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

Percentage Accurate: 95.4% → 99.9%

Time: 9.0s

Alternatives: 11

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: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

   1. remove-double-neg 95.3%

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

   2. neg-mul-1 95.3%

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

   3. associate-/l* 95.4%

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

   4. neg-mul-1 95.4%

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

   5. associate-/r* 95.4%

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

   6. div-sub 95.5%

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

   7. metadata-eval 95.5%

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

   8. associate-/l* 95.5%

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

   9. *-commutative 95.5%

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

   10. neg-mul-1 95.5%

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

   11. distribute-lft-neg-out 95.5%

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

   12. /-rgt-identity 95.5%

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

   13. div-sub 95.4%

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

   14. associate-/r* 95.4%

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

   15. neg-mul-1 95.4%

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

   16. *-rgt-identity 95.4%

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

   17. times-frac 95.4%

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

   18. /-rgt-identity 95.4%

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

   19. *-commutative 95.4%

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

   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. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 5000:\\ \;\;\;\;x + \frac{-1}{x + \frac{z \cdot -1.1283791670955126 - 1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 5000.0)
     (+
      x
      (/ -1.0 (+ x (/ (- (* z -1.1283791670955126) 1.1283791670955126) y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 5000.0) {
		tmp = x + (-1.0 / (x + (((z * -1.1283791670955126) - 1.1283791670955126) / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 5000.0d0) then
        tmp = x + ((-1.0d0) / (x + (((z * (-1.1283791670955126d0)) - 1.1283791670955126d0) / y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 5000.0) {
		tmp = x + (-1.0 / (x + (((z * -1.1283791670955126) - 1.1283791670955126) / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 5000.0:
		tmp = x + (-1.0 / (x + (((z * -1.1283791670955126) - 1.1283791670955126) / y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 5000.0)
		tmp = Float64(x + Float64(-1.0 / Float64(x + Float64(Float64(Float64(z * -1.1283791670955126) - 1.1283791670955126) / y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 5000.0)
		tmp = x + (-1.0 / (x + (((z * -1.1283791670955126) - 1.1283791670955126) / y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 5000.0], N[(x + N[(-1.0 / N[(x + N[(N[(N[(z * -1.1283791670955126), $MachinePrecision] - 1.1283791670955126), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.3%

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

      1. remove-double-neg 90.3%

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

      2. neg-mul-1 90.3%

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

      3. associate-/l* 90.5%

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

      4. neg-mul-1 90.5%

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

      5. associate-/r* 90.5%

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

      6. div-sub 90.9%

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

      7. metadata-eval 90.9%

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

      8. associate-/l* 90.9%

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

      9. *-commutative 90.9%

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

      10. neg-mul-1 90.9%

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

      11. distribute-lft-neg-out 90.9%

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

      12. /-rgt-identity 90.9%

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

      13. div-sub 90.7%

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

      14. associate-/r* 90.7%

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

      15. neg-mul-1 90.7%

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

      16. *-rgt-identity 90.7%

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

      17. times-frac 90.7%

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

      18. /-rgt-identity 90.7%

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

      19. *-commutative 90.7%

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

      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 x around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 5e3

   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 99.6%

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

   5. Taylor expanded in y around 0 99.6%

      \[\leadsto x + \frac{-1}{\left(x + -1.1283791670955126 \cdot \frac{z}{y}\right) - \color{blue}{\frac{1.1283791670955126}{y}}} \]
   6. Step-by-step derivation

      1. div-inv 99.6%

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

      2. mul-1-neg 99.6%

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

      3. associate--l+ 99.6%

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

      4. associate-*r/ 99.6%

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

      5. sub-div 99.6%

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

   7. Applied egg-rr 99.6%

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

   if 5e3 < (exp.f64 z)

   1. Initial program 92.7%

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

      1. remove-double-neg 92.7%

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

      2. neg-mul-1 92.7%

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

      3. associate-/l* 92.7%

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

      4. neg-mul-1 92.7%

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

      5. associate-/r* 92.7%

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

      6. div-sub 92.7%

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

      7. metadata-eval 92.7%

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

      8. associate-/l* 92.7%

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

      9. *-commutative 92.7%

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

      10. neg-mul-1 92.7%

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

      11. distribute-lft-neg-out 92.7%

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

      12. /-rgt-identity 92.7%

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

      13. div-sub 92.7%

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

      14. associate-/r* 92.7%

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

      15. neg-mul-1 92.7%

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

      16. *-rgt-identity 92.7%

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

      17. times-frac 92.7%

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

      18. /-rgt-identity 92.7%

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

      19. *-commutative 92.7%

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

      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 63.0%

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

      1. cancel-sign-sub-inv 63.0%

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

      2. metadata-eval 63.0%

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

      3. associate-*r/ 63.0%

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

      4. metadata-eval 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 5000:\\ \;\;\;\;x + \frac{-1}{x + \frac{z \cdot -1.1283791670955126 - 1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.3%

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

      1. remove-double-neg 90.3%

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

      2. neg-mul-1 90.3%

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

      3. associate-/l* 90.5%

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

      4. neg-mul-1 90.5%

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

      5. associate-/r* 90.5%

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

      6. div-sub 90.9%

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

      7. metadata-eval 90.9%

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

      8. associate-/l* 90.9%

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

      9. *-commutative 90.9%

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

      10. neg-mul-1 90.9%

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

      11. distribute-lft-neg-out 90.9%

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

      12. /-rgt-identity 90.9%

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

      13. div-sub 90.7%

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

      14. associate-/r* 90.7%

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

      15. neg-mul-1 90.7%

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

      16. *-rgt-identity 90.7%

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

      17. times-frac 90.7%

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

      18. /-rgt-identity 90.7%

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

      19. *-commutative 90.7%

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

      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 x around inf 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.6%

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

4. Final simplification 98.4%

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

Alternative 4: 99.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -19000000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 92:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -19000000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 92.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -19000000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 92.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-19000000000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 92.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * 1.1283791670955126d0)) - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -19000000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 92.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -19000000000.0:
		tmp = x + (-1.0 / x)
	elif z <= 92.0:
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -19000000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 92.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * 1.1283791670955126)) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -19000000000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 92.0)
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -19000000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 92.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e10

   1. Initial program 89.6%

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

      1. remove-double-neg 89.6%

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

      2. neg-mul-1 89.6%

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

      3. associate-/l* 89.8%

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

      4. neg-mul-1 89.8%

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

      5. associate-/r* 89.8%

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

      6. div-sub 90.2%

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

      7. metadata-eval 90.2%

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

      8. associate-/l* 90.2%

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

      9. *-commutative 90.2%

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

      10. neg-mul-1 90.2%

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

      11. distribute-lft-neg-out 90.2%

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

      12. /-rgt-identity 90.2%

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

      13. div-sub 90.0%

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

      14. associate-/r* 90.0%

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

      15. neg-mul-1 90.0%

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

      16. *-rgt-identity 90.0%

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

      17. times-frac 90.0%

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

      18. /-rgt-identity 90.0%

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

      19. *-commutative 90.0%

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

      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 x around inf 100.0%

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

   if -1.9e10 < z < 92

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.6%

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

   if 92 < z

   1. Initial program 92.7%

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

      1. remove-double-neg 92.7%

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

      2. neg-mul-1 92.7%

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

      3. associate-/l* 92.7%

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

      4. neg-mul-1 92.7%

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

      5. associate-/r* 92.7%

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

      6. div-sub 92.7%

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

      7. metadata-eval 92.7%

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

      8. associate-/l* 92.7%

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

      9. *-commutative 92.7%

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

      10. neg-mul-1 92.7%

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

      11. distribute-lft-neg-out 92.7%

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

      12. /-rgt-identity 92.7%

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

      13. div-sub 92.7%

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

      14. associate-/r* 92.7%

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

      15. neg-mul-1 92.7%

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

      16. *-rgt-identity 92.7%

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

      17. times-frac 92.7%

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

      18. /-rgt-identity 92.7%

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

      19. *-commutative 92.7%

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

      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 63.0%

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

      1. cancel-sign-sub-inv 63.0%

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

      2. metadata-eval 63.0%

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

      3. associate-*r/ 63.0%

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

      4. metadata-eval 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -19000000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 92:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 69.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-308}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.9e-93)
   x
   (if (<= x -2e-250)
     (/ -1.0 x)
     (if (<= x 3.6e-308)
       (* y 0.8862269254527579)
       (if (<= x 8.2e-163)
         (/ -1.0 x)
         (if (<= x 3e-113) x (if (<= x 2.7e-6) (/ -1.0 x) x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.9e-93) {
		tmp = x;
	} else if (x <= -2e-250) {
		tmp = -1.0 / x;
	} else if (x <= 3.6e-308) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 8.2e-163) {
		tmp = -1.0 / x;
	} else if (x <= 3e-113) {
		tmp = x;
	} else if (x <= 2.7e-6) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5.9d-93)) then
        tmp = x
    else if (x <= (-2d-250)) then
        tmp = (-1.0d0) / x
    else if (x <= 3.6d-308) then
        tmp = y * 0.8862269254527579d0
    else if (x <= 8.2d-163) then
        tmp = (-1.0d0) / x
    else if (x <= 3d-113) then
        tmp = x
    else if (x <= 2.7d-6) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.9e-93) {
		tmp = x;
	} else if (x <= -2e-250) {
		tmp = -1.0 / x;
	} else if (x <= 3.6e-308) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 8.2e-163) {
		tmp = -1.0 / x;
	} else if (x <= 3e-113) {
		tmp = x;
	} else if (x <= 2.7e-6) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.9e-93:
		tmp = x
	elif x <= -2e-250:
		tmp = -1.0 / x
	elif x <= 3.6e-308:
		tmp = y * 0.8862269254527579
	elif x <= 8.2e-163:
		tmp = -1.0 / x
	elif x <= 3e-113:
		tmp = x
	elif x <= 2.7e-6:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.9e-93)
		tmp = x;
	elseif (x <= -2e-250)
		tmp = Float64(-1.0 / x);
	elseif (x <= 3.6e-308)
		tmp = Float64(y * 0.8862269254527579);
	elseif (x <= 8.2e-163)
		tmp = Float64(-1.0 / x);
	elseif (x <= 3e-113)
		tmp = x;
	elseif (x <= 2.7e-6)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.9e-93)
		tmp = x;
	elseif (x <= -2e-250)
		tmp = -1.0 / x;
	elseif (x <= 3.6e-308)
		tmp = y * 0.8862269254527579;
	elseif (x <= 8.2e-163)
		tmp = -1.0 / x;
	elseif (x <= 3e-113)
		tmp = x;
	elseif (x <= 2.7e-6)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.9e-93], x, If[LessEqual[x, -2e-250], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 3.6e-308], N[(y * 0.8862269254527579), $MachinePrecision], If[LessEqual[x, 8.2e-163], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 3e-113], x, If[LessEqual[x, 2.7e-6], N[(-1.0 / x), $MachinePrecision], x]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.9e-93 or 8.19999999999999965e-163 < x < 3.0000000000000001e-113 or 2.69999999999999998e-6 < x

   1. Initial program 97.5%

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

      1. remove-double-neg 97.5%

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

      2. neg-mul-1 97.5%

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

      3. associate-/l* 97.5%

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

      4. neg-mul-1 97.5%

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

      5. associate-/r* 97.5%

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

      6. div-sub 97.5%

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

      7. metadata-eval 97.5%

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

      8. associate-/l* 97.5%

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

      9. *-commutative 97.5%

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

      10. neg-mul-1 97.5%

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

      11. distribute-lft-neg-out 97.5%

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

      12. /-rgt-identity 97.5%

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

      13. div-sub 97.5%

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

      14. associate-/r* 97.5%

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

      15. neg-mul-1 97.5%

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

      16. *-rgt-identity 97.5%

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

      17. times-frac 97.5%

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

      18. /-rgt-identity 97.5%

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

      19. *-commutative 97.5%

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

      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 95.8%

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

      1. cancel-sign-sub-inv 95.8%

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

      2. metadata-eval 95.8%

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

      3. associate-*r/ 95.8%

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

      4. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in x around inf 92.6%

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

   if -5.9e-93 < x < -2.0000000000000001e-250 or 3.5999999999999999e-308 < x < 8.19999999999999965e-163 or 3.0000000000000001e-113 < x < 2.69999999999999998e-6

   1. Initial program 92.0%

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

      1. remove-double-neg 92.0%

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

      2. neg-mul-1 92.0%

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

      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 92.4%

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

      14. associate-/r* 92.4%

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

      15. neg-mul-1 92.4%

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

      16. *-rgt-identity 92.4%

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

      17. times-frac 92.4%

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

      18. /-rgt-identity 92.4%

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

      19. *-commutative 92.4%

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

      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 x around inf 56.8%

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

   5. Taylor expanded in x around 0 56.8%

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

   if -2.0000000000000001e-250 < x < 3.5999999999999999e-308

   1. Initial program 89.7%

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

      1. remove-double-neg 89.7%

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

      2. neg-mul-1 89.7%

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

      3. associate-/l* 89.9%

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

      4. neg-mul-1 89.9%

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

      5. associate-/r* 89.9%

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

      6. div-sub 90.6%

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

      7. metadata-eval 90.6%

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

      8. associate-/l* 90.6%

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

      9. *-commutative 90.6%

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

      10. neg-mul-1 90.6%

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

      11. distribute-lft-neg-out 90.6%

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

      12. /-rgt-identity 90.6%

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

      13. div-sub 89.9%

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

      14. associate-/r* 89.9%

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

      15. neg-mul-1 89.9%

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

      16. *-rgt-identity 89.9%

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

      17. times-frac 89.9%

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

      18. /-rgt-identity 89.9%

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

      19. *-commutative 89.9%

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

      20. associate-*r/ 99.8%

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

   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 70.7%

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

      1. cancel-sign-sub-inv 70.7%

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

      2. metadata-eval 70.7%

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

      3. associate-*r/ 70.5%

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

      4. metadata-eval 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in x around 0 61.1%

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

      1. *-commutative 61.1%

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

   9. Simplified 61.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-308}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 69.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-263}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-160}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-114}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.6e-93)
   x
   (if (<= x -2.6e-263)
     (/ -1.0 x)
     (if (<= x 6.8e-307)
       (/ y 1.1283791670955126)
       (if (<= x 5.6e-160)
         (/ -1.0 x)
         (if (<= x 3.9e-114) x (if (<= x 3.2e-9) (/ -1.0 x) x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.6e-93) {
		tmp = x;
	} else if (x <= -2.6e-263) {
		tmp = -1.0 / x;
	} else if (x <= 6.8e-307) {
		tmp = y / 1.1283791670955126;
	} else if (x <= 5.6e-160) {
		tmp = -1.0 / x;
	} else if (x <= 3.9e-114) {
		tmp = x;
	} else if (x <= 3.2e-9) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.6d-93)) then
        tmp = x
    else if (x <= (-2.6d-263)) then
        tmp = (-1.0d0) / x
    else if (x <= 6.8d-307) then
        tmp = y / 1.1283791670955126d0
    else if (x <= 5.6d-160) then
        tmp = (-1.0d0) / x
    else if (x <= 3.9d-114) then
        tmp = x
    else if (x <= 3.2d-9) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.6e-93) {
		tmp = x;
	} else if (x <= -2.6e-263) {
		tmp = -1.0 / x;
	} else if (x <= 6.8e-307) {
		tmp = y / 1.1283791670955126;
	} else if (x <= 5.6e-160) {
		tmp = -1.0 / x;
	} else if (x <= 3.9e-114) {
		tmp = x;
	} else if (x <= 3.2e-9) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.6e-93:
		tmp = x
	elif x <= -2.6e-263:
		tmp = -1.0 / x
	elif x <= 6.8e-307:
		tmp = y / 1.1283791670955126
	elif x <= 5.6e-160:
		tmp = -1.0 / x
	elif x <= 3.9e-114:
		tmp = x
	elif x <= 3.2e-9:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.6e-93)
		tmp = x;
	elseif (x <= -2.6e-263)
		tmp = Float64(-1.0 / x);
	elseif (x <= 6.8e-307)
		tmp = Float64(y / 1.1283791670955126);
	elseif (x <= 5.6e-160)
		tmp = Float64(-1.0 / x);
	elseif (x <= 3.9e-114)
		tmp = x;
	elseif (x <= 3.2e-9)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.6e-93)
		tmp = x;
	elseif (x <= -2.6e-263)
		tmp = -1.0 / x;
	elseif (x <= 6.8e-307)
		tmp = y / 1.1283791670955126;
	elseif (x <= 5.6e-160)
		tmp = -1.0 / x;
	elseif (x <= 3.9e-114)
		tmp = x;
	elseif (x <= 3.2e-9)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.6e-93], x, If[LessEqual[x, -2.6e-263], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 6.8e-307], N[(y / 1.1283791670955126), $MachinePrecision], If[LessEqual[x, 5.6e-160], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 3.9e-114], x, If[LessEqual[x, 3.2e-9], N[(-1.0 / x), $MachinePrecision], x]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5999999999999996e-93 or 5.60000000000000032e-160 < x < 3.90000000000000002e-114 or 3.20000000000000012e-9 < x

   1. Initial program 97.5%

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

      1. remove-double-neg 97.5%

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

      2. neg-mul-1 97.5%

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

      3. associate-/l* 97.5%

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

      4. neg-mul-1 97.5%

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

      5. associate-/r* 97.5%

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

      6. div-sub 97.5%

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

      7. metadata-eval 97.5%

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

      8. associate-/l* 97.5%

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

      9. *-commutative 97.5%

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

      10. neg-mul-1 97.5%

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

      11. distribute-lft-neg-out 97.5%

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

      12. /-rgt-identity 97.5%

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

      13. div-sub 97.5%

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

      14. associate-/r* 97.5%

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

      15. neg-mul-1 97.5%

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

      16. *-rgt-identity 97.5%

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

      17. times-frac 97.5%

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

      18. /-rgt-identity 97.5%

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

      19. *-commutative 97.5%

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

      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 95.8%

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

      1. cancel-sign-sub-inv 95.8%

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

      2. metadata-eval 95.8%

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

      3. associate-*r/ 95.8%

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

      4. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in x around inf 92.6%

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

   if -4.5999999999999996e-93 < x < -2.6e-263 or 6.79999999999999978e-307 < x < 5.60000000000000032e-160 or 3.90000000000000002e-114 < x < 3.20000000000000012e-9

   1. Initial program 92.0%

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

      1. remove-double-neg 92.0%

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

      2. neg-mul-1 92.0%

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

      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 92.4%

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

      14. associate-/r* 92.4%

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

      15. neg-mul-1 92.4%

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

      16. *-rgt-identity 92.4%

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

      17. times-frac 92.4%

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

      18. /-rgt-identity 92.4%

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

      19. *-commutative 92.4%

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

      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 x around inf 56.8%

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

   5. Taylor expanded in x around 0 56.8%

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

   if -2.6e-263 < x < 6.79999999999999978e-307

   1. Initial program 89.7%

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

      1. remove-double-neg 89.7%

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

      2. neg-mul-1 89.7%

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

      3. associate-/l* 89.9%

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

      4. neg-mul-1 89.9%

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

      5. associate-/r* 89.9%

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

      6. div-sub 90.6%

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

      7. metadata-eval 90.6%

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

      8. associate-/l* 90.6%

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

      9. *-commutative 90.6%

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

      10. neg-mul-1 90.6%

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

      11. distribute-lft-neg-out 90.6%

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

      12. /-rgt-identity 90.6%

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

      13. div-sub 89.9%

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

      14. associate-/r* 89.9%

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

      15. neg-mul-1 89.9%

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

      16. *-rgt-identity 89.9%

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

      17. times-frac 89.9%

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

      18. /-rgt-identity 89.9%

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

      19. *-commutative 89.9%

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

      20. associate-*r/ 99.8%

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

   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 70.7%

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

      1. cancel-sign-sub-inv 70.7%

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

      2. metadata-eval 70.7%

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

      3. associate-*r/ 70.5%

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

      4. metadata-eval 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in x around 0 61.1%

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

      1. *-commutative 61.1%

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

   9. Simplified 61.1%

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

       1. metadata-eval 61.2%

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

       2. div-inv 61.2%

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

   11. Applied egg-rr 61.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-263}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-160}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-114}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 84.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -7.3 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-248}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;z \leq 7.1 \cdot 10^{-246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-107}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{elif}\;z \leq 19:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z -7.3e-56)
     t_0
     (if (<= z -7.5e-248)
       (+ x (/ y 1.1283791670955126))
       (if (<= z 7.1e-246)
         t_0
         (if (<= z 4.4e-107)
           (- x (* y -0.8862269254527579))
           (if (<= z 19.0) t_0 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -7.3e-56) {
		tmp = t_0;
	} else if (z <= -7.5e-248) {
		tmp = x + (y / 1.1283791670955126);
	} else if (z <= 7.1e-246) {
		tmp = t_0;
	} else if (z <= 4.4e-107) {
		tmp = x - (y * -0.8862269254527579);
	} else if (z <= 19.0) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    if (z <= (-7.3d-56)) then
        tmp = t_0
    else if (z <= (-7.5d-248)) then
        tmp = x + (y / 1.1283791670955126d0)
    else if (z <= 7.1d-246) then
        tmp = t_0
    else if (z <= 4.4d-107) then
        tmp = x - (y * (-0.8862269254527579d0))
    else if (z <= 19.0d0) then
        tmp = t_0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -7.3e-56) {
		tmp = t_0;
	} else if (z <= -7.5e-248) {
		tmp = x + (y / 1.1283791670955126);
	} else if (z <= 7.1e-246) {
		tmp = t_0;
	} else if (z <= 4.4e-107) {
		tmp = x - (y * -0.8862269254527579);
	} else if (z <= 19.0) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if z <= -7.3e-56:
		tmp = t_0
	elif z <= -7.5e-248:
		tmp = x + (y / 1.1283791670955126)
	elif z <= 7.1e-246:
		tmp = t_0
	elif z <= 4.4e-107:
		tmp = x - (y * -0.8862269254527579)
	elif z <= 19.0:
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if (z <= -7.3e-56)
		tmp = t_0;
	elseif (z <= -7.5e-248)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	elseif (z <= 7.1e-246)
		tmp = t_0;
	elseif (z <= 4.4e-107)
		tmp = Float64(x - Float64(y * -0.8862269254527579));
	elseif (z <= 19.0)
		tmp = t_0;
	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 <= -7.3e-56)
		tmp = t_0;
	elseif (z <= -7.5e-248)
		tmp = x + (y / 1.1283791670955126);
	elseif (z <= 7.1e-246)
		tmp = t_0;
	elseif (z <= 4.4e-107)
		tmp = x - (y * -0.8862269254527579);
	elseif (z <= 19.0)
		tmp = t_0;
	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, -7.3e-56], t$95$0, If[LessEqual[z, -7.5e-248], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.1e-246], t$95$0, If[LessEqual[z, 4.4e-107], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 19.0], t$95$0, x]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.30000000000000045e-56 or -7.4999999999999994e-248 < z < 7.10000000000000036e-246 or 4.40000000000000025e-107 < z < 19

   1. Initial program 94.3%

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

      1. remove-double-neg 94.3%

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

      2. neg-mul-1 94.3%

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

      3. associate-/l* 94.5%

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

      4. neg-mul-1 94.5%

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

      5. associate-/r* 94.5%

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

      6. div-sub 94.7%

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

      7. metadata-eval 94.7%

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

      8. associate-/l* 94.7%

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

      9. *-commutative 94.7%

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

      10. neg-mul-1 94.7%

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

      11. distribute-lft-neg-out 94.7%

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

      12. /-rgt-identity 94.7%

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

      13. div-sub 94.6%

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

      14. associate-/r* 94.6%

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

      15. neg-mul-1 94.6%

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

      16. *-rgt-identity 94.6%

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

      17. times-frac 94.6%

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

      18. /-rgt-identity 94.6%

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

      19. *-commutative 94.6%

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

      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 x around inf 92.2%

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

   if -7.30000000000000045e-56 < z < -7.4999999999999994e-248

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.9%

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

   3. Taylor expanded in x around 0 72.8%

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

   if 7.10000000000000036e-246 < z < 4.40000000000000025e-107

   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* 99.7%

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

      4. neg-mul-1 99.7%

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

      5. associate-/r* 99.7%

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

      6. div-sub 99.7%

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

      7. metadata-eval 99.7%

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

      8. associate-/l* 99.7%

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

      9. *-commutative 99.7%

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

      10. neg-mul-1 99.7%

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

      11. distribute-lft-neg-out 99.7%

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

      12. /-rgt-identity 99.7%

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

      13. div-sub 99.7%

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

      14. associate-/r* 99.7%

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

      15. neg-mul-1 99.7%

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

      16. *-rgt-identity 99.7%

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

      17. times-frac 99.7%

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

      18. /-rgt-identity 99.7%

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

      19. *-commutative 99.7%

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

      20. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*r/ 99.7%

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

      4. metadata-eval 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 77.9%

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

   if 19 < z

   1. Initial program 92.7%

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

      1. remove-double-neg 92.7%

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

      2. neg-mul-1 92.7%

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

      3. associate-/l* 92.7%

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

      4. neg-mul-1 92.7%

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

      5. associate-/r* 92.7%

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

      6. div-sub 92.7%

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

      7. metadata-eval 92.7%

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

      8. associate-/l* 92.7%

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

      9. *-commutative 92.7%

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

      10. neg-mul-1 92.7%

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

      11. distribute-lft-neg-out 92.7%

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

      12. /-rgt-identity 92.7%

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

      13. div-sub 92.7%

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

      14. associate-/r* 92.7%

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

      15. neg-mul-1 92.7%

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

      16. *-rgt-identity 92.7%

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

      17. times-frac 92.7%

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

      18. /-rgt-identity 92.7%

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

      19. *-commutative 92.7%

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

      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 63.0%

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

      1. cancel-sign-sub-inv 63.0%

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

      2. metadata-eval 63.0%

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

      3. associate-*r/ 63.0%

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

      4. metadata-eval 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.3 \cdot 10^{-56}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-248}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;z \leq 7.1 \cdot 10^{-246}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-107}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{elif}\;z \leq 19:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 99.4% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -19000000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 210:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -19000000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 210.0) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -19000000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 210.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-19000000000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 210.0d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -19000000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 210.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -19000000000.0:
		tmp = x + (-1.0 / x)
	elif z <= 210.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -19000000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 210.0)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -19000000000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 210.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -19000000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 210.0], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e10

   1. Initial program 89.6%

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

      1. remove-double-neg 89.6%

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

      2. neg-mul-1 89.6%

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

      3. associate-/l* 89.8%

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

      4. neg-mul-1 89.8%

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

      5. associate-/r* 89.8%

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

      6. div-sub 90.2%

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

      7. metadata-eval 90.2%

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

      8. associate-/l* 90.2%

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

      9. *-commutative 90.2%

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

      10. neg-mul-1 90.2%

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

      11. distribute-lft-neg-out 90.2%

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

      12. /-rgt-identity 90.2%

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

      13. div-sub 90.0%

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

      14. associate-/r* 90.0%

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

      15. neg-mul-1 90.0%

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

      16. *-rgt-identity 90.0%

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

      17. times-frac 90.0%

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

      18. /-rgt-identity 90.0%

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

      19. *-commutative 90.0%

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

      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 x around inf 100.0%

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

   if -1.9e10 < z < 210

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.4%

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

   if 210 < z

   1. Initial program 92.7%

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

      1. remove-double-neg 92.7%

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

      2. neg-mul-1 92.7%

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

      3. associate-/l* 92.7%

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

      4. neg-mul-1 92.7%

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

      5. associate-/r* 92.7%

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

      6. div-sub 92.7%

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

      7. metadata-eval 92.7%

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

      8. associate-/l* 92.7%

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

      9. *-commutative 92.7%

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

      10. neg-mul-1 92.7%

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

      11. distribute-lft-neg-out 92.7%

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

      12. /-rgt-identity 92.7%

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

      13. div-sub 92.7%

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

      14. associate-/r* 92.7%

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

      15. neg-mul-1 92.7%

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

      16. *-rgt-identity 92.7%

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

      17. times-frac 92.7%

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

      18. /-rgt-identity 92.7%

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

      19. *-commutative 92.7%

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

      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 63.0%

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

      1. cancel-sign-sub-inv 63.0%

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

      2. metadata-eval 63.0%

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

      3. associate-*r/ 63.0%

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

      4. metadata-eval 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -19000000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 210:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 99.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -19000000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -19000000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 190.0) (+ x (/ -1.0 (+ x (/ -1.1283791670955126 y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -19000000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 190.0) {
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-19000000000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 190.0d0) then
        tmp = x + ((-1.0d0) / (x + ((-1.1283791670955126d0) / y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -19000000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 190.0) {
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -19000000000.0:
		tmp = x + (-1.0 / x)
	elif z <= 190.0:
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -19000000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 190.0)
		tmp = Float64(x + Float64(-1.0 / Float64(x + Float64(-1.1283791670955126 / y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -19000000000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 190.0)
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -19000000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 190.0], N[(x + N[(-1.0 / N[(x + N[(-1.1283791670955126 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e10

   1. Initial program 89.6%

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

      1. remove-double-neg 89.6%

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

      2. neg-mul-1 89.6%

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

      3. associate-/l* 89.8%

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

      4. neg-mul-1 89.8%

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

      5. associate-/r* 89.8%

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

      6. div-sub 90.2%

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

      7. metadata-eval 90.2%

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

      8. associate-/l* 90.2%

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

      9. *-commutative 90.2%

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

      10. neg-mul-1 90.2%

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

      11. distribute-lft-neg-out 90.2%

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

      12. /-rgt-identity 90.2%

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

      13. div-sub 90.0%

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

      14. associate-/r* 90.0%

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

      15. neg-mul-1 90.0%

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

      16. *-rgt-identity 90.0%

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

      17. times-frac 90.0%

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

      18. /-rgt-identity 90.0%

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

      19. *-commutative 90.0%

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

      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 x around inf 100.0%

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

   if -1.9e10 < z < 190

   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 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. associate-*r/ 99.4%

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

      4. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   if 190 < z

   1. Initial program 92.7%

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

      1. remove-double-neg 92.7%

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

      2. neg-mul-1 92.7%

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

      3. associate-/l* 92.7%

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

      4. neg-mul-1 92.7%

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

      5. associate-/r* 92.7%

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

      6. div-sub 92.7%

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

      7. metadata-eval 92.7%

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

      8. associate-/l* 92.7%

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

      9. *-commutative 92.7%

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

      10. neg-mul-1 92.7%

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

      11. distribute-lft-neg-out 92.7%

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

      12. /-rgt-identity 92.7%

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

      13. div-sub 92.7%

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

      14. associate-/r* 92.7%

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

      15. neg-mul-1 92.7%

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

      16. *-rgt-identity 92.7%

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

      17. times-frac 92.7%

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

      18. /-rgt-identity 92.7%

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

      19. *-commutative 92.7%

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

      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 63.0%

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

      1. cancel-sign-sub-inv 63.0%

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

      2. metadata-eval 63.0%

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

      3. associate-*r/ 63.0%

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

      4. metadata-eval 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -19000000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 72.2% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-253}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-114}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.6e-96)
   x
   (if (<= x -8e-253)
     (/ -1.0 x)
     (if (<= x 4.3e-114)
       (+ x (/ y 1.1283791670955126))
       (if (<= x 2.6e-6) (/ -1.0 x) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.6e-96) {
		tmp = x;
	} else if (x <= -8e-253) {
		tmp = -1.0 / x;
	} else if (x <= 4.3e-114) {
		tmp = x + (y / 1.1283791670955126);
	} else if (x <= 2.6e-6) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8.6d-96)) then
        tmp = x
    else if (x <= (-8d-253)) then
        tmp = (-1.0d0) / x
    else if (x <= 4.3d-114) then
        tmp = x + (y / 1.1283791670955126d0)
    else if (x <= 2.6d-6) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.6e-96) {
		tmp = x;
	} else if (x <= -8e-253) {
		tmp = -1.0 / x;
	} else if (x <= 4.3e-114) {
		tmp = x + (y / 1.1283791670955126);
	} else if (x <= 2.6e-6) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.6e-96:
		tmp = x
	elif x <= -8e-253:
		tmp = -1.0 / x
	elif x <= 4.3e-114:
		tmp = x + (y / 1.1283791670955126)
	elif x <= 2.6e-6:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.6e-96)
		tmp = x;
	elseif (x <= -8e-253)
		tmp = Float64(-1.0 / x);
	elseif (x <= 4.3e-114)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	elseif (x <= 2.6e-6)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.6e-96)
		tmp = x;
	elseif (x <= -8e-253)
		tmp = -1.0 / x;
	elseif (x <= 4.3e-114)
		tmp = x + (y / 1.1283791670955126);
	elseif (x <= 2.6e-6)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.6e-96], x, If[LessEqual[x, -8e-253], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 4.3e-114], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e-6], N[(-1.0 / x), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.59999999999999961e-96 or 2.60000000000000009e-6 < x

   1. Initial program 97.3%

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

      1. remove-double-neg 97.3%

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

      2. neg-mul-1 97.3%

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

      3. associate-/l* 97.3%

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

      4. neg-mul-1 97.3%

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

      5. associate-/r* 97.3%

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

      6. div-sub 97.3%

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

      7. metadata-eval 97.3%

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

      8. associate-/l* 97.3%

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

      9. *-commutative 97.3%

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

      10. neg-mul-1 97.3%

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

      11. distribute-lft-neg-out 97.3%

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

      12. /-rgt-identity 97.3%

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

      13. div-sub 97.3%

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

      14. associate-/r* 97.3%

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

      15. neg-mul-1 97.3%

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

      16. *-rgt-identity 97.3%

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

      17. times-frac 97.3%

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

      18. /-rgt-identity 97.3%

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

      19. *-commutative 97.3%

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

      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 z around 0 96.4%

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

      1. cancel-sign-sub-inv 96.4%

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

      2. metadata-eval 96.4%

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

      3. associate-*r/ 96.4%

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

      4. metadata-eval 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in x around inf 94.2%

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

   if -8.59999999999999961e-96 < x < -8.0000000000000005e-253 or 4.3e-114 < x < 2.60000000000000009e-6

   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* 95.1%

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

      4. neg-mul-1 95.1%

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

      5. associate-/r* 95.1%

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

      6. div-sub 95.2%

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

      7. metadata-eval 95.2%

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

      8. associate-/l* 95.2%

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

      9. *-commutative 95.2%

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

      10. neg-mul-1 95.2%

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

      11. distribute-lft-neg-out 95.2%

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

      12. /-rgt-identity 95.2%

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

      13. div-sub 95.1%

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

      14. associate-/r* 95.1%

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

      15. neg-mul-1 95.1%

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

      16. *-rgt-identity 95.1%

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

      17. times-frac 95.1%

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

      18. /-rgt-identity 95.1%

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

      19. *-commutative 95.1%

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

      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 x around inf 61.3%

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

   5. Taylor expanded in x around 0 61.3%

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

   if -8.0000000000000005e-253 < x < 4.3e-114

   1. Initial program 89.1%

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

   2. Taylor expanded in z around 0 59.8%

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

   3. Taylor expanded in x around 0 50.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-253}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-114}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 69.1% accurate, 111.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 95.3%

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

   1. remove-double-neg 95.3%

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

   2. neg-mul-1 95.3%

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

   3. associate-/l* 95.4%

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

   4. neg-mul-1 95.4%

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

   5. associate-/r* 95.4%

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

   6. div-sub 95.5%

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

   7. metadata-eval 95.5%

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

   8. associate-/l* 95.5%

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

   9. *-commutative 95.5%

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

   10. neg-mul-1 95.5%

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

   11. distribute-lft-neg-out 95.5%

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

   12. /-rgt-identity 95.5%

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

   13. div-sub 95.4%

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

   14. associate-/r* 95.4%

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

   15. neg-mul-1 95.4%

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

   16. *-rgt-identity 95.4%

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

   17. times-frac 95.4%

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

   18. /-rgt-identity 95.4%

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

   19. *-commutative 95.4%

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

   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 81.4%

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

   1. cancel-sign-sub-inv 81.4%

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

   2. metadata-eval 81.4%

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

   3. associate-*r/ 81.4%

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

   4. metadata-eval 81.4%

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

6. Simplified 81.4%

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

7. Taylor expanded in x around inf 68.9%

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

8. Final simplification 68.9%

   \[\leadsto x \]

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 2023290 
(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)))))