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

Percentage Accurate: 95.9% → 98.8%

Time: 9.9s

Alternatives: 10

Speedup: 8.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% 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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 10^{-43}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{0.8862269254527579}{\frac{e^{z}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1200.0)
   (+ x (/ -1.0 x))
   (if (<= z 1e-43)
     (+ x (/ y (- (+ 1.1283791670955126 (* 1.1283791670955126 z)) (* x y))))
     (+ x (/ 0.8862269254527579 (/ (exp z) y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1200.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1e-43) {
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)));
	} else {
		tmp = x + (0.8862269254527579 / (exp(z) / 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 (z <= (-1200.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1d-43) then
        tmp = x + (y / ((1.1283791670955126d0 + (1.1283791670955126d0 * z)) - (x * y)))
    else
        tmp = x + (0.8862269254527579d0 / (exp(z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1200.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1e-43) {
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)));
	} else {
		tmp = x + (0.8862269254527579 / (Math.exp(z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1200.0:
		tmp = x + (-1.0 / x)
	elif z <= 1e-43:
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)))
	else:
		tmp = x + (0.8862269254527579 / (math.exp(z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1200.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1e-43)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(1.1283791670955126 * z)) - Float64(x * y))));
	else
		tmp = Float64(x + Float64(0.8862269254527579 / Float64(exp(z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1200.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 1e-43)
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)));
	else
		tmp = x + (0.8862269254527579 / (exp(z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1200.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e-43], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(1.1283791670955126 * z), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(0.8862269254527579 / N[(N[Exp[z], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1200

   1. Initial program 91.8%

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

      1. *-lft-identity 91.8%

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

      2. associate-/l* 92.0%

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

      3. div-sub 92.1%

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

      4. associate-*r/ 92.1%

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

      5. /-rgt-identity 92.1%

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

      6. metadata-eval 92.1%

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

      7. associate-/l* 92.1%

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

      8. *-commutative 92.1%

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

      9. neg-mul-1 92.1%

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

      10. associate-/l* 92.1%

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

      11. associate-*r* 92.1%

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

      12. *-commutative 92.1%

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

      13. neg-mul-1 92.1%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 -1200 < z < 1.00000000000000008e-43

   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%

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

   if 1.00000000000000008e-43 < z

   1. Initial program 95.7%

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

      1. *-lft-identity 95.7%

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

      2. associate-/l* 95.7%

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

      3. div-sub 95.7%

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

      4. associate-*r/ 95.7%

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

      5. /-rgt-identity 95.7%

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

      6. metadata-eval 95.7%

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

      7. associate-/l* 95.7%

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

      8. *-commutative 95.7%

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

      9. neg-mul-1 95.7%

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

      10. associate-/l* 95.7%

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

      11. associate-*r* 95.7%

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

      12. *-commutative 95.7%

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

      13. neg-mul-1 95.7%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 y around 0 100.0%

      \[\leadsto x + \color{blue}{0.8862269254527579 \cdot \frac{y}{e^{z}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

   7. Simplified 100.0%

      \[\leadsto x + \color{blue}{\frac{0.8862269254527579}{\frac{e^{z}}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 10^{-43}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{0.8862269254527579}{\frac{e^{z}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 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 96.4%

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

   1. *-lft-identity 96.4%

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

   2. associate-/l* 96.4%

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

   3. div-sub 96.5%

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

   4. associate-*r/ 96.5%

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

   5. /-rgt-identity 96.5%

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

   6. metadata-eval 96.5%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. neg-mul-1 96.5%

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

   10. associate-/l* 96.5%

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

   11. associate-*r* 96.5%

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

   12. *-commutative 96.5%

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

   13. neg-mul-1 96.5%

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

   14. associate-/l* 99.9%

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

   15. *-inverses 99.9%

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

   16. /-rgt-identity 99.9%

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

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 3: 99.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 44:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1200.0)
   (+ x (/ -1.0 x))
   (if (<= z 44.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* 1.1283791670955126 z)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1200.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 44.0) {
		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 (z <= (-1200.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 44.0d0) 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 (z <= -1200.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 44.0) {
		tmp = x + (y / ((1.1283791670955126 + (1.1283791670955126 * z)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1200.0:
		tmp = x + (-1.0 / x)
	elif z <= 44.0:
		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 (z <= -1200.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 44.0)
		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 (z <= -1200.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 44.0)
		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[z, -1200.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 44.0], 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 z < -1200

   1. Initial program 91.8%

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

      1. *-lft-identity 91.8%

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

      2. associate-/l* 92.0%

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

      3. div-sub 92.1%

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

      4. associate-*r/ 92.1%

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

      5. /-rgt-identity 92.1%

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

      6. metadata-eval 92.1%

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

      7. associate-/l* 92.1%

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

      8. *-commutative 92.1%

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

      9. neg-mul-1 92.1%

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

      10. associate-/l* 92.1%

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

      11. associate-*r* 92.1%

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

      12. *-commutative 92.1%

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

      13. neg-mul-1 92.1%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 -1200 < z < 44

   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%

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

   if 44 < z

   1. Initial program 95.4%

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

      1. *-lft-identity 95.4%

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

      2. associate-/l* 95.4%

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

      3. div-sub 95.4%

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

      4. associate-*r/ 95.4%

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

      5. /-rgt-identity 95.4%

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

      6. metadata-eval 95.4%

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

      7. associate-/l* 95.4%

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

      8. *-commutative 95.4%

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

      9. neg-mul-1 95.4%

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

      10. associate-/l* 95.4%

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

      11. associate-*r* 95.4%

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

      12. *-commutative 95.4%

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

      13. neg-mul-1 95.4%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 44:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{1.1283791670955126}\\ \mathbf{if}\;z \leq -0.72:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))) (t_1 (+ x (/ y 1.1283791670955126))))
   (if (<= z -0.72)
     t_0
     (if (<= z -1.22e-266)
       t_1
       (if (<= z 5.2e-181) t_0 (if (<= z 2.7e-9) t_1 x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / 1.1283791670955126);
	double tmp;
	if (z <= -0.72) {
		tmp = t_0;
	} else if (z <= -1.22e-266) {
		tmp = t_1;
	} else if (z <= 5.2e-181) {
		tmp = t_0;
	} else if (z <= 2.7e-9) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x + (y / 1.1283791670955126d0)
    if (z <= (-0.72d0)) then
        tmp = t_0
    else if (z <= (-1.22d-266)) then
        tmp = t_1
    else if (z <= 5.2d-181) then
        tmp = t_0
    else if (z <= 2.7d-9) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / 1.1283791670955126);
	double tmp;
	if (z <= -0.72) {
		tmp = t_0;
	} else if (z <= -1.22e-266) {
		tmp = t_1;
	} else if (z <= 5.2e-181) {
		tmp = t_0;
	} else if (z <= 2.7e-9) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x + (y / 1.1283791670955126)
	tmp = 0
	if z <= -0.72:
		tmp = t_0
	elif z <= -1.22e-266:
		tmp = t_1
	elif z <= 5.2e-181:
		tmp = t_0
	elif z <= 2.7e-9:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / 1.1283791670955126))
	tmp = 0.0
	if (z <= -0.72)
		tmp = t_0;
	elseif (z <= -1.22e-266)
		tmp = t_1;
	elseif (z <= 5.2e-181)
		tmp = t_0;
	elseif (z <= 2.7e-9)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x + (y / 1.1283791670955126);
	tmp = 0.0;
	if (z <= -0.72)
		tmp = t_0;
	elseif (z <= -1.22e-266)
		tmp = t_1;
	elseif (z <= 5.2e-181)
		tmp = t_0;
	elseif (z <= 2.7e-9)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.72], t$95$0, If[LessEqual[z, -1.22e-266], t$95$1, If[LessEqual[z, 5.2e-181], t$95$0, If[LessEqual[z, 2.7e-9], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.71999999999999997 or -1.22000000000000002e-266 < z < 5.19999999999999998e-181

   1. Initial program 94.7%

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

      1. *-lft-identity 94.7%

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

      2. associate-/l* 94.7%

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

      3. div-sub 94.8%

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

      4. associate-*r/ 94.8%

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

      5. /-rgt-identity 94.8%

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

      6. metadata-eval 94.8%

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

      7. associate-/l* 94.8%

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

      8. *-commutative 94.8%

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

      9. neg-mul-1 94.8%

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

      10. associate-/l* 94.8%

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

      11. associate-*r* 94.8%

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

      12. *-commutative 94.8%

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

      13. neg-mul-1 94.8%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

   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 92.9%

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

   if -0.71999999999999997 < z < -1.22000000000000002e-266 or 5.19999999999999998e-181 < z < 2.7000000000000002e-9

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 99.7%

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

      1. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 73.9%

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

   if 2.7000000000000002e-9 < z

   1. Initial program 95.4%

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

      1. *-lft-identity 95.4%

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

      2. associate-/l* 95.4%

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

      3. div-sub 95.4%

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

      4. associate-*r/ 95.4%

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

      5. /-rgt-identity 95.4%

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

      6. metadata-eval 95.4%

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

      7. associate-/l* 95.4%

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

      8. *-commutative 95.4%

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

      9. neg-mul-1 95.4%

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

      10. associate-/l* 95.4%

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

      11. associate-*r* 95.4%

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

      12. *-commutative 95.4%

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

      13. neg-mul-1 95.4%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.72:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-266}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-181}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -0.058:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-265}:\\ \;\;\;\;x + 0.8862269254527579 \cdot \frac{y}{1 + z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z -0.058)
     t_0
     (if (<= z -5e-265)
       (+ x (* 0.8862269254527579 (/ y (+ 1.0 z))))
       (if (<= z 7.5e-181)
         t_0
         (if (<= z 1.7e-7) (+ x (/ y 1.1283791670955126)) x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -0.058) {
		tmp = t_0;
	} else if (z <= -5e-265) {
		tmp = x + (0.8862269254527579 * (y / (1.0 + z)));
	} else if (z <= 7.5e-181) {
		tmp = t_0;
	} else if (z <= 1.7e-7) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    if (z <= (-0.058d0)) then
        tmp = t_0
    else if (z <= (-5d-265)) then
        tmp = x + (0.8862269254527579d0 * (y / (1.0d0 + z)))
    else if (z <= 7.5d-181) then
        tmp = t_0
    else if (z <= 1.7d-7) then
        tmp = x + (y / 1.1283791670955126d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -0.058) {
		tmp = t_0;
	} else if (z <= -5e-265) {
		tmp = x + (0.8862269254527579 * (y / (1.0 + z)));
	} else if (z <= 7.5e-181) {
		tmp = t_0;
	} else if (z <= 1.7e-7) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if z <= -0.058:
		tmp = t_0
	elif z <= -5e-265:
		tmp = x + (0.8862269254527579 * (y / (1.0 + z)))
	elif z <= 7.5e-181:
		tmp = t_0
	elif z <= 1.7e-7:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if (z <= -0.058)
		tmp = t_0;
	elseif (z <= -5e-265)
		tmp = Float64(x + Float64(0.8862269254527579 * Float64(y / Float64(1.0 + z))));
	elseif (z <= 7.5e-181)
		tmp = t_0;
	elseif (z <= 1.7e-7)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	tmp = 0.0;
	if (z <= -0.058)
		tmp = t_0;
	elseif (z <= -5e-265)
		tmp = x + (0.8862269254527579 * (y / (1.0 + z)));
	elseif (z <= 7.5e-181)
		tmp = t_0;
	elseif (z <= 1.7e-7)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.058], t$95$0, If[LessEqual[z, -5e-265], N[(x + N[(0.8862269254527579 * N[(y / N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.5e-181], t$95$0, If[LessEqual[z, 1.7e-7], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -0.0580000000000000029 or -5.0000000000000001e-265 < z < 7.5000000000000002e-181

   1. Initial program 94.7%

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

      1. *-lft-identity 94.7%

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

      2. associate-/l* 94.7%

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

      3. div-sub 94.8%

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

      4. associate-*r/ 94.8%

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

      5. /-rgt-identity 94.8%

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

      6. metadata-eval 94.8%

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

      7. associate-/l* 94.8%

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

      8. *-commutative 94.8%

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

      9. neg-mul-1 94.8%

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

      10. associate-/l* 94.8%

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

      11. associate-*r* 94.8%

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

      12. *-commutative 94.8%

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

      13. neg-mul-1 94.8%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

   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 92.9%

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

   if -0.0580000000000000029 < z < -5.0000000000000001e-265

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-/l* 99.9%

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

      3. div-sub 99.8%

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

      4. associate-*r/ 99.8%

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

      5. /-rgt-identity 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-/l* 99.8%

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

      8. *-commutative 99.8%

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

      9. neg-mul-1 99.8%

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

      10. associate-/l* 99.8%

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

      11. associate-*r* 99.8%

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

      12. *-commutative 99.8%

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

      13. neg-mul-1 99.8%

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

      14. associate-/l* 99.8%

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

      15. *-inverses 99.8%

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

      16. /-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.8%

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

      1. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 73.2%

      \[\leadsto x + \color{blue}{0.8862269254527579 \cdot \frac{y}{1 + z}} \]
   9. Step-by-step derivation

      1. +-commutative 73.2%

         \[\leadsto x + 0.8862269254527579 \cdot \frac{y}{\color{blue}{z + 1}} \]

   10. Simplified 73.2%

       \[\leadsto x + \color{blue}{0.8862269254527579 \cdot \frac{y}{z + 1}} \]

   if 7.5000000000000002e-181 < z < 1.69999999999999987e-7

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 99.7%

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

      1. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 75.7%

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

   if 1.69999999999999987e-7 < z

   1. Initial program 95.4%

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

      1. *-lft-identity 95.4%

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

      2. associate-/l* 95.4%

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

      3. div-sub 95.4%

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

      4. associate-*r/ 95.4%

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

      5. /-rgt-identity 95.4%

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

      6. metadata-eval 95.4%

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

      7. associate-/l* 95.4%

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

      8. *-commutative 95.4%

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

      9. neg-mul-1 95.4%

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

      10. associate-/l* 95.4%

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

      11. associate-*r* 95.4%

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

      12. *-commutative 95.4%

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

      13. neg-mul-1 95.4%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 4 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.058:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-265}:\\ \;\;\;\;x + 0.8862269254527579 \cdot \frac{y}{1 + z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-181}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 44:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1200.0)
   (+ x (/ -1.0 x))
   (if (<= z 44.0) (+ x (/ 1.0 (- (/ 1.1283791670955126 y) x))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1200.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 44.0) {
		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 <= (-1200.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 44.0d0) 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 <= -1200.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 44.0) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1200.0:
		tmp = x + (-1.0 / x)
	elif z <= 44.0:
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1200.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 44.0)
		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 <= -1200.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 44.0)
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1200.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 44.0], 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 < -1200

   1. Initial program 91.8%

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

      1. *-lft-identity 91.8%

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

      2. associate-/l* 92.0%

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

      3. div-sub 92.1%

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

      4. associate-*r/ 92.1%

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

      5. /-rgt-identity 92.1%

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

      6. metadata-eval 92.1%

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

      7. associate-/l* 92.1%

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

      8. *-commutative 92.1%

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

      9. neg-mul-1 92.1%

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

      10. associate-/l* 92.1%

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

      11. associate-*r* 92.1%

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

      12. *-commutative 92.1%

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

      13. neg-mul-1 92.1%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 -1200 < z < 44

   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* 99.8%

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

      3. div-sub 99.8%

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

      4. associate-*r/ 99.8%

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

      5. /-rgt-identity 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-/l* 99.8%

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

      8. *-commutative 99.8%

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

      9. neg-mul-1 99.8%

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

      10. associate-/l* 99.8%

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

      11. associate-*r* 99.8%

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

      12. *-commutative 99.8%

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

      13. neg-mul-1 99.8%

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

      14. associate-/l* 99.8%

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

      15. *-inverses 99.8%

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

      16. /-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.7%

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

   if 44 < z

   1. Initial program 95.4%

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

      1. *-lft-identity 95.4%

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

      2. associate-/l* 95.4%

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

      3. div-sub 95.4%

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

      4. associate-*r/ 95.4%

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

      5. /-rgt-identity 95.4%

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

      6. metadata-eval 95.4%

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

      7. associate-/l* 95.4%

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

      8. *-commutative 95.4%

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

      9. neg-mul-1 95.4%

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

      10. associate-/l* 95.4%

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

      11. associate-*r* 95.4%

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

      12. *-commutative 95.4%

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

      13. neg-mul-1 95.4%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 44:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 44:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1200.0)
   (+ x (/ -1.0 x))
   (if (<= z 44.0) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1200.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 44.0) {
		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 <= (-1200.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 44.0d0) 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 <= -1200.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 44.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1200.0:
		tmp = x + (-1.0 / x)
	elif z <= 44.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1200.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 44.0)
		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 <= -1200.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 44.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1200.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 44.0], 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 < -1200

   1. Initial program 91.8%

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

      1. *-lft-identity 91.8%

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

      2. associate-/l* 92.0%

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

      3. div-sub 92.1%

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

      4. associate-*r/ 92.1%

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

      5. /-rgt-identity 92.1%

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

      6. metadata-eval 92.1%

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

      7. associate-/l* 92.1%

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

      8. *-commutative 92.1%

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

      9. neg-mul-1 92.1%

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

      10. associate-/l* 92.1%

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

      11. associate-*r* 92.1%

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

      12. *-commutative 92.1%

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

      13. neg-mul-1 92.1%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 -1200 < z < 44

   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 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   if 44 < z

   1. Initial program 95.4%

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

      1. *-lft-identity 95.4%

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

      2. associate-/l* 95.4%

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

      3. div-sub 95.4%

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

      4. associate-*r/ 95.4%

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

      5. /-rgt-identity 95.4%

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

      6. metadata-eval 95.4%

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

      7. associate-/l* 95.4%

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

      8. *-commutative 95.4%

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

      9. neg-mul-1 95.4%

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

      10. associate-/l* 95.4%

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

      11. associate-*r* 95.4%

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

      12. *-commutative 95.4%

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

      13. neg-mul-1 95.4%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1200:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 44:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-94}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.5e-66) x (if (<= x 8e-94) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.5e-66) {
		tmp = x;
	} else if (x <= 8e-94) {
		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 (x <= (-5.5d-66)) then
        tmp = x
    else if (x <= 8d-94) 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 (x <= -5.5e-66) {
		tmp = x;
	} else if (x <= 8e-94) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.5e-66:
		tmp = x
	elif x <= 8e-94:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.5e-66)
		tmp = x;
	elseif (x <= 8e-94)
		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 (x <= -5.5e-66)
		tmp = x;
	elseif (x <= 8e-94)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.5e-66], x, If[LessEqual[x, 8e-94], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.50000000000000053e-66 or 7.9999999999999996e-94 < x

   1. Initial program 98.0%

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

      1. *-lft-identity 98.0%

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

      2. associate-/l* 97.9%

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

      3. div-sub 97.9%

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

      4. associate-*r/ 97.9%

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

      5. /-rgt-identity 97.9%

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

      6. metadata-eval 97.9%

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

      7. associate-/l* 97.9%

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

      8. *-commutative 97.9%

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

      9. neg-mul-1 97.9%

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

      10. associate-/l* 97.9%

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

      11. associate-*r* 97.9%

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

      12. *-commutative 97.9%

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

      13. neg-mul-1 97.9%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 92.9%

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

   if -5.50000000000000053e-66 < x < 7.9999999999999996e-94

   1. Initial program 94.3%

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

      1. *-lft-identity 94.3%

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

      2. associate-/l* 94.3%

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

      3. div-sub 94.4%

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

      4. associate-*r/ 94.4%

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

      5. /-rgt-identity 94.4%

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

      6. metadata-eval 94.4%

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

      7. associate-/l* 94.4%

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

      8. *-commutative 94.4%

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

      9. neg-mul-1 94.4%

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

      10. associate-/l* 94.4%

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

      11. associate-*r* 94.4%

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

      12. *-commutative 94.4%

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

      13. neg-mul-1 94.4%

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

      14. associate-/l* 99.8%

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

      15. *-inverses 99.8%

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

      16. /-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 63.5%

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

   6. Taylor expanded in y around 0 43.7%

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

      1. *-commutative 43.7%

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

   8. Simplified 43.7%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-94}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-94}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.8e-52) x (if (<= x 9.2e-94) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.8e-52) {
		tmp = x;
	} else if (x <= 9.2e-94) {
		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 (x <= (-5.8d-52)) then
        tmp = x
    else if (x <= 9.2d-94) 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 (x <= -5.8e-52) {
		tmp = x;
	} else if (x <= 9.2e-94) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.8e-52:
		tmp = x
	elif x <= 9.2e-94:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.8e-52)
		tmp = x;
	elseif (x <= 9.2e-94)
		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 (x <= -5.8e-52)
		tmp = x;
	elseif (x <= 9.2e-94)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.8e-52], x, If[LessEqual[x, 9.2e-94], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8000000000000003e-52 or 9.1999999999999997e-94 < x

   1. Initial program 98.0%

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

      1. *-lft-identity 98.0%

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

      2. associate-/l* 97.9%

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

      3. div-sub 97.9%

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

      4. associate-*r/ 97.9%

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

      5. /-rgt-identity 97.9%

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

      6. metadata-eval 97.9%

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

      7. associate-/l* 97.9%

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

      8. *-commutative 97.9%

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

      9. neg-mul-1 97.9%

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

      10. associate-/l* 97.9%

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

      11. associate-*r* 97.9%

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

      12. *-commutative 97.9%

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

      13. neg-mul-1 97.9%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

   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 92.9%

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

   if -5.8000000000000003e-52 < x < 9.1999999999999997e-94

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0 63.5%

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

      1. *-commutative 63.5%

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

   5. Simplified 63.5%

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

   6. Taylor expanded in y around 0 43.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-94}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.2% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 96.4%

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

   1. *-lft-identity 96.4%

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

   2. associate-/l* 96.4%

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

   3. div-sub 96.5%

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

   4. associate-*r/ 96.5%

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

   5. /-rgt-identity 96.5%

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

   6. metadata-eval 96.5%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. neg-mul-1 96.5%

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

   10. associate-/l* 96.5%

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

   11. associate-*r* 96.5%

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

   12. *-commutative 96.5%

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

   13. neg-mul-1 96.5%

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

   14. associate-/l* 99.9%

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

   15. *-inverses 99.9%

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

   16. /-rgt-identity 99.9%

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

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 66.1%

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

6. Final simplification 66.1%

   \[\leadsto x \]
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 2024011 
(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)))))