x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \]

Alternative 2: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (exp(z) <= 1.0) {
		tmp = x - (1.0 / (x + (-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 (exp(z) <= 0.0d0) then
        tmp = x - (1.0d0 / x)
    else if (exp(z) <= 1.0d0) then
        tmp = x - (1.0d0 / (x + ((-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 (Math.exp(z) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (Math.exp(z) <= 1.0) {
		tmp = x - (1.0 / (x + (-1.1283791670955126 / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.1%

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

      1. *-lft-identity 89.1%

         \[\leadsto x + \color{blue}{1 \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      2. metadata-eval 89.1%

         \[\leadsto x + \color{blue}{\frac{-1}{-1}} \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

      3. times-frac 89.1%

         \[\leadsto x + \color{blue}{\frac{-1 \cdot y}{-1 \cdot \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      4. neg-mul-1 89.1%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{-\left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      5. sub0-neg 89.4%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 89.4%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 89.6%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(-1.1283791670955126 \cdot e^{z}\right)} + x \cdot y} \]

      8. +-commutative 89.6%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y + \left(-1.1283791670955126 \cdot e^{z}\right)}} \]

      9. sub-neg 89.6%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]

      10. associate-/l* 89.7%

          \[\leadsto x + \color{blue}{\frac{-1}{\frac{x \cdot y - 1.1283791670955126 \cdot e^{z}}{y}}} \]

      11. div-sub 89.4%

          \[\leadsto x + \frac{-1}{\color{blue}{\frac{x \cdot y}{y} - \frac{1.1283791670955126 \cdot e^{z}}{y}}} \]

      12. associate-*r/ 100.0%

          \[\leadsto x + \frac{-1}{\color{blue}{x \cdot \frac{y}{y}} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      13. *-inverses 100.0%

          \[\leadsto x + \frac{-1}{x \cdot \color{blue}{1} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      14. *-rgt-identity 100.0%

          \[\leadsto x + \frac{-1}{\color{blue}{x} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      15. associate-*l/ 100.0%

          \[\leadsto x + \frac{-1}{x - \color{blue}{\frac{1.1283791670955126}{y} \cdot e^{z}}} \]

      16. cancel-sign-sub-inv 100.0%

          \[\leadsto x + \frac{-1}{\color{blue}{x + \left(-\frac{1.1283791670955126}{y}\right) \cdot e^{z}}} \]

      17. distribute-lft-neg-in 100.0%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\left(-\frac{1.1283791670955126}{y} \cdot e^{z}\right)}} \]

      18. distribute-rgt-neg-in 100.0%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126}{y} \cdot \left(-e^{z}\right)}} \]

      19. associate-*l/ 100.0%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126 \cdot \left(-e^{z}\right)}{y}}} \]

      20. distribute-rgt-neg-in 100.0%

          \[\leadsto x + \frac{-1}{x + \frac{\color{blue}{-1.1283791670955126 \cdot e^{z}}}{y}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}} \]

   4. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

         \[\leadsto x + \color{blue}{1 \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      2. metadata-eval 99.8%

         \[\leadsto x + \color{blue}{\frac{-1}{-1}} \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

      3. times-frac 99.8%

         \[\leadsto x + \color{blue}{\frac{-1 \cdot y}{-1 \cdot \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      4. neg-mul-1 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{-\left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      5. sub0-neg 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(-1.1283791670955126 \cdot e^{z}\right)} + x \cdot y} \]

      8. +-commutative 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y + \left(-1.1283791670955126 \cdot e^{z}\right)}} \]

      9. sub-neg 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]

      10. associate-/l* 99.9%

          \[\leadsto x + \color{blue}{\frac{-1}{\frac{x \cdot y - 1.1283791670955126 \cdot e^{z}}{y}}} \]

      11. div-sub 99.9%

          \[\leadsto x + \frac{-1}{\color{blue}{\frac{x \cdot y}{y} - \frac{1.1283791670955126 \cdot e^{z}}{y}}} \]

      12. associate-*r/ 99.9%

          \[\leadsto x + \frac{-1}{\color{blue}{x \cdot \frac{y}{y}} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      13. *-inverses 99.9%

          \[\leadsto x + \frac{-1}{x \cdot \color{blue}{1} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      14. *-rgt-identity 99.9%

          \[\leadsto x + \frac{-1}{\color{blue}{x} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      15. associate-*l/ 99.9%

          \[\leadsto x + \frac{-1}{x - \color{blue}{\frac{1.1283791670955126}{y} \cdot e^{z}}} \]

      16. cancel-sign-sub-inv 99.9%

          \[\leadsto x + \frac{-1}{\color{blue}{x + \left(-\frac{1.1283791670955126}{y}\right) \cdot e^{z}}} \]

      17. distribute-lft-neg-in 99.9%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\left(-\frac{1.1283791670955126}{y} \cdot e^{z}\right)}} \]

      18. distribute-rgt-neg-in 99.9%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126}{y} \cdot \left(-e^{z}\right)}} \]

      19. associate-*l/ 99.9%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126 \cdot \left(-e^{z}\right)}{y}}} \]

      20. distribute-rgt-neg-in 99.9%

          \[\leadsto x + \frac{-1}{x + \frac{\color{blue}{-1.1283791670955126 \cdot e^{z}}}{y}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}} \]

   4. Taylor expanded in z around 0 99.8%

      \[\leadsto \color{blue}{x - \frac{1}{x - 1.1283791670955126 \cdot \frac{1}{y}}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 99.8%

         \[\leadsto x - \frac{1}{\color{blue}{x + \left(-1.1283791670955126\right) \cdot \frac{1}{y}}} \]

      2. metadata-eval 99.8%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   if 1 < (exp.f64 z)

   1. Initial program 94.5%

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

   2. Taylor expanded in z around 0 58.9%

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

   3. Taylor expanded in y around 0 39.6%

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

   4. Taylor expanded in y around 0 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 85.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{1}{x}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-213}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-31}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (/ 1.0 x))))
   (if (<= z -3.3e-32)
     t_0
     (if (<= z 3.8e-213)
       (- x (* y -0.8862269254527579))
       (if (<= z 6.2e-141)
         t_0
         (if (<= z 3.2e-31) (+ x (/ y 1.1283791670955126)) x))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (1.0 / x);
	double tmp;
	if (z <= -3.3e-32) {
		tmp = t_0;
	} else if (z <= 3.8e-213) {
		tmp = x - (y * -0.8862269254527579);
	} else if (z <= 6.2e-141) {
		tmp = t_0;
	} else if (z <= 3.2e-31) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (1.0d0 / x)
    if (z <= (-3.3d-32)) then
        tmp = t_0
    else if (z <= 3.8d-213) then
        tmp = x - (y * (-0.8862269254527579d0))
    else if (z <= 6.2d-141) then
        tmp = t_0
    else if (z <= 3.2d-31) then
        tmp = x + (y / 1.1283791670955126d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (1.0 / x);
	double tmp;
	if (z <= -3.3e-32) {
		tmp = t_0;
	} else if (z <= 3.8e-213) {
		tmp = x - (y * -0.8862269254527579);
	} else if (z <= 6.2e-141) {
		tmp = t_0;
	} else if (z <= 3.2e-31) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (1.0 / x)
	tmp = 0
	if z <= -3.3e-32:
		tmp = t_0
	elif z <= 3.8e-213:
		tmp = x - (y * -0.8862269254527579)
	elif z <= 6.2e-141:
		tmp = t_0
	elif z <= 3.2e-31:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(1.0 / x))
	tmp = 0.0
	if (z <= -3.3e-32)
		tmp = t_0;
	elseif (z <= 3.8e-213)
		tmp = Float64(x - Float64(y * -0.8862269254527579));
	elseif (z <= 6.2e-141)
		tmp = t_0;
	elseif (z <= 3.2e-31)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (1.0 / x);
	tmp = 0.0;
	if (z <= -3.3e-32)
		tmp = t_0;
	elseif (z <= 3.8e-213)
		tmp = x - (y * -0.8862269254527579);
	elseif (z <= 6.2e-141)
		tmp = t_0;
	elseif (z <= 3.2e-31)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e-32], t$95$0, If[LessEqual[z, 3.8e-213], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.2e-141], t$95$0, If[LessEqual[z, 3.2e-31], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.30000000000000025e-32 or 3.8e-213 < z < 6.20000000000000055e-141

   1. Initial program 91.8%

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

      1. *-lft-identity 91.8%

         \[\leadsto x + \color{blue}{1 \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      2. metadata-eval 91.8%

         \[\leadsto x + \color{blue}{\frac{-1}{-1}} \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

      3. times-frac 91.8%

         \[\leadsto x + \color{blue}{\frac{-1 \cdot y}{-1 \cdot \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      4. neg-mul-1 91.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{-\left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      5. sub0-neg 92.0%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 92.0%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 92.1%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(-1.1283791670955126 \cdot e^{z}\right)} + x \cdot y} \]

      8. +-commutative 92.1%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y + \left(-1.1283791670955126 \cdot e^{z}\right)}} \]

      9. sub-neg 92.1%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]

      10. associate-/l* 92.2%

          \[\leadsto x + \color{blue}{\frac{-1}{\frac{x \cdot y - 1.1283791670955126 \cdot e^{z}}{y}}} \]

      11. div-sub 92.0%

          \[\leadsto x + \frac{-1}{\color{blue}{\frac{x \cdot y}{y} - \frac{1.1283791670955126 \cdot e^{z}}{y}}} \]

      12. associate-*r/ 100.0%

          \[\leadsto x + \frac{-1}{\color{blue}{x \cdot \frac{y}{y}} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      13. *-inverses 100.0%

          \[\leadsto x + \frac{-1}{x \cdot \color{blue}{1} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      14. *-rgt-identity 100.0%

          \[\leadsto x + \frac{-1}{\color{blue}{x} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      15. associate-*l/ 100.0%

          \[\leadsto x + \frac{-1}{x - \color{blue}{\frac{1.1283791670955126}{y} \cdot e^{z}}} \]

      16. cancel-sign-sub-inv 100.0%

          \[\leadsto x + \frac{-1}{\color{blue}{x + \left(-\frac{1.1283791670955126}{y}\right) \cdot e^{z}}} \]

      17. distribute-lft-neg-in 100.0%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\left(-\frac{1.1283791670955126}{y} \cdot e^{z}\right)}} \]

      18. distribute-rgt-neg-in 100.0%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126}{y} \cdot \left(-e^{z}\right)}} \]

      19. associate-*l/ 100.0%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126 \cdot \left(-e^{z}\right)}{y}}} \]

      20. distribute-rgt-neg-in 100.0%

          \[\leadsto x + \frac{-1}{x + \frac{\color{blue}{-1.1283791670955126 \cdot e^{z}}}{y}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}} \]

   4. Taylor expanded in x around inf 97.8%

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

   if -3.30000000000000025e-32 < z < 3.8e-213

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

         \[\leadsto x + \color{blue}{1 \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      2. metadata-eval 99.8%

         \[\leadsto x + \color{blue}{\frac{-1}{-1}} \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

      3. times-frac 99.8%

         \[\leadsto x + \color{blue}{\frac{-1 \cdot y}{-1 \cdot \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      4. neg-mul-1 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{-\left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      5. sub0-neg 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(-1.1283791670955126 \cdot e^{z}\right)} + x \cdot y} \]

      8. +-commutative 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y + \left(-1.1283791670955126 \cdot e^{z}\right)}} \]

      9. sub-neg 99.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]

      10. associate-/l* 99.9%

          \[\leadsto x + \color{blue}{\frac{-1}{\frac{x \cdot y - 1.1283791670955126 \cdot e^{z}}{y}}} \]

      11. div-sub 99.9%

          \[\leadsto x + \frac{-1}{\color{blue}{\frac{x \cdot y}{y} - \frac{1.1283791670955126 \cdot e^{z}}{y}}} \]

      12. associate-*r/ 99.9%

          \[\leadsto x + \frac{-1}{\color{blue}{x \cdot \frac{y}{y}} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      13. *-inverses 99.9%

          \[\leadsto x + \frac{-1}{x \cdot \color{blue}{1} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      14. *-rgt-identity 99.9%

          \[\leadsto x + \frac{-1}{\color{blue}{x} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      15. associate-*l/ 99.9%

          \[\leadsto x + \frac{-1}{x - \color{blue}{\frac{1.1283791670955126}{y} \cdot e^{z}}} \]

      16. cancel-sign-sub-inv 99.9%

          \[\leadsto x + \frac{-1}{\color{blue}{x + \left(-\frac{1.1283791670955126}{y}\right) \cdot e^{z}}} \]

      17. distribute-lft-neg-in 99.9%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\left(-\frac{1.1283791670955126}{y} \cdot e^{z}\right)}} \]

      18. distribute-rgt-neg-in 99.9%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126}{y} \cdot \left(-e^{z}\right)}} \]

      19. associate-*l/ 99.9%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126 \cdot \left(-e^{z}\right)}{y}}} \]

      20. distribute-rgt-neg-in 99.9%

          \[\leadsto x + \frac{-1}{x + \frac{\color{blue}{-1.1283791670955126 \cdot e^{z}}}{y}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}} \]

   4. Taylor expanded in z around 0 99.8%

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

   5. Taylor expanded in x around 0 80.3%

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

      1. *-commutative 80.3%

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

   7. Simplified 80.3%

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

   if 6.20000000000000055e-141 < z < 3.20000000000000018e-31

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 99.7%

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

   3. Taylor expanded in y around 0 78.2%

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

   if 3.20000000000000018e-31 < z

   1. Initial program 94.7%

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

   2. Taylor expanded in z around 0 60.0%

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

   3. Taylor expanded in y around 0 39.9%

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

   4. Taylor expanded in y around 0 100.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-32}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-213}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-141}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-31}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 98.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -88:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -88.0)
   (- x (/ 1.0 x))
   (if (<= z 3.7e-26) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -88.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 3.7e-26) {
		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 <= (-88.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 3.7d-26) 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 <= -88.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 3.7e-26) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -88.0:
		tmp = x - (1.0 / x)
	elif z <= 3.7e-26:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -88.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 3.7e-26)
		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 <= -88.0)
		tmp = x - (1.0 / x);
	elseif (z <= 3.7e-26)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -88.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.7e-26], 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 < -88

   1. Initial program 89.1%

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

      1. *-lft-identity 89.1%

         \[\leadsto x + \color{blue}{1 \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

      2. metadata-eval 89.1%

         \[\leadsto x + \color{blue}{\frac{-1}{-1}} \cdot \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

      3. times-frac 89.1%

         \[\leadsto x + \color{blue}{\frac{-1 \cdot y}{-1 \cdot \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      4. neg-mul-1 89.1%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{-\left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      5. sub0-neg 89.4%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 89.4%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 89.6%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(-1.1283791670955126 \cdot e^{z}\right)} + x \cdot y} \]

      8. +-commutative 89.6%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y + \left(-1.1283791670955126 \cdot e^{z}\right)}} \]

      9. sub-neg 89.6%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]

      10. associate-/l* 89.7%

          \[\leadsto x + \color{blue}{\frac{-1}{\frac{x \cdot y - 1.1283791670955126 \cdot e^{z}}{y}}} \]

      11. div-sub 89.4%

          \[\leadsto x + \frac{-1}{\color{blue}{\frac{x \cdot y}{y} - \frac{1.1283791670955126 \cdot e^{z}}{y}}} \]

      12. associate-*r/ 100.0%

          \[\leadsto x + \frac{-1}{\color{blue}{x \cdot \frac{y}{y}} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      13. *-inverses 100.0%

          \[\leadsto x + \frac{-1}{x \cdot \color{blue}{1} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      14. *-rgt-identity 100.0%

          \[\leadsto x + \frac{-1}{\color{blue}{x} - \frac{1.1283791670955126 \cdot e^{z}}{y}} \]

      15. associate-*l/ 100.0%

          \[\leadsto x + \frac{-1}{x - \color{blue}{\frac{1.1283791670955126}{y} \cdot e^{z}}} \]

      16. cancel-sign-sub-inv 100.0%

          \[\leadsto x + \frac{-1}{\color{blue}{x + \left(-\frac{1.1283791670955126}{y}\right) \cdot e^{z}}} \]

      17. distribute-lft-neg-in 100.0%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\left(-\frac{1.1283791670955126}{y} \cdot e^{z}\right)}} \]

      18. distribute-rgt-neg-in 100.0%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126}{y} \cdot \left(-e^{z}\right)}} \]

      19. associate-*l/ 100.0%

          \[\leadsto x + \frac{-1}{x + \color{blue}{\frac{1.1283791670955126 \cdot \left(-e^{z}\right)}{y}}} \]

      20. distribute-rgt-neg-in 100.0%

          \[\leadsto x + \frac{-1}{x + \frac{\color{blue}{-1.1283791670955126 \cdot e^{z}}}{y}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}} \]

   4. Taylor expanded in x around inf 100.0%

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

   if -88 < z < 3.6999999999999999e-26

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.8%

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

   if 3.6999999999999999e-26 < z

   1. Initial program 94.7%

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

   2. Taylor expanded in z around 0 60.0%

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

   3. Taylor expanded in y around 0 39.9%

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

   4. Taylor expanded in y around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -88:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 73.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.5e-31) x (if (<= z 4.8e-27) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.5e-31) {
		tmp = x;
	} else if (z <= 4.8e-27) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.5d-31)) then
        tmp = x
    else if (z <= 4.8d-27) then
        tmp = x + (y / 1.1283791670955126d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.5e-31) {
		tmp = x;
	} else if (z <= 4.8e-27) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.5e-31:
		tmp = x
	elif z <= 4.8e-27:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.5e-31)
		tmp = x;
	elseif (z <= 4.8e-27)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.5e-31)
		tmp = x;
	elseif (z <= 4.8e-27)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.5e-31], x, If[LessEqual[z, 4.8e-27], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5e-31 or 4.80000000000000004e-27 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in z around 0 62.8%

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

   3. Taylor expanded in y around 0 35.5%

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

   4. Taylor expanded in y around 0 74.7%

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

   if -2.5e-31 < z < 4.80000000000000004e-27

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.8%

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

   3. Taylor expanded in y around 0 73.8%

      \[\leadsto \frac{y}{\color{blue}{1.1283791670955126}} + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 68.7% 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 95.6%

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

2. Taylor expanded in z around 0 78.2%

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

3. Taylor expanded in y around 0 51.5%

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

4. Taylor expanded in y around 0 67.4%

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

5. Final simplification 67.4%

   \[\leadsto x \]

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / (((1.1283791670955126d0 / y) * exp(z)) - x))
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * Math.exp(z)) - x));
}

Python

def code(x, y, z):
	return x + (1.0 / (((1.1283791670955126 / y) * math.exp(z)) - x))

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / Float64(Float64(Float64(1.1283791670955126 / y) * exp(z)) - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
end

Wolfram

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

Reproduce

herbie shell --seed 2023192 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

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

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