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

Percentage Accurate: 95.7% → 99.9%

Time: 9.3s

Alternatives: 8

Speedup: 5.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.7% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. *-lft-identity 97.5%

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

   2. associate-/l* 97.5%

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

   3. remove-double-neg 97.5%

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

   4. neg-mul-1 97.5%

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

   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. associate-*l* 97.5%

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

   11. neg-mul-1 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%

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

   15. neg-mul-1 97.5%

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

   16. remove-double-neg 97.5%

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

   17. associate-*r/ 97.5%

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

   18. distribute-lft-neg-out 97.5%

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

   19. neg-mul-1 97.5%

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

   20. *-commutative 97.5%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

   \[\leadsto x + \frac{1}{1.1283791670955126 \cdot \frac{e^{z}}{y} - x} \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 10:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 10.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* 1.1283791670955126 z)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 10.0) {
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 10.0d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (1.1283791670955126d0 * z)) - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 10.0) {
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 10.0:
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 10.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(1.1283791670955126 * z)) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 10.0)
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 10.0], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(1.1283791670955126 * z), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 93.7%

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

      1. *-lft-identity 93.7%

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

      2. associate-/l* 93.8%

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

      3. remove-double-neg 93.8%

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

      4. neg-mul-1 93.8%

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

      5. associate-/r* 93.8%

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

      6. div-sub 93.9%

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

      7. metadata-eval 93.9%

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

      8. associate-/l* 93.9%

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

      9. *-commutative 93.9%

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

      10. associate-*l* 93.9%

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

      11. neg-mul-1 93.9%

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

      12. /-rgt-identity 93.9%

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

      13. div-sub 93.9%

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

      14. associate-/r* 93.9%

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

      15. neg-mul-1 93.9%

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

      16. remove-double-neg 93.9%

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

      17. associate-*r/ 93.9%

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

      18. distribute-lft-neg-out 93.9%

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

      19. neg-mul-1 93.9%

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

      20. *-commutative 93.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 10

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 98.8%

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

   if 10 < (exp.f64 z)

   1. Initial program 95.8%

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

      1. *-lft-identity 95.8%

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

      2. associate-/l* 95.8%

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

      3. remove-double-neg 95.8%

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

      4. neg-mul-1 95.8%

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

      5. associate-/r* 95.8%

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

      6. div-sub 95.8%

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

      7. metadata-eval 95.8%

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

      8. associate-/l* 95.8%

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

      9. *-commutative 95.8%

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

      10. associate-*l* 95.8%

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

      11. neg-mul-1 95.8%

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

      12. /-rgt-identity 95.8%

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

      13. div-sub 95.8%

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

      14. associate-/r* 95.8%

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

      15. neg-mul-1 95.8%

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

      16. remove-double-neg 95.8%

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

      17. associate-*r/ 95.8%

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

      18. distribute-lft-neg-out 95.8%

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

      19. neg-mul-1 95.8%

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

      20. *-commutative 95.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 10:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -230:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 17.5:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -230.0)
   (+ x (/ -1.0 x))
   (if (<= z 17.5) (+ x (/ 1.0 (- (/ 1.1283791670955126 y) x))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -230.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 17.5) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-230.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 17.5d0) then
        tmp = x + (1.0d0 / ((1.1283791670955126d0 / y) - x))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -230.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 17.5) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -230.0:
		tmp = x + (-1.0 / x)
	elif z <= 17.5:
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -230.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 17.5)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(1.1283791670955126 / y) - x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -230.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 17.5)
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -230.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 17.5], N[(x + N[(1.0 / N[(N[(1.1283791670955126 / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -230

   1. Initial program 93.7%

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

      1. *-lft-identity 93.7%

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

      2. associate-/l* 93.8%

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

      3. remove-double-neg 93.8%

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

      4. neg-mul-1 93.8%

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

      5. associate-/r* 93.8%

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

      6. div-sub 93.9%

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

      7. metadata-eval 93.9%

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

      8. associate-/l* 93.9%

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

      9. *-commutative 93.9%

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

      10. associate-*l* 93.9%

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

      11. neg-mul-1 93.9%

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

      12. /-rgt-identity 93.9%

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

      13. div-sub 93.9%

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

      14. associate-/r* 93.9%

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

      15. neg-mul-1 93.9%

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

      16. remove-double-neg 93.9%

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

      17. associate-*r/ 93.9%

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

      18. distribute-lft-neg-out 93.9%

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

      19. neg-mul-1 93.9%

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

      20. *-commutative 93.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -230 < z < 17.5

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

      3. remove-double-neg 99.9%

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

      4. neg-mul-1 99.9%

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

      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. associate-*l* 99.9%

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

      11. neg-mul-1 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%

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

      15. neg-mul-1 99.9%

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

      16. remove-double-neg 99.9%

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

      17. associate-*r/ 99.9%

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

      18. distribute-lft-neg-out 99.9%

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

      19. neg-mul-1 99.9%

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

      20. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 98.7%

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

   if 17.5 < z

   1. Initial program 95.8%

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

      1. *-lft-identity 95.8%

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

      2. associate-/l* 95.8%

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

      3. remove-double-neg 95.8%

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

      4. neg-mul-1 95.8%

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

      5. associate-/r* 95.8%

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

      6. div-sub 95.8%

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

      7. metadata-eval 95.8%

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

      8. associate-/l* 95.8%

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

      9. *-commutative 95.8%

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

      10. associate-*l* 95.8%

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

      11. neg-mul-1 95.8%

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

      12. /-rgt-identity 95.8%

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

      13. div-sub 95.8%

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

      14. associate-/r* 95.8%

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

      15. neg-mul-1 95.8%

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

      16. remove-double-neg 95.8%

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

      17. associate-*r/ 95.8%

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

      18. distribute-lft-neg-out 95.8%

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

      19. neg-mul-1 95.8%

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

      20. *-commutative 95.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -230:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 17.5:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-19}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.52e-105) x (if (<= z 5.6e-19) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.52e-105) {
		tmp = x;
	} else if (z <= 5.6e-19) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.52d-105)) then
        tmp = x
    else if (z <= 5.6d-19) then
        tmp = x + (y * 0.8862269254527579d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.52e-105) {
		tmp = x;
	} else if (z <= 5.6e-19) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.52e-105:
		tmp = x
	elif z <= 5.6e-19:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.52e-105)
		tmp = x;
	elseif (z <= 5.6e-19)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.52e-105)
		tmp = x;
	elseif (z <= 5.6e-19)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.52e-105], x, If[LessEqual[z, 5.6e-19], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.52e-105 or 5.60000000000000005e-19 < z

   1. Initial program 95.9%

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

      1. *-lft-identity 95.9%

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

      2. associate-/l* 96.0%

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

      3. remove-double-neg 96.0%

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

      4. neg-mul-1 96.0%

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

      5. associate-/r* 96.0%

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

      6. div-sub 96.0%

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

      7. metadata-eval 96.0%

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

      8. associate-/l* 96.0%

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

      9. *-commutative 96.0%

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

      10. associate-*l* 96.0%

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

      11. neg-mul-1 96.0%

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

      12. /-rgt-identity 96.0%

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

      13. div-sub 96.0%

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

      14. associate-/r* 96.0%

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

      15. neg-mul-1 96.0%

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

      16. remove-double-neg 96.0%

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

      17. associate-*r/ 96.0%

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

      18. distribute-lft-neg-out 96.0%

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

      19. neg-mul-1 96.0%

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

      20. *-commutative 96.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 82.8%

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

   if -1.52e-105 < z < 5.60000000000000005e-19

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

      3. remove-double-neg 99.9%

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

      4. neg-mul-1 99.9%

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

      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. associate-*l* 99.9%

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

      11. neg-mul-1 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%

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

      15. neg-mul-1 99.9%

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

      16. remove-double-neg 99.9%

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

      17. associate-*r/ 99.9%

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

      18. distribute-lft-neg-out 99.9%

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

      19. neg-mul-1 99.9%

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

      20. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.9%

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

   6. Taylor expanded in y around 0 81.0%

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

      1. *-commutative 81.0%

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

   8. Simplified 81.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-19}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 10^{-18}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.35e-105) x (if (<= z 1e-18) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e-105) {
		tmp = x;
	} else if (z <= 1e-18) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.35d-105)) then
        tmp = x
    else if (z <= 1d-18) then
        tmp = x + (y / 1.1283791670955126d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e-105) {
		tmp = x;
	} else if (z <= 1e-18) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.35e-105:
		tmp = x
	elif z <= 1e-18:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.35e-105)
		tmp = x;
	elseif (z <= 1e-18)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.35e-105)
		tmp = x;
	elseif (z <= 1e-18)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.35e-105], x, If[LessEqual[z, 1e-18], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.34999999999999996e-105 or 1.0000000000000001e-18 < z

   1. Initial program 95.9%

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

      1. *-lft-identity 95.9%

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

      2. associate-/l* 96.0%

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

      3. remove-double-neg 96.0%

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

      4. neg-mul-1 96.0%

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

      5. associate-/r* 96.0%

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

      6. div-sub 96.0%

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

      7. metadata-eval 96.0%

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

      8. associate-/l* 96.0%

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

      9. *-commutative 96.0%

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

      10. associate-*l* 96.0%

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

      11. neg-mul-1 96.0%

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

      12. /-rgt-identity 96.0%

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

      13. div-sub 96.0%

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

      14. associate-/r* 96.0%

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

      15. neg-mul-1 96.0%

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

      16. remove-double-neg 96.0%

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

      17. associate-*r/ 96.0%

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

      18. distribute-lft-neg-out 96.0%

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

      19. neg-mul-1 96.0%

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

      20. *-commutative 96.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 82.8%

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

   if -1.34999999999999996e-105 < z < 1.0000000000000001e-18

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 81.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 10^{-18}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-30)
   (+ x (/ -1.0 x))
   (if (<= z 5.5e-19) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-30) {
		tmp = x + (-1.0 / x);
	} else if (z <= 5.5e-19) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1d-30)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 5.5d-19) then
        tmp = x + (y / 1.1283791670955126d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-30) {
		tmp = x + (-1.0 / x);
	} else if (z <= 5.5e-19) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1e-30:
		tmp = x + (-1.0 / x)
	elif z <= 5.5e-19:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-30)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 5.5e-19)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e-30)
		tmp = x + (-1.0 / x);
	elseif (z <= 5.5e-19)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e-30], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.5e-19], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1e-30

   1. Initial program 94.5%

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

      1. *-lft-identity 94.5%

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

      2. associate-/l* 94.6%

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

      3. remove-double-neg 94.6%

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

      4. neg-mul-1 94.6%

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

      5. associate-/r* 94.6%

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

      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. associate-*l* 94.7%

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

      11. neg-mul-1 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 94.7%

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

      14. associate-/r* 94.7%

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

      15. neg-mul-1 94.7%

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

      16. remove-double-neg 94.7%

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

      17. associate-*r/ 94.7%

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

      18. distribute-lft-neg-out 94.7%

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

      19. neg-mul-1 94.7%

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

      20. *-commutative 94.7%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.4%

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

   if -1e-30 < z < 5.4999999999999996e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 99.9%

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

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 80.0%

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

   if 5.4999999999999996e-19 < z

   1. Initial program 96.1%

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

      1. *-lft-identity 96.1%

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

      2. associate-/l* 96.1%

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

      3. remove-double-neg 96.1%

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

      4. neg-mul-1 96.1%

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

      5. associate-/r* 96.1%

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

      6. div-sub 96.1%

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

      7. metadata-eval 96.1%

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

      8. associate-/l* 96.1%

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

      9. *-commutative 96.1%

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

      10. associate-*l* 96.1%

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

      11. neg-mul-1 96.1%

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

      12. /-rgt-identity 96.1%

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

      13. div-sub 96.1%

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

      14. associate-/r* 96.1%

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

      15. neg-mul-1 96.1%

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

      16. remove-double-neg 96.1%

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

      17. associate-*r/ 96.1%

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

      18. distribute-lft-neg-out 96.1%

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

      19. neg-mul-1 96.1%

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

      20. *-commutative 96.1%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-228}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-194}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.9e-228) x (if (<= x 4.1e-194) (/ y 1.1283791670955126) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e-228) {
		tmp = x;
	} else if (x <= 4.1e-194) {
		tmp = y / 1.1283791670955126;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.9d-228)) then
        tmp = x
    else if (x <= 4.1d-194) then
        tmp = y / 1.1283791670955126d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e-228) {
		tmp = x;
	} else if (x <= 4.1e-194) {
		tmp = y / 1.1283791670955126;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.9e-228:
		tmp = x
	elif x <= 4.1e-194:
		tmp = y / 1.1283791670955126
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.9e-228)
		tmp = x;
	elseif (x <= 4.1e-194)
		tmp = Float64(y / 1.1283791670955126);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.9e-228)
		tmp = x;
	elseif (x <= 4.1e-194)
		tmp = y / 1.1283791670955126;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.9e-228], x, If[LessEqual[x, 4.1e-194], N[(y / 1.1283791670955126), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.90000000000000029e-228 or 4.1000000000000003e-194 < x

   1. Initial program 97.7%

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

      1. *-lft-identity 97.7%

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

      2. associate-/l* 97.7%

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

      3. remove-double-neg 97.7%

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

      4. neg-mul-1 97.7%

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

      5. associate-/r* 97.7%

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

      6. div-sub 97.7%

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

      7. metadata-eval 97.7%

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

      8. associate-/l* 97.7%

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

      9. *-commutative 97.7%

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

      10. associate-*l* 97.7%

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

      11. neg-mul-1 97.7%

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

      12. /-rgt-identity 97.7%

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

      13. div-sub 97.7%

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

      14. associate-/r* 97.7%

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

      15. neg-mul-1 97.7%

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

      16. remove-double-neg 97.7%

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

      17. associate-*r/ 97.7%

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

      18. distribute-lft-neg-out 97.7%

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

      19. neg-mul-1 97.7%

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

      20. *-commutative 97.7%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 82.6%

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

   if -3.90000000000000029e-228 < x < 4.1000000000000003e-194

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0 72.2%

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

   4. Taylor expanded in y around 0 61.9%

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

   5. Taylor expanded in x around 0 52.2%

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

   6. Taylor expanded in z around 0 51.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-228}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-194}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.7% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 97.5%

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

   1. *-lft-identity 97.5%

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

   2. associate-/l* 97.5%

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

   3. remove-double-neg 97.5%

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

   4. neg-mul-1 97.5%

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

   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. associate-*l* 97.5%

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

   11. neg-mul-1 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%

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

   15. neg-mul-1 97.5%

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

   16. remove-double-neg 97.5%

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

   17. associate-*r/ 97.5%

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

   18. distribute-lft-neg-out 97.5%

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

   19. neg-mul-1 97.5%

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

   20. *-commutative 97.5%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 74.4%

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

6. Final simplification 74.4%

   \[\leadsto x \]
7. Add Preprocessing

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 2024021 
(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)))))