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

Percentage Accurate: 95.5% → 99.9%

Time: 8.2s

Alternatives: 10

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: 95.5% 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 94.9%

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

   1. remove-double-neg 94.9%

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

   2. neg-mul-1 94.9%

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

   3. associate-/l* 94.9%

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

   4. neg-mul-1 94.9%

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

   5. associate-/r* 94.9%

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

   6. div-sub 95.0%

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

   7. metadata-eval 95.0%

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

   8. associate-/l* 95.0%

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

   9. *-commutative 95.0%

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

   10. neg-mul-1 95.0%

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

   11. distribute-lft-neg-out 95.0%

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

   12. /-rgt-identity 95.0%

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

   13. div-sub 95.0%

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

   14. associate-/r* 95.0%

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

   15. neg-mul-1 95.0%

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

   16. *-rgt-identity 95.0%

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

   17. times-frac 95.0%

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

   18. /-rgt-identity 95.0%

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

   19. *-commutative 95.0%

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

   20. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.4%

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

      1. remove-double-neg 89.4%

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

      2. neg-mul-1 89.4%

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

      3. associate-/l* 89.4%

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

      4. neg-mul-1 89.4%

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

      5. associate-/r* 89.4%

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

      6. div-sub 89.7%

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

      7. metadata-eval 89.7%

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

      8. associate-/l* 89.7%

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

      9. *-commutative 89.7%

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

      10. neg-mul-1 89.7%

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

      11. distribute-lft-neg-out 89.7%

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

      12. /-rgt-identity 89.7%

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

      13. div-sub 89.5%

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

      14. associate-/r* 89.5%

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

      15. neg-mul-1 89.5%

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

      16. *-rgt-identity 89.5%

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

      17. times-frac 89.5%

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

      18. /-rgt-identity 89.5%

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

      19. *-commutative 89.5%

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

      20. associate-*r/ 100.0%

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

   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 y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.8%

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

   if 1 < (exp.f64 z)

   1. Initial program 92.6%

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

      1. remove-double-neg 92.6%

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

      2. neg-mul-1 92.6%

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

      3. associate-/l* 92.6%

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

      4. neg-mul-1 92.6%

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

      5. associate-/r* 92.6%

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

      6. div-sub 92.6%

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

      7. metadata-eval 92.6%

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

      8. associate-/l* 92.6%

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

      9. *-commutative 92.6%

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

      10. neg-mul-1 92.6%

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

      11. distribute-lft-neg-out 92.6%

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

      12. /-rgt-identity 92.6%

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

      13. div-sub 92.6%

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

      14. associate-/r* 92.6%

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

      15. neg-mul-1 92.6%

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

      16. *-rgt-identity 92.6%

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

      17. times-frac 92.6%

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

      18. /-rgt-identity 92.6%

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

      19. *-commutative 92.6%

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

      20. associate-*r/ 100.0%

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

   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 z around 0 60.5%

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

      1. associate-*r/ 60.5%

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

      2. metadata-eval 60.5%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in x around 0 47.3%

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

      1. *-commutative 47.3%

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

   9. Simplified 47.3%

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

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.9%

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

Alternative 3: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.4%

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

      1. remove-double-neg 89.4%

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

      2. neg-mul-1 89.4%

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

      3. associate-/l* 89.4%

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

      4. neg-mul-1 89.4%

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

      5. associate-/r* 89.4%

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

      6. div-sub 89.7%

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

      7. metadata-eval 89.7%

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

      8. associate-/l* 89.7%

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

      9. *-commutative 89.7%

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

      10. neg-mul-1 89.7%

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

      11. distribute-lft-neg-out 89.7%

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

      12. /-rgt-identity 89.7%

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

      13. div-sub 89.5%

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

      14. associate-/r* 89.5%

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

      15. neg-mul-1 89.5%

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

      16. *-rgt-identity 89.5%

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

      17. times-frac 89.5%

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

      18. /-rgt-identity 89.5%

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

      19. *-commutative 89.5%

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

      20. associate-*r/ 100.0%

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

   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 y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.1%

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

   if 1 < (exp.f64 z)

   1. Initial program 92.6%

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

      1. remove-double-neg 92.6%

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

      2. neg-mul-1 92.6%

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

      3. associate-/l* 92.6%

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

      4. neg-mul-1 92.6%

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

      5. associate-/r* 92.6%

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

      6. div-sub 92.6%

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

      7. metadata-eval 92.6%

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

      8. associate-/l* 92.6%

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

      9. *-commutative 92.6%

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

      10. neg-mul-1 92.6%

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

      11. distribute-lft-neg-out 92.6%

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

      12. /-rgt-identity 92.6%

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

      13. div-sub 92.6%

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

      14. associate-/r* 92.6%

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

      15. neg-mul-1 92.6%

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

      16. *-rgt-identity 92.6%

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

      17. times-frac 92.6%

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

      18. /-rgt-identity 92.6%

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

      19. *-commutative 92.6%

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

      20. associate-*r/ 100.0%

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

   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 z around 0 60.5%

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

      1. associate-*r/ 60.5%

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

      2. metadata-eval 60.5%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in x around 0 47.3%

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

      1. *-commutative 47.3%

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

   9. Simplified 47.3%

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

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

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

Alternative 4: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. remove-double-neg 94.9%

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

   2. neg-mul-1 94.9%

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

   3. associate-/l* 94.9%

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

   4. neg-mul-1 94.9%

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

   5. associate-/r* 94.9%

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

   6. div-sub 95.0%

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

   7. metadata-eval 95.0%

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

   8. associate-/l* 95.0%

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

   9. *-commutative 95.0%

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

   10. neg-mul-1 95.0%

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

   11. distribute-lft-neg-out 95.0%

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

   12. /-rgt-identity 95.0%

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

   13. div-sub 95.0%

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

   14. associate-/r* 95.0%

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

   15. neg-mul-1 95.0%

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

   16. *-rgt-identity 95.0%

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

   17. times-frac 95.0%

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

   18. /-rgt-identity 95.0%

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

   19. *-commutative 95.0%

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

   20. associate-*r/ 99.9%

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

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 inf 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-def 99.9%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

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

Alternative 5: 85.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + y \cdot 0.8862269254527579\\ \mathbf{if}\;z \leq -6.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))) (t_1 (+ x (* y 0.8862269254527579))))
   (if (<= z -6.5)
     t_0
     (if (<= z 6.5e-237)
       t_1
       (if (<= z 5.5e-113) t_0 (if (<= z 5.6e-27) t_1 x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y * 0.8862269254527579);
	double tmp;
	if (z <= -6.5) {
		tmp = t_0;
	} else if (z <= 6.5e-237) {
		tmp = t_1;
	} else if (z <= 5.5e-113) {
		tmp = t_0;
	} else if (z <= 5.6e-27) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x + (y * 0.8862269254527579d0)
    if (z <= (-6.5d0)) then
        tmp = t_0
    else if (z <= 6.5d-237) then
        tmp = t_1
    else if (z <= 5.5d-113) then
        tmp = t_0
    else if (z <= 5.6d-27) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y * 0.8862269254527579);
	double tmp;
	if (z <= -6.5) {
		tmp = t_0;
	} else if (z <= 6.5e-237) {
		tmp = t_1;
	} else if (z <= 5.5e-113) {
		tmp = t_0;
	} else if (z <= 5.6e-27) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x + (y * 0.8862269254527579)
	tmp = 0
	if z <= -6.5:
		tmp = t_0
	elif z <= 6.5e-237:
		tmp = t_1
	elif z <= 5.5e-113:
		tmp = t_0
	elif z <= 5.6e-27:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y * 0.8862269254527579))
	tmp = 0.0
	if (z <= -6.5)
		tmp = t_0;
	elseif (z <= 6.5e-237)
		tmp = t_1;
	elseif (z <= 5.5e-113)
		tmp = t_0;
	elseif (z <= 5.6e-27)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x + (y * 0.8862269254527579);
	tmp = 0.0;
	if (z <= -6.5)
		tmp = t_0;
	elseif (z <= 6.5e-237)
		tmp = t_1;
	elseif (z <= 5.5e-113)
		tmp = t_0;
	elseif (z <= 5.6e-27)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5], t$95$0, If[LessEqual[z, 6.5e-237], t$95$1, If[LessEqual[z, 5.5e-113], t$95$0, If[LessEqual[z, 5.6e-27], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5 or 6.5000000000000001e-237 < z < 5.50000000000000053e-113

   1. Initial program 92.2%

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

      1. remove-double-neg 92.2%

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

      2. neg-mul-1 92.2%

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

      3. associate-/l* 92.2%

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

      4. neg-mul-1 92.2%

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

      5. associate-/r* 92.2%

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

      6. div-sub 92.4%

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

      7. metadata-eval 92.4%

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

      8. associate-/l* 92.4%

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

      9. *-commutative 92.4%

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

      10. neg-mul-1 92.4%

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

      11. distribute-lft-neg-out 92.4%

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

      12. /-rgt-identity 92.4%

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

      13. div-sub 92.3%

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

      14. associate-/r* 92.3%

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

      15. neg-mul-1 92.3%

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

      16. *-rgt-identity 92.3%

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

      17. times-frac 92.3%

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

      18. /-rgt-identity 92.3%

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

      19. *-commutative 92.3%

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

      20. associate-*r/ 100.0%

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

   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 y around inf 95.5%

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

   if -6.5 < z < 6.5000000000000001e-237 or 5.50000000000000053e-113 < z < 5.5999999999999999e-27

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      20. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 98.6%

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

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around 0 76.4%

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

      1. *-commutative 76.4%

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

   9. Simplified 76.4%

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

   if 5.5999999999999999e-27 < z

   1. Initial program 92.8%

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

      1. remove-double-neg 92.8%

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

      2. neg-mul-1 92.8%

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

      3. associate-/l* 92.8%

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

      4. neg-mul-1 92.8%

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

      5. associate-/r* 92.8%

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

      6. div-sub 92.8%

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

      7. metadata-eval 92.8%

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

      8. associate-/l* 92.8%

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

      9. *-commutative 92.8%

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

      10. neg-mul-1 92.8%

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

      11. distribute-lft-neg-out 92.8%

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

      12. /-rgt-identity 92.8%

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

      13. div-sub 92.8%

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

      14. associate-/r* 92.8%

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

      15. neg-mul-1 92.8%

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

      16. *-rgt-identity 92.8%

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

      17. times-frac 92.8%

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

      18. /-rgt-identity 92.8%

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

      19. *-commutative 92.8%

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

      20. associate-*r/ 100.0%

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

   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 z around 0 61.4%

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

      1. associate-*r/ 61.4%

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

      2. metadata-eval 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in x around 0 48.6%

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

      1. *-commutative 48.6%

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

   9. Simplified 48.6%

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

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-237}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-113}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-27}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 85.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -0.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-237}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-27}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z -0.45)
     t_0
     (if (<= z 1.5e-237)
       (+ x (/ y 1.1283791670955126))
       (if (<= z 7.2e-113)
         t_0
         (if (<= z 5.5e-27) (+ x (* y 0.8862269254527579)) x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -0.45) {
		tmp = t_0;
	} else if (z <= 1.5e-237) {
		tmp = x + (y / 1.1283791670955126);
	} else if (z <= 7.2e-113) {
		tmp = t_0;
	} else if (z <= 5.5e-27) {
		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) :: t_0
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    if (z <= (-0.45d0)) then
        tmp = t_0
    else if (z <= 1.5d-237) then
        tmp = x + (y / 1.1283791670955126d0)
    else if (z <= 7.2d-113) then
        tmp = t_0
    else if (z <= 5.5d-27) 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 t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -0.45) {
		tmp = t_0;
	} else if (z <= 1.5e-237) {
		tmp = x + (y / 1.1283791670955126);
	} else if (z <= 7.2e-113) {
		tmp = t_0;
	} else if (z <= 5.5e-27) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if z <= -0.45:
		tmp = t_0
	elif z <= 1.5e-237:
		tmp = x + (y / 1.1283791670955126)
	elif z <= 7.2e-113:
		tmp = t_0
	elif z <= 5.5e-27:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if (z <= -0.45)
		tmp = t_0;
	elseif (z <= 1.5e-237)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	elseif (z <= 7.2e-113)
		tmp = t_0;
	elseif (z <= 5.5e-27)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	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 <= -0.45)
		tmp = t_0;
	elseif (z <= 1.5e-237)
		tmp = x + (y / 1.1283791670955126);
	elseif (z <= 7.2e-113)
		tmp = t_0;
	elseif (z <= 5.5e-27)
		tmp = x + (y * 0.8862269254527579);
	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, -0.45], t$95$0, If[LessEqual[z, 1.5e-237], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.2e-113], t$95$0, If[LessEqual[z, 5.5e-27], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -0.450000000000000011 or 1.50000000000000012e-237 < z < 7.1999999999999995e-113

   1. Initial program 92.2%

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

      1. remove-double-neg 92.2%

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

      2. neg-mul-1 92.2%

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

      3. associate-/l* 92.2%

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

      4. neg-mul-1 92.2%

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

      5. associate-/r* 92.2%

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

      6. div-sub 92.4%

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

      7. metadata-eval 92.4%

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

      8. associate-/l* 92.4%

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

      9. *-commutative 92.4%

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

      10. neg-mul-1 92.4%

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

      11. distribute-lft-neg-out 92.4%

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

      12. /-rgt-identity 92.4%

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

      13. div-sub 92.3%

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

      14. associate-/r* 92.3%

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

      15. neg-mul-1 92.3%

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

      16. *-rgt-identity 92.3%

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

      17. times-frac 92.3%

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

      18. /-rgt-identity 92.3%

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

      19. *-commutative 92.3%

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

      20. associate-*r/ 100.0%

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

   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 y around inf 95.5%

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

   if -0.450000000000000011 < z < 1.50000000000000012e-237

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      20. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 98.3%

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

      1. associate-*r/ 98.4%

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

      2. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in x around 0 75.5%

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

      1. *-commutative 75.5%

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

   9. Simplified 75.5%

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

       1. metadata-eval 75.5%

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

       2. div-inv 75.5%

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

   11. Applied egg-rr 75.5%

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

   if 7.1999999999999995e-113 < z < 5.5000000000000002e-27

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      20. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around 0 79.5%

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

      1. *-commutative 79.5%

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

   9. Simplified 79.5%

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

   if 5.5000000000000002e-27 < z

   1. Initial program 92.8%

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

      1. remove-double-neg 92.8%

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

      2. neg-mul-1 92.8%

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

      3. associate-/l* 92.8%

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

      4. neg-mul-1 92.8%

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

      5. associate-/r* 92.8%

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

      6. div-sub 92.8%

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

      7. metadata-eval 92.8%

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

      8. associate-/l* 92.8%

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

      9. *-commutative 92.8%

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

      10. neg-mul-1 92.8%

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

      11. distribute-lft-neg-out 92.8%

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

      12. /-rgt-identity 92.8%

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

      13. div-sub 92.8%

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

      14. associate-/r* 92.8%

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

      15. neg-mul-1 92.8%

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

      16. *-rgt-identity 92.8%

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

      17. times-frac 92.8%

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

      18. /-rgt-identity 92.8%

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

      19. *-commutative 92.8%

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

      20. associate-*r/ 100.0%

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

   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 z around 0 61.4%

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

      1. associate-*r/ 61.4%

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

      2. metadata-eval 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in x around 0 48.6%

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

      1. *-commutative 48.6%

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

   9. Simplified 48.6%

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

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.45:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-237}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-27}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 85.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -1.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-238}:\\ \;\;\;\;x + y \cdot \left(0.8862269254527579 + z \cdot -0.8862269254527579\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-27}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z -1.1)
     t_0
     (if (<= z 8.5e-238)
       (+ x (* y (+ 0.8862269254527579 (* z -0.8862269254527579))))
       (if (<= z 8.4e-113)
         t_0
         (if (<= z 3e-27) (+ x (* y 0.8862269254527579)) x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -1.1) {
		tmp = t_0;
	} else if (z <= 8.5e-238) {
		tmp = x + (y * (0.8862269254527579 + (z * -0.8862269254527579)));
	} else if (z <= 8.4e-113) {
		tmp = t_0;
	} else if (z <= 3e-27) {
		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) :: t_0
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    if (z <= (-1.1d0)) then
        tmp = t_0
    else if (z <= 8.5d-238) then
        tmp = x + (y * (0.8862269254527579d0 + (z * (-0.8862269254527579d0))))
    else if (z <= 8.4d-113) then
        tmp = t_0
    else if (z <= 3d-27) 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 t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -1.1) {
		tmp = t_0;
	} else if (z <= 8.5e-238) {
		tmp = x + (y * (0.8862269254527579 + (z * -0.8862269254527579)));
	} else if (z <= 8.4e-113) {
		tmp = t_0;
	} else if (z <= 3e-27) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if z <= -1.1:
		tmp = t_0
	elif z <= 8.5e-238:
		tmp = x + (y * (0.8862269254527579 + (z * -0.8862269254527579)))
	elif z <= 8.4e-113:
		tmp = t_0
	elif z <= 3e-27:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if (z <= -1.1)
		tmp = t_0;
	elseif (z <= 8.5e-238)
		tmp = Float64(x + Float64(y * Float64(0.8862269254527579 + Float64(z * -0.8862269254527579))));
	elseif (z <= 8.4e-113)
		tmp = t_0;
	elseif (z <= 3e-27)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	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 <= -1.1)
		tmp = t_0;
	elseif (z <= 8.5e-238)
		tmp = x + (y * (0.8862269254527579 + (z * -0.8862269254527579)));
	elseif (z <= 8.4e-113)
		tmp = t_0;
	elseif (z <= 3e-27)
		tmp = x + (y * 0.8862269254527579);
	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, -1.1], t$95$0, If[LessEqual[z, 8.5e-238], N[(x + N[(y * N[(0.8862269254527579 + N[(z * -0.8862269254527579), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.4e-113], t$95$0, If[LessEqual[z, 3e-27], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.1000000000000001 or 8.5000000000000006e-238 < z < 8.39999999999999999e-113

   1. Initial program 92.2%

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

      1. remove-double-neg 92.2%

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

      2. neg-mul-1 92.2%

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

      3. associate-/l* 92.2%

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

      4. neg-mul-1 92.2%

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

      5. associate-/r* 92.2%

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

      6. div-sub 92.4%

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

      7. metadata-eval 92.4%

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

      8. associate-/l* 92.4%

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

      9. *-commutative 92.4%

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

      10. neg-mul-1 92.4%

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

      11. distribute-lft-neg-out 92.4%

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

      12. /-rgt-identity 92.4%

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

      13. div-sub 92.3%

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

      14. associate-/r* 92.3%

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

      15. neg-mul-1 92.3%

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

      16. *-rgt-identity 92.3%

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

      17. times-frac 92.3%

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

      18. /-rgt-identity 92.3%

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

      19. *-commutative 92.3%

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

      20. associate-*r/ 100.0%

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

   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 y around inf 95.5%

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

   if -1.1000000000000001 < z < 8.5000000000000006e-238

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 98.6%

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

   3. Taylor expanded in y around 0 75.8%

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

   4. Taylor expanded in z around 0 75.7%

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

      1. +-commutative 75.7%

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

      2. *-commutative 75.7%

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

      3. *-commutative 75.7%

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

      4. associate-*r* 75.7%

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

      5. *-commutative 75.7%

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

      6. distribute-lft-in 75.7%

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

   6. Simplified 75.7%

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

   if 8.39999999999999999e-113 < z < 3.0000000000000001e-27

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      20. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around 0 79.5%

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

      1. *-commutative 79.5%

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

   9. Simplified 79.5%

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

   if 3.0000000000000001e-27 < z

   1. Initial program 92.8%

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

      1. remove-double-neg 92.8%

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

      2. neg-mul-1 92.8%

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

      3. associate-/l* 92.8%

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

      4. neg-mul-1 92.8%

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

      5. associate-/r* 92.8%

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

      6. div-sub 92.8%

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

      7. metadata-eval 92.8%

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

      8. associate-/l* 92.8%

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

      9. *-commutative 92.8%

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

      10. neg-mul-1 92.8%

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

      11. distribute-lft-neg-out 92.8%

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

      12. /-rgt-identity 92.8%

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

      13. div-sub 92.8%

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

      14. associate-/r* 92.8%

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

      15. neg-mul-1 92.8%

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

      16. *-rgt-identity 92.8%

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

      17. times-frac 92.8%

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

      18. /-rgt-identity 92.8%

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

      19. *-commutative 92.8%

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

      20. associate-*r/ 100.0%

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

   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 z around 0 61.4%

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

      1. associate-*r/ 61.4%

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

      2. metadata-eval 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in x around 0 48.6%

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

      1. *-commutative 48.6%

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

   9. Simplified 48.6%

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

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-238}:\\ \;\;\;\;x + y \cdot \left(0.8862269254527579 + z \cdot -0.8862269254527579\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-113}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-27}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 98.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -350000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{-1}{x - \frac{1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -350000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 8.5e-26) (+ x (/ -1.0 (- x (/ 1.1283791670955126 y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -350000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 8.5e-26) {
		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 (z <= (-350000000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 8.5d-26) 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 (z <= -350000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 8.5e-26) {
		tmp = x + (-1.0 / (x - (1.1283791670955126 / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -350000000.0:
		tmp = x + (-1.0 / x)
	elif z <= 8.5e-26:
		tmp = x + (-1.0 / (x - (1.1283791670955126 / y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -350000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 8.5e-26)
		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 (z <= -350000000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 8.5e-26)
		tmp = x + (-1.0 / (x - (1.1283791670955126 / y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -350000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e-26], 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 z < -3.5e8

   1. Initial program 89.0%

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

      1. remove-double-neg 89.0%

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

      2. neg-mul-1 89.0%

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

      3. associate-/l* 89.1%

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

      4. neg-mul-1 89.1%

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

      5. associate-/r* 89.1%

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

      6. div-sub 89.4%

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

      7. metadata-eval 89.4%

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

      8. associate-/l* 89.4%

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

      9. *-commutative 89.4%

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

      10. neg-mul-1 89.4%

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

      11. distribute-lft-neg-out 89.4%

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

      12. /-rgt-identity 89.4%

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

      13. div-sub 89.2%

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

      14. associate-/r* 89.2%

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

      15. neg-mul-1 89.2%

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

      16. *-rgt-identity 89.2%

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

      17. times-frac 89.2%

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

      18. /-rgt-identity 89.2%

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

      19. *-commutative 89.2%

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

      20. associate-*r/ 100.0%

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

   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 y around inf 100.0%

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

   if -3.5e8 < z < 8.50000000000000004e-26

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      20. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 98.9%

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

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   if 8.50000000000000004e-26 < z

   1. Initial program 92.8%

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

      1. remove-double-neg 92.8%

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

      2. neg-mul-1 92.8%

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

      3. associate-/l* 92.8%

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

      4. neg-mul-1 92.8%

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

      5. associate-/r* 92.8%

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

      6. div-sub 92.8%

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

      7. metadata-eval 92.8%

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

      8. associate-/l* 92.8%

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

      9. *-commutative 92.8%

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

      10. neg-mul-1 92.8%

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

      11. distribute-lft-neg-out 92.8%

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

      12. /-rgt-identity 92.8%

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

      13. div-sub 92.8%

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

      14. associate-/r* 92.8%

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

      15. neg-mul-1 92.8%

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

      16. *-rgt-identity 92.8%

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

      17. times-frac 92.8%

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

      18. /-rgt-identity 92.8%

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

      19. *-commutative 92.8%

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

      20. associate-*r/ 100.0%

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

   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 z around 0 61.4%

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

      1. associate-*r/ 61.4%

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

      2. metadata-eval 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in x around 0 48.6%

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

      1. *-commutative 48.6%

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

   9. Simplified 48.6%

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

   10. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -350000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{-1}{x - \frac{1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 73.3% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-137}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-26}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.8e-137) x (if (<= z 6e-26) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.8e-137) {
		tmp = x;
	} else if (z <= 6e-26) {
		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-137)) then
        tmp = x
    else if (z <= 6d-26) 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-137) {
		tmp = x;
	} else if (z <= 6e-26) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.8e-137:
		tmp = x
	elif z <= 6e-26:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.8e-137)
		tmp = x;
	elseif (z <= 6e-26)
		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-137)
		tmp = x;
	elseif (z <= 6e-26)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.8e-137], x, If[LessEqual[z, 6e-26], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -6.80000000000000028e-137 or 6.00000000000000023e-26 < z

   1. Initial program 92.5%

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

      1. remove-double-neg 92.5%

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

      2. neg-mul-1 92.5%

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

      3. associate-/l* 92.5%

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

      4. neg-mul-1 92.5%

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

      5. associate-/r* 92.5%

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

      6. div-sub 92.6%

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

      7. metadata-eval 92.6%

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

      8. associate-/l* 92.6%

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

      9. *-commutative 92.6%

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

      10. neg-mul-1 92.6%

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

      11. distribute-lft-neg-out 92.6%

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

      12. /-rgt-identity 92.6%

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

      13. div-sub 92.5%

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

      14. associate-/r* 92.5%

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

      15. neg-mul-1 92.5%

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

      16. *-rgt-identity 92.5%

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

      17. times-frac 92.5%

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

      18. /-rgt-identity 92.5%

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

      19. *-commutative 92.5%

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

      20. associate-*r/ 100.0%

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

   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 z around 0 69.0%

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

      1. associate-*r/ 69.1%

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

      2. metadata-eval 69.1%

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

   6. Simplified 69.1%

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

   7. Taylor expanded in x around 0 51.2%

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

      1. *-commutative 51.2%

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

   9. Simplified 51.2%

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

   10. Taylor expanded in x around inf 81.0%

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

   if -6.80000000000000028e-137 < z < 6.00000000000000023e-26

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. div-sub 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/l* 99.9%

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

      9. *-commutative 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-lft-neg-out 99.9%

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

      12. /-rgt-identity 99.9%

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

      13. div-sub 99.9%

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

      14. associate-/r* 99.9%

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

      15. neg-mul-1 99.9%

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

      16. *-rgt-identity 99.9%

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

      17. times-frac 99.9%

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

      18. /-rgt-identity 99.9%

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

      19. *-commutative 99.9%

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

      20. associate-*r/ 99.9%

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

   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%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 73.6%

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

      1. *-commutative 73.6%

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

   9. Simplified 73.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-137}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-26}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 69.2% 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 94.9%

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

   1. remove-double-neg 94.9%

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

   2. neg-mul-1 94.9%

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

   3. associate-/l* 94.9%

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

   4. neg-mul-1 94.9%

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

   5. associate-/r* 94.9%

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

   6. div-sub 95.0%

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

   7. metadata-eval 95.0%

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

   8. associate-/l* 95.0%

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

   9. *-commutative 95.0%

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

   10. neg-mul-1 95.0%

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

   11. distribute-lft-neg-out 95.0%

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

   12. /-rgt-identity 95.0%

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

   13. div-sub 95.0%

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

   14. associate-/r* 95.0%

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

   15. neg-mul-1 95.0%

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

   16. *-rgt-identity 95.0%

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

   17. times-frac 95.0%

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

   18. /-rgt-identity 95.0%

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

   19. *-commutative 95.0%

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

   20. associate-*r/ 99.9%

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

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 79.4%

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

   1. associate-*r/ 79.4%

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

   2. metadata-eval 79.4%

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

6. Simplified 79.4%

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

7. Taylor expanded in x around 0 58.7%

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

   1. *-commutative 58.7%

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

9. Simplified 58.7%

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

10. Taylor expanded in x around inf 74.4%

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

11. Final simplification 74.4%

    \[\leadsto x \]

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 2023271 
(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)))))