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

Percentage Accurate: 95.4% → 99.6%

Time: 7.0s

Alternatives: 9

Speedup: 8.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.000000002:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot \left(1 + z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot 0.8862269254527579}{e^{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.000000002)
     (+ x (/ y (- (* 1.1283791670955126 (+ 1.0 z)) (* x y))))
     (+ x (/ (* y 0.8862269254527579) (exp z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.000000002) {
		tmp = x + (y / ((1.1283791670955126 * (1.0 + z)) - (x * y)));
	} else {
		tmp = x + ((y * 0.8862269254527579) / exp(z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.000000002d0) then
        tmp = x + (y / ((1.1283791670955126d0 * (1.0d0 + z)) - (x * y)))
    else
        tmp = x + ((y * 0.8862269254527579d0) / exp(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 1.000000002) {
		tmp = x + (y / ((1.1283791670955126 * (1.0 + z)) - (x * y)));
	} else {
		tmp = x + ((y * 0.8862269254527579) / Math.exp(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.000000002:
		tmp = x + (y / ((1.1283791670955126 * (1.0 + z)) - (x * y)))
	else:
		tmp = x + ((y * 0.8862269254527579) / math.exp(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.000000002)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * Float64(1.0 + z)) - Float64(x * y))));
	else
		tmp = Float64(x + Float64(Float64(y * 0.8862269254527579) / exp(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 1.000000002)
		tmp = x + (y / ((1.1283791670955126 * (1.0 + z)) - (x * y)));
	else
		tmp = x + ((y * 0.8862269254527579) / exp(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.000000002], N[(x + N[(y / N[(N[(1.1283791670955126 * N[(1.0 + z), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * 0.8862269254527579), $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 88.9%

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

      1. *-lft-identity 88.9%

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

      2. associate-/l* 88.9%

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

      3. div-sub 89.1%

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

      4. associate-*r/ 89.1%

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

      5. /-rgt-identity 89.1%

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

      6. metadata-eval 89.1%

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

      7. associate-/l* 89.1%

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

      8. *-commutative 89.1%

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

      9. neg-mul-1 89.1%

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

      10. associate-/l* 89.1%

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

      11. associate-*r* 89.1%

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

      12. *-commutative 89.1%

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

      13. neg-mul-1 89.1%

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

      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 y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1.00000000199999994

   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%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 1.00000000199999994 < (exp.f64 z)

   1. Initial program 92.9%

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

      1. *-lft-identity 92.9%

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

      2. associate-/l* 92.9%

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

      3. div-sub 92.9%

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

      4. associate-*r/ 92.9%

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

      5. /-rgt-identity 92.9%

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

      6. metadata-eval 92.9%

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

      7. associate-/l* 92.9%

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

      8. *-commutative 92.9%

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

      9. neg-mul-1 92.9%

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

      10. associate-/l* 92.9%

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

      11. associate-*r* 92.9%

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

      12. *-commutative 92.9%

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

      13. neg-mul-1 92.9%

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

      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 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%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.000000002:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot \left(1 + z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot 0.8862269254527579}{e^{z}}\\ \end{array} \]
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 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* 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: 99.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.02:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot \left(1 + z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e+14)
   (+ x (/ -1.0 x))
   (if (<= z 0.02)
     (+ x (/ y (- (* 1.1283791670955126 (+ 1.0 z)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+14) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.02) {
		tmp = x + (y / ((1.1283791670955126 * (1.0 + 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 <= (-2.1d+14)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.02d0) then
        tmp = x + (y / ((1.1283791670955126d0 * (1.0d0 + 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 <= -2.1e+14) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.02) {
		tmp = x + (y / ((1.1283791670955126 * (1.0 + z)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.1e+14:
		tmp = x + (-1.0 / x)
	elif z <= 0.02:
		tmp = x + (y / ((1.1283791670955126 * (1.0 + z)) - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e+14)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.02)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * Float64(1.0 + z)) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.1e+14)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.02)
		tmp = x + (y / ((1.1283791670955126 * (1.0 + z)) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e+14], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.02], N[(x + N[(y / N[(N[(1.1283791670955126 * N[(1.0 + z), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1e14

   1. Initial program 88.3%

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

      1. *-lft-identity 88.3%

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

      2. associate-/l* 88.3%

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

      3. div-sub 88.6%

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

      4. associate-*r/ 88.6%

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

      5. /-rgt-identity 88.6%

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

      6. metadata-eval 88.6%

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

      7. associate-/l* 88.6%

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

      8. *-commutative 88.6%

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

      9. neg-mul-1 88.6%

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

      10. associate-/l* 88.6%

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

      11. associate-*r* 88.6%

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

      12. *-commutative 88.6%

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

      13. neg-mul-1 88.6%

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

      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 inf 100.0%

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

   if -2.1e14 < z < 0.0200000000000000004

   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 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-rgt1-in 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 0.0200000000000000004 < z

   1. Initial program 92.6%

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

      1. *-lft-identity 92.6%

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

      2. associate-/l* 92.6%

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

      3. div-sub 92.6%

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

      4. associate-*r/ 92.6%

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

      5. /-rgt-identity 92.6%

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

      6. metadata-eval 92.6%

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

      7. associate-/l* 92.6%

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

      8. *-commutative 92.6%

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

      9. neg-mul-1 92.6%

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

      10. associate-/l* 92.6%

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

      11. associate-*r* 92.6%

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

      12. *-commutative 92.6%

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

      13. neg-mul-1 92.6%

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

      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 z around 0 62.9%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.02:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot \left(1 + z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.02:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e+14)
   (+ x (/ -1.0 x))
   (if (<= z 0.02) (+ x (/ 1.0 (- (/ 1.1283791670955126 y) x))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+14) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.02) {
		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 <= (-2.1d+14)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.02d0) 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 <= -2.1e+14) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.02) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.1e+14:
		tmp = x + (-1.0 / x)
	elif z <= 0.02:
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e+14)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.02)
		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 <= -2.1e+14)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.02)
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e+14], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.02], 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 < -2.1e14

   1. Initial program 88.3%

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

      1. *-lft-identity 88.3%

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

      2. associate-/l* 88.3%

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

      3. div-sub 88.6%

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

      4. associate-*r/ 88.6%

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

      5. /-rgt-identity 88.6%

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

      6. metadata-eval 88.6%

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

      7. associate-/l* 88.6%

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

      8. *-commutative 88.6%

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

      9. neg-mul-1 88.6%

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

      10. associate-/l* 88.6%

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

      11. associate-*r* 88.6%

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

      12. *-commutative 88.6%

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

      13. neg-mul-1 88.6%

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

      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 inf 100.0%

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

   if -2.1e14 < z < 0.0200000000000000004

   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. div-sub 99.9%

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

      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 99.1%

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

   if 0.0200000000000000004 < z

   1. Initial program 92.6%

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

      1. *-lft-identity 92.6%

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

      2. associate-/l* 92.6%

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

      3. div-sub 92.6%

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

      4. associate-*r/ 92.6%

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

      5. /-rgt-identity 92.6%

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

      6. metadata-eval 92.6%

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

      7. associate-/l* 92.6%

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

      8. *-commutative 92.6%

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

      9. neg-mul-1 92.6%

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

      10. associate-/l* 92.6%

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

      11. associate-*r* 92.6%

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

      12. *-commutative 92.6%

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

      13. neg-mul-1 92.6%

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

      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 z around 0 62.9%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.02:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.02:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e+14)
   (+ x (/ -1.0 x))
   (if (<= z 0.02) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+14) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.02) {
		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 <= (-2.1d+14)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.02d0) 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 <= -2.1e+14) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.02) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.1e+14:
		tmp = x + (-1.0 / x)
	elif z <= 0.02:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e+14)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.02)
		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 <= -2.1e+14)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.02)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e+14], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.02], 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 < -2.1e14

   1. Initial program 88.3%

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

      1. *-lft-identity 88.3%

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

      2. associate-/l* 88.3%

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

      3. div-sub 88.6%

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

      4. associate-*r/ 88.6%

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

      5. /-rgt-identity 88.6%

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

      6. metadata-eval 88.6%

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

      7. associate-/l* 88.6%

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

      8. *-commutative 88.6%

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

      9. neg-mul-1 88.6%

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

      10. associate-/l* 88.6%

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

      11. associate-*r* 88.6%

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

      12. *-commutative 88.6%

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

      13. neg-mul-1 88.6%

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

      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 inf 100.0%

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

   if -2.1e14 < z < 0.0200000000000000004

   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 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

   if 0.0200000000000000004 < z

   1. Initial program 92.6%

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

      1. *-lft-identity 92.6%

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

      2. associate-/l* 92.6%

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

      3. div-sub 92.6%

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

      4. associate-*r/ 92.6%

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

      5. /-rgt-identity 92.6%

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

      6. metadata-eval 92.6%

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

      7. associate-/l* 92.6%

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

      8. *-commutative 92.6%

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

      9. neg-mul-1 92.6%

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

      10. associate-/l* 92.6%

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

      11. associate-*r* 92.6%

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

      12. *-commutative 92.6%

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

      13. neg-mul-1 92.6%

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

      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 z around 0 62.9%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.02:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.000106:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.000106) x (if (<= z 6.2e-59) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.000106) {
		tmp = x;
	} else if (z <= 6.2e-59) {
		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 <= (-0.000106d0)) then
        tmp = x
    else if (z <= 6.2d-59) 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 <= -0.000106) {
		tmp = x;
	} else if (z <= 6.2e-59) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.000106:
		tmp = x
	elif z <= 6.2e-59:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.000106)
		tmp = x;
	elseif (z <= 6.2e-59)
		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 <= -0.000106)
		tmp = x;
	elseif (z <= 6.2e-59)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.000106], x, If[LessEqual[z, 6.2e-59], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.06e-4 or 6.19999999999999998e-59 < z

   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* 91.8%

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

      3. div-sub 91.9%

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

      4. associate-*r/ 91.9%

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

      5. /-rgt-identity 91.9%

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

      6. metadata-eval 91.9%

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

      7. associate-/l* 91.9%

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

      8. *-commutative 91.9%

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

      9. neg-mul-1 91.9%

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

      10. associate-/l* 91.9%

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

      11. associate-*r* 91.9%

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

      12. *-commutative 91.9%

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

      13. neg-mul-1 91.9%

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

      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 z around 0 69.4%

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

   6. Taylor expanded in x around inf 81.2%

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

   if -1.06e-4 < z < 6.19999999999999998e-59

   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. div-sub 99.9%

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

      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 99.9%

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

   6. Taylor expanded in y around 0 80.7%

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

      1. *-commutative 80.7%

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

   8. Simplified 80.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.000106:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-79}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-58}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e-79)
   (+ x (/ -1.0 x))
   (if (<= z 1.1e-58) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e-79) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.1e-58) {
		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 <= (-4.8d-79)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.1d-58) 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 <= -4.8e-79) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.1e-58) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.8e-79:
		tmp = x + (-1.0 / x)
	elif z <= 1.1e-58:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e-79)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.1e-58)
		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 <= -4.8e-79)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.1e-58)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e-79], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-58], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -4.80000000000000011e-79

   1. Initial program 90.6%

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

      1. *-lft-identity 90.6%

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

      2. associate-/l* 90.6%

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

      3. div-sub 90.8%

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

      4. associate-*r/ 90.7%

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

      5. /-rgt-identity 90.7%

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

      6. metadata-eval 90.7%

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

      7. associate-/l* 90.7%

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

      8. *-commutative 90.7%

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

      9. neg-mul-1 90.7%

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

      10. associate-/l* 90.7%

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

      11. associate-*r* 90.7%

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

      12. *-commutative 90.7%

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

      13. neg-mul-1 90.7%

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

      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 y around inf 94.0%

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

   if -4.80000000000000011e-79 < z < 1.10000000000000003e-58

   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. div-sub 99.9%

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

      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 99.9%

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

   6. Taylor expanded in y around 0 82.8%

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

      1. *-commutative 82.8%

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

   8. Simplified 82.8%

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

   if 1.10000000000000003e-58 < z

   1. Initial program 94.0%

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

      1. *-lft-identity 94.0%

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

      2. associate-/l* 94.0%

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

      3. div-sub 94.0%

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

      4. associate-*r/ 93.9%

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

      5. /-rgt-identity 93.9%

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

      6. metadata-eval 93.9%

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

      7. associate-/l* 93.9%

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

      8. *-commutative 93.9%

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

      9. neg-mul-1 93.9%

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

      10. associate-/l* 93.9%

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

      11. associate-*r* 93.9%

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

      12. *-commutative 93.9%

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

      13. neg-mul-1 93.9%

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

      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 z around 0 68.4%

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

   6. Taylor expanded in x around inf 95.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-79}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-58}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-75}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-60}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.5e-75)
   (+ x (/ -1.0 x))
   (if (<= z 1.15e-60) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.5e-75) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.15e-60) {
		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 <= (-3.5d-75)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.15d-60) 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 <= -3.5e-75) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.15e-60) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.5e-75:
		tmp = x + (-1.0 / x)
	elif z <= 1.15e-60:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.5e-75)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.15e-60)
		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 <= -3.5e-75)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.15e-60)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.5e-75], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e-60], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -3.49999999999999985e-75

   1. Initial program 90.6%

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

      1. *-lft-identity 90.6%

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

      2. associate-/l* 90.6%

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

      3. div-sub 90.8%

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

      4. associate-*r/ 90.7%

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

      5. /-rgt-identity 90.7%

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

      6. metadata-eval 90.7%

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

      7. associate-/l* 90.7%

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

      8. *-commutative 90.7%

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

      9. neg-mul-1 90.7%

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

      10. associate-/l* 90.7%

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

      11. associate-*r* 90.7%

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

      12. *-commutative 90.7%

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

      13. neg-mul-1 90.7%

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

      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 y around inf 94.0%

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

   if -3.49999999999999985e-75 < z < 1.1500000000000001e-60

   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. div-sub 99.9%

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

      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 99.9%

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

   6. Taylor expanded in y around 0 82.8%

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

      1. *-commutative 82.8%

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

   8. Simplified 82.8%

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

      1. metadata-eval 82.8%

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

      2. div-inv 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 1.1500000000000001e-60 < z

   1. Initial program 94.0%

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

      1. *-lft-identity 94.0%

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

      2. associate-/l* 94.0%

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

      3. div-sub 94.0%

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

      4. associate-*r/ 93.9%

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

      5. /-rgt-identity 93.9%

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

      6. metadata-eval 93.9%

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

      7. associate-/l* 93.9%

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

      8. *-commutative 93.9%

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

      9. neg-mul-1 93.9%

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

      10. associate-/l* 93.9%

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

      11. associate-*r* 93.9%

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

      12. *-commutative 93.9%

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

      13. neg-mul-1 93.9%

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

      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 z around 0 68.4%

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

   6. Taylor expanded in x around inf 95.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-75}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-60}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.1% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 95.4%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
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* 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 z around 0 82.8%

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

6. Taylor expanded in x around inf 74.4%

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

7. Final simplification 74.4%

   \[\leadsto x \]
8. 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 2024010 
(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)))))