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

Percentage Accurate: 95.2% → 99.5%

Time: 5.8s

Alternatives: 8

Speedup: 5.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.2% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.0000000000002:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.0000000000002)
     (+ x (/ 1.0 (- (/ 1.1283791670955126 y) x)))
     (+ x (/ y (* 1.1283791670955126 (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) <= 1.0000000000002) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x + (y / (1.1283791670955126 * 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) <= 1.0000000000002d0) then
        tmp = x + (1.0d0 / ((1.1283791670955126d0 / y) - x))
    else
        tmp = x + (y / (1.1283791670955126d0 * 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) <= 1.0000000000002) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x + (y / (1.1283791670955126 * 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) <= 1.0000000000002:
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x))
	else:
		tmp = x + (y / (1.1283791670955126 * 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) <= 1.0000000000002)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(1.1283791670955126 / y) - x)));
	else
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 * 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) <= 1.0000000000002)
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	else
		tmp = x + (y / (1.1283791670955126 * 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], 1.0000000000002], N[(x + N[(1.0 / N[(N[(1.1283791670955126 / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 92.5%

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

      1. *-lft-identity 92.5%

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

      2. associate-/l* 92.6%

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

      3. remove-double-neg 92.6%

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

      4. neg-mul-1 92.6%

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

      5. associate-/r* 92.6%

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

      6. div-sub 92.7%

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

      7. metadata-eval 92.7%

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

      8. associate-/l* 92.7%

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

      9. *-commutative 92.7%

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

      10. associate-*l* 92.7%

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

      11. neg-mul-1 92.7%

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

      12. /-rgt-identity 92.7%

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

      13. div-sub 92.6%

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

      14. associate-/r* 92.6%

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

      15. neg-mul-1 92.6%

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

      16. remove-double-neg 92.6%

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

      17. associate-*r/ 92.6%

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

      18. distribute-lft-neg-out 92.6%

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

      19. neg-mul-1 92.6%

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

      20. *-commutative 92.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1.00000000000020006

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

      3. remove-double-neg 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. div-sub 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/l* 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

      11. neg-mul-1 99.9%

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

      12. /-rgt-identity 99.9%

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

      13. div-sub 99.9%

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

      14. associate-/r* 99.9%

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

      15. neg-mul-1 99.9%

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

      16. remove-double-neg 99.9%

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

      17. associate-*r/ 99.9%

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

      18. distribute-lft-neg-out 99.9%

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

      19. neg-mul-1 99.9%

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

      20. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.9%

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

   if 1.00000000000020006 < (exp.f64 z)

   1. Initial program 92.5%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.0000000000002:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z}}\\ \end{array} \]
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 95.9%

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

   1. *-lft-identity 95.9%

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

   2. associate-/l* 95.9%

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

   3. remove-double-neg 95.9%

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

   4. neg-mul-1 95.9%

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

   5. associate-/r* 95.9%

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

   6. div-sub 96.0%

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

   7. metadata-eval 96.0%

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

   8. associate-/l* 96.0%

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

   9. *-commutative 96.0%

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

   10. associate-*l* 96.0%

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

   11. neg-mul-1 96.0%

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

   12. /-rgt-identity 96.0%

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

   13. div-sub 95.9%

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

   14. associate-/r* 95.9%

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

   15. neg-mul-1 95.9%

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

   16. remove-double-neg 95.9%

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

   17. associate-*r/ 95.9%

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

   18. distribute-lft-neg-out 95.9%

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

   19. neg-mul-1 95.9%

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

   20. *-commutative 95.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 85.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + y \cdot 0.8862269254527579\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-281}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))) (t_1 (+ x (* y 0.8862269254527579))))
   (if (<= z -4.5e-7)
     t_0
     (if (<= z -4.4e-119)
       t_1
       (if (<= z -1.45e-281) t_0 (if (<= z 2.5e-5) t_1 x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y * 0.8862269254527579);
	double tmp;
	if (z <= -4.5e-7) {
		tmp = t_0;
	} else if (z <= -4.4e-119) {
		tmp = t_1;
	} else if (z <= -1.45e-281) {
		tmp = t_0;
	} else if (z <= 2.5e-5) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x + (y * 0.8862269254527579d0)
    if (z <= (-4.5d-7)) then
        tmp = t_0
    else if (z <= (-4.4d-119)) then
        tmp = t_1
    else if (z <= (-1.45d-281)) then
        tmp = t_0
    else if (z <= 2.5d-5) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y * 0.8862269254527579);
	double tmp;
	if (z <= -4.5e-7) {
		tmp = t_0;
	} else if (z <= -4.4e-119) {
		tmp = t_1;
	} else if (z <= -1.45e-281) {
		tmp = t_0;
	} else if (z <= 2.5e-5) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x + (y * 0.8862269254527579)
	tmp = 0
	if z <= -4.5e-7:
		tmp = t_0
	elif z <= -4.4e-119:
		tmp = t_1
	elif z <= -1.45e-281:
		tmp = t_0
	elif z <= 2.5e-5:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y * 0.8862269254527579))
	tmp = 0.0
	if (z <= -4.5e-7)
		tmp = t_0;
	elseif (z <= -4.4e-119)
		tmp = t_1;
	elseif (z <= -1.45e-281)
		tmp = t_0;
	elseif (z <= 2.5e-5)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x + (y * 0.8862269254527579);
	tmp = 0.0;
	if (z <= -4.5e-7)
		tmp = t_0;
	elseif (z <= -4.4e-119)
		tmp = t_1;
	elseif (z <= -1.45e-281)
		tmp = t_0;
	elseif (z <= 2.5e-5)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.5e-7], t$95$0, If[LessEqual[z, -4.4e-119], t$95$1, If[LessEqual[z, -1.45e-281], t$95$0, If[LessEqual[z, 2.5e-5], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.4999999999999998e-7 or -4.4000000000000001e-119 < z < -1.44999999999999995e-281

   1. Initial program 94.5%

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

      1. *-lft-identity 94.5%

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

      2. associate-/l* 94.5%

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

      3. remove-double-neg 94.5%

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

      4. neg-mul-1 94.5%

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

      5. associate-/r* 94.5%

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

      6. div-sub 94.7%

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

      7. metadata-eval 94.7%

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

      8. associate-/l* 94.7%

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

      9. *-commutative 94.7%

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

      10. associate-*l* 94.7%

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

      11. neg-mul-1 94.7%

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

      12. /-rgt-identity 94.7%

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

      13. div-sub 94.6%

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

      14. associate-/r* 94.6%

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

      15. neg-mul-1 94.6%

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

      16. remove-double-neg 94.6%

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

      17. associate-*r/ 94.6%

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

      18. distribute-lft-neg-out 94.6%

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

      19. neg-mul-1 94.6%

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

      20. *-commutative 94.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 99.0%

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

   if -4.4999999999999998e-7 < z < -4.4000000000000001e-119 or -1.44999999999999995e-281 < z < 2.50000000000000012e-5

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

      3. remove-double-neg 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. div-sub 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/l* 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

      11. neg-mul-1 99.9%

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

      12. /-rgt-identity 99.9%

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

      13. div-sub 99.9%

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

      14. associate-/r* 99.9%

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

      15. neg-mul-1 99.9%

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

      16. remove-double-neg 99.9%

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

      17. associate-*r/ 99.8%

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

      18. distribute-lft-neg-out 99.8%

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

      19. neg-mul-1 99.8%

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

      20. *-commutative 99.8%

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

   3. Simplified 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%

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

   6. Taylor expanded in y around 0 76.8%

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

      1. *-commutative 76.8%

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

   8. Simplified 76.8%

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

   if 2.50000000000000012e-5 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-119}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-281}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-117}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-281}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z -2.3e-5)
     t_0
     (if (<= z -4.6e-117)
       (+ x (* y 0.8862269254527579))
       (if (<= z -1.45e-281)
         t_0
         (if (<= z 2.7e-5) (+ 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 <= -2.3e-5) {
		tmp = t_0;
	} else if (z <= -4.6e-117) {
		tmp = x + (y * 0.8862269254527579);
	} else if (z <= -1.45e-281) {
		tmp = t_0;
	} else if (z <= 2.7e-5) {
		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 <= (-2.3d-5)) then
        tmp = t_0
    else if (z <= (-4.6d-117)) then
        tmp = x + (y * 0.8862269254527579d0)
    else if (z <= (-1.45d-281)) then
        tmp = t_0
    else if (z <= 2.7d-5) 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 <= -2.3e-5) {
		tmp = t_0;
	} else if (z <= -4.6e-117) {
		tmp = x + (y * 0.8862269254527579);
	} else if (z <= -1.45e-281) {
		tmp = t_0;
	} else if (z <= 2.7e-5) {
		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 <= -2.3e-5:
		tmp = t_0
	elif z <= -4.6e-117:
		tmp = x + (y * 0.8862269254527579)
	elif z <= -1.45e-281:
		tmp = t_0
	elif z <= 2.7e-5:
		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 <= -2.3e-5)
		tmp = t_0;
	elseif (z <= -4.6e-117)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	elseif (z <= -1.45e-281)
		tmp = t_0;
	elseif (z <= 2.7e-5)
		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 <= -2.3e-5)
		tmp = t_0;
	elseif (z <= -4.6e-117)
		tmp = x + (y * 0.8862269254527579);
	elseif (z <= -1.45e-281)
		tmp = t_0;
	elseif (z <= 2.7e-5)
		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, -2.3e-5], t$95$0, If[LessEqual[z, -4.6e-117], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.45e-281], t$95$0, If[LessEqual[z, 2.7e-5], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.3e-5 or -4.59999999999999989e-117 < z < -1.44999999999999995e-281

   1. Initial program 94.5%

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

      1. *-lft-identity 94.5%

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

      2. associate-/l* 94.5%

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

      3. remove-double-neg 94.5%

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

      4. neg-mul-1 94.5%

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

      5. associate-/r* 94.5%

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

      6. div-sub 94.7%

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

      7. metadata-eval 94.7%

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

      8. associate-/l* 94.7%

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

      9. *-commutative 94.7%

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

      10. associate-*l* 94.7%

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

      11. neg-mul-1 94.7%

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

      12. /-rgt-identity 94.7%

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

      13. div-sub 94.6%

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

      14. associate-/r* 94.6%

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

      15. neg-mul-1 94.6%

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

      16. remove-double-neg 94.6%

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

      17. associate-*r/ 94.6%

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

      18. distribute-lft-neg-out 94.6%

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

      19. neg-mul-1 94.6%

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

      20. *-commutative 94.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 99.0%

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

   if -2.3e-5 < z < -4.59999999999999989e-117

   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* 100.0%

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

      3. remove-double-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. div-sub 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l* 100.0%

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

      11. neg-mul-1 100.0%

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

      12. /-rgt-identity 100.0%

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

      13. div-sub 100.0%

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

      14. associate-/r* 100.0%

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

      15. neg-mul-1 100.0%

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

      16. remove-double-neg 100.0%

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

      17. associate-*r/ 100.0%

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

      18. distribute-lft-neg-out 100.0%

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

      19. neg-mul-1 100.0%

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

      20. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 100.0%

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

   6. Taylor expanded in y around 0 77.9%

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

      1. *-commutative 77.9%

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

   8. Simplified 77.9%

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

   if -1.44999999999999995e-281 < z < 2.6999999999999999e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 77.5%

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

   4. Taylor expanded in z around 0 76.5%

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

   if 2.6999999999999999e-5 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-117}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-281}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -115:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -115.0)
   (+ x (/ -1.0 x))
   (if (<= z 6.2e-5)
     (+ x (/ y (- (+ 1.1283791670955126 (* 1.1283791670955126 z)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -115.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 6.2e-5) {
		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 <= (-115.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 6.2d-5) 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 <= -115.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 6.2e-5) {
		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 <= -115.0:
		tmp = x + (-1.0 / x)
	elif z <= 6.2e-5:
		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 <= -115.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 6.2e-5)
		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 <= -115.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 6.2e-5)
		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, -115.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.2e-5], 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 < -115

   1. Initial program 92.5%

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

      1. *-lft-identity 92.5%

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

      2. associate-/l* 92.6%

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

      3. remove-double-neg 92.6%

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

      4. neg-mul-1 92.6%

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

      5. associate-/r* 92.6%

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

      6. div-sub 92.7%

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

      7. metadata-eval 92.7%

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

      8. associate-/l* 92.7%

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

      9. *-commutative 92.7%

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

      10. associate-*l* 92.7%

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

      11. neg-mul-1 92.7%

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

      12. /-rgt-identity 92.7%

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

      13. div-sub 92.6%

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

      14. associate-/r* 92.6%

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

      15. neg-mul-1 92.6%

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

      16. remove-double-neg 92.6%

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

      17. associate-*r/ 92.6%

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

      18. distribute-lft-neg-out 92.6%

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

      19. neg-mul-1 92.6%

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

      20. *-commutative 92.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

   if -115 < z < 6.20000000000000027e-5

   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 99.6%

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

   if 6.20000000000000027e-5 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -115:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + 1.1283791670955126 \cdot z\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -375:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -375.0)
   (+ x (/ -1.0 x))
   (if (<= z 6.2e-5) (+ x (/ 1.0 (- (/ 1.1283791670955126 y) x))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -375.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 6.2e-5) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-375.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 6.2d-5) 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 <= -375.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 6.2e-5) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -375.0:
		tmp = x + (-1.0 / x)
	elif z <= 6.2e-5:
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -375.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 6.2e-5)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(1.1283791670955126 / y) - x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -375.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 6.2e-5)
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -375.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.2e-5], N[(x + N[(1.0 / N[(N[(1.1283791670955126 / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -375

   1. Initial program 92.5%

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

      1. *-lft-identity 92.5%

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

      2. associate-/l* 92.6%

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

      3. remove-double-neg 92.6%

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

      4. neg-mul-1 92.6%

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

      5. associate-/r* 92.6%

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

      6. div-sub 92.7%

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

      7. metadata-eval 92.7%

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

      8. associate-/l* 92.7%

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

      9. *-commutative 92.7%

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

      10. associate-*l* 92.7%

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

      11. neg-mul-1 92.7%

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

      12. /-rgt-identity 92.7%

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

      13. div-sub 92.6%

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

      14. associate-/r* 92.6%

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

      15. neg-mul-1 92.6%

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

      16. remove-double-neg 92.6%

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

      17. associate-*r/ 92.6%

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

      18. distribute-lft-neg-out 92.6%

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

      19. neg-mul-1 92.6%

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

      20. *-commutative 92.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

   if -375 < z < 6.20000000000000027e-5

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

      3. remove-double-neg 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. div-sub 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/l* 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

      11. neg-mul-1 99.9%

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

      12. /-rgt-identity 99.9%

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

      13. div-sub 99.9%

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

      14. associate-/r* 99.9%

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

      15. neg-mul-1 99.9%

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

      16. remove-double-neg 99.9%

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

      17. associate-*r/ 99.9%

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

      18. distribute-lft-neg-out 99.9%

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

      19. neg-mul-1 99.9%

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

      20. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.3%

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

   if 6.20000000000000027e-5 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -375:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-122}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6e-97) x (if (<= x 3.4e-122) (+ x (* y 0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6e-97) {
		tmp = x;
	} else if (x <= 3.4e-122) {
		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 <= (-6d-97)) then
        tmp = x
    else if (x <= 3.4d-122) 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 <= -6e-97) {
		tmp = x;
	} else if (x <= 3.4e-122) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6e-97:
		tmp = x
	elif x <= 3.4e-122:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6e-97)
		tmp = x;
	elseif (x <= 3.4e-122)
		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 <= -6e-97)
		tmp = x;
	elseif (x <= 3.4e-122)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6e-97], x, If[LessEqual[x, 3.4e-122], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000048e-97 or 3.3999999999999998e-122 < x

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0 57.5%

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

   4. Taylor expanded in x around inf 93.2%

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

   if -6.00000000000000048e-97 < x < 3.3999999999999998e-122

   1. Initial program 93.1%

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

      1. *-lft-identity 93.1%

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

      2. associate-/l* 93.2%

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

      3. remove-double-neg 93.2%

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

      4. neg-mul-1 93.2%

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

      5. associate-/r* 93.2%

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

      6. div-sub 93.4%

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

      7. metadata-eval 93.4%

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

      8. associate-/l* 93.4%

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

      9. *-commutative 93.4%

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

      10. associate-*l* 93.4%

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

      11. neg-mul-1 93.4%

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

      12. /-rgt-identity 93.4%

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

      13. div-sub 93.2%

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

      14. associate-/r* 93.2%

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

      15. neg-mul-1 93.2%

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

      16. remove-double-neg 93.2%

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

      17. associate-*r/ 93.2%

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

      18. distribute-lft-neg-out 93.2%

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

      19. neg-mul-1 93.2%

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

      20. *-commutative 93.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 59.6%

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

   6. Taylor expanded in y around 0 47.6%

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

      1. *-commutative 47.6%

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

   8. Simplified 47.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-122}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 95.9%

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

3. Taylor expanded in x around 0 59.8%

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

4. Taylor expanded in x around inf 73.7%

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

5. Final simplification 73.7%

   \[\leadsto x \]
6. 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 2024036 
(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)))))