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

Percentage Accurate: 95.7% → 99.9%

Time: 11.7s

Alternatives: 10

Speedup: 8.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

   1. remove-double-neg 96.9%

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

   2. neg-mul-1 96.9%

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

   3. associate-/l* 96.9%

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

   4. neg-mul-1 96.9%

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

   5. associate-/r* 96.9%

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

   6. div-sub 97.0%

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

   7. metadata-eval 97.0%

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

   8. associate-/l* 97.0%

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

   9. *-commutative 97.0%

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

   10. neg-mul-1 97.0%

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

   11. distribute-lft-neg-out 97.0%

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

   12. /-rgt-identity 97.0%

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

   13. div-sub 97.0%

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

   14. associate-/r* 97.0%

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

   15. neg-mul-1 97.0%

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

   16. *-rgt-identity 97.0%

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

   17. times-frac 97.0%

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

   18. /-rgt-identity 97.0%

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

   19. *-commutative 97.0%

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

   20. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_0 2e+194) t_0 (+ x (/ -1.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 2e+194) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / 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 + (y / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    if (t_0 <= 2d+194) then
        tmp = t_0
    else
        tmp = x + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 2e+194) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	tmp = 0
	if t_0 <= 2e+194:
		tmp = t_0
	else:
		tmp = x + (-1.0 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_0 <= 2e+194)
		tmp = t_0;
	else
		tmp = Float64(x + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	tmp = 0.0;
	if (t_0 <= 2e+194)
		tmp = t_0;
	else
		tmp = x + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+194], t$95$0, N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))) < 1.99999999999999989e194

   1. Initial program 99.5%

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

   if 1.99999999999999989e194 < (+.f64 x (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))

   1. Initial program 83.7%

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

      1. remove-double-neg 83.7%

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

      2. neg-mul-1 83.7%

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

      3. associate-/l* 83.7%

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

      4. neg-mul-1 83.7%

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

      5. associate-/r* 83.7%

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

      6. div-sub 84.0%

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

      7. metadata-eval 84.0%

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

      8. associate-/l* 84.0%

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

      9. *-commutative 84.0%

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

      10. neg-mul-1 84.0%

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

      11. distribute-lft-neg-out 84.0%

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

      12. /-rgt-identity 84.0%

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

      13. div-sub 84.0%

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

      14. associate-/r* 84.0%

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

      15. neg-mul-1 84.0%

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

      16. *-rgt-identity 84.0%

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

      17. times-frac 84.0%

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

      18. /-rgt-identity 84.0%

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

      19. *-commutative 84.0%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{x - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 2 \cdot 10^{+194}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.005:\\ \;\;\;\;x + \frac{-1}{\frac{-1.1283791670955126}{y} + \left(x + \frac{z \cdot -1.1283791670955126}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.005)
     (+
      x
      (/
       -1.0
       (+ (/ -1.1283791670955126 y) (+ x (/ (* z -1.1283791670955126) y)))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.005) {
		tmp = x + (-1.0 / ((-1.1283791670955126 / y) + (x + ((z * -1.1283791670955126) / y))));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.005d0) then
        tmp = x + ((-1.0d0) / (((-1.1283791670955126d0) / y) + (x + ((z * (-1.1283791670955126d0)) / y))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 1.005) {
		tmp = x + (-1.0 / ((-1.1283791670955126 / y) + (x + ((z * -1.1283791670955126) / y))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.005:
		tmp = x + (-1.0 / ((-1.1283791670955126 / y) + (x + ((z * -1.1283791670955126) / y))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.005)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(-1.1283791670955126 / y) + Float64(x + Float64(Float64(z * -1.1283791670955126) / y)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 1.005)
		tmp = x + (-1.0 / ((-1.1283791670955126 / y) + (x + ((z * -1.1283791670955126) / y))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.005], N[(x + N[(-1.0 / N[(N[(-1.1283791670955126 / y), $MachinePrecision] + N[(x + N[(N[(z * -1.1283791670955126), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 90.4%

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

      1. remove-double-neg 90.4%

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

      2. neg-mul-1 90.4%

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

      3. associate-/l* 90.5%

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

      4. neg-mul-1 90.5%

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

      5. associate-/r* 90.5%

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

      6. div-sub 90.7%

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

      7. metadata-eval 90.7%

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

      8. associate-/l* 90.7%

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

      9. *-commutative 90.7%

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

      10. neg-mul-1 90.7%

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

      11. distribute-lft-neg-out 90.7%

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

      12. /-rgt-identity 90.7%

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

      13. div-sub 90.7%

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

      14. associate-/r* 90.7%

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

      15. neg-mul-1 90.7%

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

      16. *-rgt-identity 90.7%

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

      17. times-frac 90.7%

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

      18. /-rgt-identity 90.7%

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

      19. *-commutative 90.7%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1.0049999999999999

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.9%

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

      14. associate-/r* 99.9%

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

      15. neg-mul-1 99.9%

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

      16. *-rgt-identity 99.9%

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

      17. times-frac 99.9%

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

      18. /-rgt-identity 99.9%

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

      19. *-commutative 99.9%

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

      20. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. associate-*r/ 99.5%

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

      3. metadata-eval 99.5%

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

      4. associate-*r/ 99.5%

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

      5. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

   if 1.0049999999999999 < (exp.f64 z)

   1. Initial program 97.0%

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

      1. remove-double-neg 97.0%

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

      2. neg-mul-1 97.0%

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

      3. associate-/l* 97.0%

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

      4. neg-mul-1 97.0%

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

      5. associate-/r* 97.0%

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

      6. div-sub 97.0%

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

      7. metadata-eval 97.0%

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

      8. associate-/l* 97.0%

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

      9. *-commutative 97.0%

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

      10. neg-mul-1 97.0%

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

      11. distribute-lft-neg-out 97.0%

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

      12. /-rgt-identity 97.0%

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

      13. div-sub 97.0%

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

      14. associate-/r* 97.0%

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

      15. neg-mul-1 97.0%

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

      16. *-rgt-identity 97.0%

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

      17. times-frac 97.0%

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

      18. /-rgt-identity 97.0%

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

      19. *-commutative 97.0%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 42.7%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.005:\\ \;\;\;\;x + \frac{-1}{\frac{-1.1283791670955126}{y} + \left(x + \frac{z \cdot -1.1283791670955126}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 99.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -65000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.008:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -65000.0)
   (+ x (/ -1.0 x))
   (if (<= z 0.008)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -65000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.008) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-65000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.008d0) then
        tmp = x + (y / ((1.1283791670955126d0 + (z * 1.1283791670955126d0)) - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -65000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.008) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -65000.0:
		tmp = x + (-1.0 / x)
	elif z <= 0.008:
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -65000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.008)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 + Float64(z * 1.1283791670955126)) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -65000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.008)
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -65000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.008], N[(x + N[(y / N[(N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -65000

   1. Initial program 90.1%

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

      1. remove-double-neg 90.1%

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

      2. neg-mul-1 90.1%

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

      3. associate-/l* 90.1%

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

      4. neg-mul-1 90.1%

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

      5. associate-/r* 90.1%

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

      6. div-sub 90.3%

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

      7. metadata-eval 90.3%

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

      8. associate-/l* 90.3%

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

      9. *-commutative 90.3%

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

      10. neg-mul-1 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. /-rgt-identity 90.3%

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

      13. div-sub 90.3%

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

      14. associate-/r* 90.3%

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

      15. neg-mul-1 90.3%

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

      16. *-rgt-identity 90.3%

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

      17. times-frac 90.3%

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

      18. /-rgt-identity 90.3%

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

      19. *-commutative 90.3%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -65000 < z < 0.0080000000000000002

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.5%

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

   if 0.0080000000000000002 < z

   1. Initial program 97.0%

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

      1. remove-double-neg 97.0%

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

      2. neg-mul-1 97.0%

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

      3. associate-/l* 97.0%

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

      4. neg-mul-1 97.0%

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

      5. associate-/r* 97.0%

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

      6. div-sub 97.0%

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

      7. metadata-eval 97.0%

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

      8. associate-/l* 97.0%

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

      9. *-commutative 97.0%

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

      10. neg-mul-1 97.0%

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

      11. distribute-lft-neg-out 97.0%

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

      12. /-rgt-identity 97.0%

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

      13. div-sub 97.0%

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

      14. associate-/r* 97.0%

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

      15. neg-mul-1 97.0%

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

      16. *-rgt-identity 97.0%

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

      17. times-frac 97.0%

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

      18. /-rgt-identity 97.0%

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

      19. *-commutative 97.0%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 42.7%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -65000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.008:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 99.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -65000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.008:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -65000.0)
   (+ x (/ -1.0 x))
   (if (<= z 0.008) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -65000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.008) {
		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 <= (-65000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.008d0) 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 <= -65000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.008) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -65000.0:
		tmp = x + (-1.0 / x)
	elif z <= 0.008:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -65000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.008)
		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 <= -65000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.008)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -65000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.008], 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 < -65000

   1. Initial program 90.1%

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

      1. remove-double-neg 90.1%

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

      2. neg-mul-1 90.1%

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

      3. associate-/l* 90.1%

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

      4. neg-mul-1 90.1%

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

      5. associate-/r* 90.1%

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

      6. div-sub 90.3%

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

      7. metadata-eval 90.3%

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

      8. associate-/l* 90.3%

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

      9. *-commutative 90.3%

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

      10. neg-mul-1 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. /-rgt-identity 90.3%

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

      13. div-sub 90.3%

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

      14. associate-/r* 90.3%

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

      15. neg-mul-1 90.3%

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

      16. *-rgt-identity 90.3%

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

      17. times-frac 90.3%

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

      18. /-rgt-identity 90.3%

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

      19. *-commutative 90.3%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -65000 < z < 0.0080000000000000002

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.3%

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

   if 0.0080000000000000002 < z

   1. Initial program 97.0%

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

      1. remove-double-neg 97.0%

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

      2. neg-mul-1 97.0%

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

      3. associate-/l* 97.0%

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

      4. neg-mul-1 97.0%

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

      5. associate-/r* 97.0%

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

      6. div-sub 97.0%

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

      7. metadata-eval 97.0%

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

      8. associate-/l* 97.0%

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

      9. *-commutative 97.0%

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

      10. neg-mul-1 97.0%

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

      11. distribute-lft-neg-out 97.0%

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

      12. /-rgt-identity 97.0%

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

      13. div-sub 97.0%

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

      14. associate-/r* 97.0%

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

      15. neg-mul-1 97.0%

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

      16. *-rgt-identity 97.0%

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

      17. times-frac 97.0%

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

      18. /-rgt-identity 97.0%

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

      19. *-commutative 97.0%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 42.7%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -65000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.008:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 99.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -65000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.008:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -65000.0)
   (+ x (/ -1.0 x))
   (if (<= z 0.008) (+ x (/ -1.0 (+ x (/ -1.1283791670955126 y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -65000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.008) {
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-65000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 0.008d0) then
        tmp = x + ((-1.0d0) / (x + ((-1.1283791670955126d0) / y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -65000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 0.008) {
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -65000.0:
		tmp = x + (-1.0 / x)
	elif z <= 0.008:
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -65000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 0.008)
		tmp = Float64(x + Float64(-1.0 / Float64(x + Float64(-1.1283791670955126 / y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -65000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 0.008)
		tmp = x + (-1.0 / (x + (-1.1283791670955126 / y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -65000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.008], N[(x + N[(-1.0 / N[(x + N[(-1.1283791670955126 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -65000

   1. Initial program 90.1%

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

      1. remove-double-neg 90.1%

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

      2. neg-mul-1 90.1%

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

      3. associate-/l* 90.1%

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

      4. neg-mul-1 90.1%

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

      5. associate-/r* 90.1%

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

      6. div-sub 90.3%

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

      7. metadata-eval 90.3%

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

      8. associate-/l* 90.3%

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

      9. *-commutative 90.3%

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

      10. neg-mul-1 90.3%

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

      11. distribute-lft-neg-out 90.3%

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

      12. /-rgt-identity 90.3%

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

      13. div-sub 90.3%

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

      14. associate-/r* 90.3%

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

      15. neg-mul-1 90.3%

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

      16. *-rgt-identity 90.3%

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

      17. times-frac 90.3%

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

      18. /-rgt-identity 90.3%

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

      19. *-commutative 90.3%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -65000 < z < 0.0080000000000000002

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.9%

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

      14. associate-/r* 99.9%

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

      15. neg-mul-1 99.9%

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

      16. *-rgt-identity 99.9%

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

      17. times-frac 99.9%

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

      18. /-rgt-identity 99.9%

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

      19. *-commutative 99.9%

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

      20. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. associate-*r/ 99.4%

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

      4. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   if 0.0080000000000000002 < z

   1. Initial program 97.0%

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

      1. remove-double-neg 97.0%

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

      2. neg-mul-1 97.0%

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

      3. associate-/l* 97.0%

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

      4. neg-mul-1 97.0%

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

      5. associate-/r* 97.0%

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

      6. div-sub 97.0%

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

      7. metadata-eval 97.0%

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

      8. associate-/l* 97.0%

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

      9. *-commutative 97.0%

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

      10. neg-mul-1 97.0%

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

      11. distribute-lft-neg-out 97.0%

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

      12. /-rgt-identity 97.0%

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

      13. div-sub 97.0%

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

      14. associate-/r* 97.0%

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

      15. neg-mul-1 97.0%

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

      16. *-rgt-identity 97.0%

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

      17. times-frac 97.0%

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

      18. /-rgt-identity 97.0%

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

      19. *-commutative 97.0%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 42.7%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -65000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 0.008:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 73.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-81}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.8e-58) x (if (<= z 1.05e-81) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.8e-58) {
		tmp = x;
	} else if (z <= 1.05e-81) {
		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 <= (-3.8d-58)) then
        tmp = x
    else if (z <= 1.05d-81) 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 <= -3.8e-58) {
		tmp = x;
	} else if (z <= 1.05e-81) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.8e-58:
		tmp = x
	elif z <= 1.05e-81:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.8e-58)
		tmp = x;
	elseif (z <= 1.05e-81)
		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 <= -3.8e-58)
		tmp = x;
	elseif (z <= 1.05e-81)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.8e-58], x, If[LessEqual[z, 1.05e-81], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999997e-58 or 1.05e-81 < z

   1. Initial program 95.0%

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

      1. remove-double-neg 95.0%

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

      2. neg-mul-1 95.0%

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

      3. associate-/l* 95.0%

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

      4. neg-mul-1 95.0%

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

      5. associate-/r* 95.0%

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

      6. div-sub 95.1%

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

      7. metadata-eval 95.1%

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

      8. associate-/l* 95.1%

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

      9. *-commutative 95.1%

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

      10. neg-mul-1 95.1%

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

      11. distribute-lft-neg-out 95.1%

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

      12. /-rgt-identity 95.1%

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

      13. div-sub 95.1%

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

      14. associate-/r* 95.1%

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

      15. neg-mul-1 95.1%

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

      16. *-rgt-identity 95.1%

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

      17. times-frac 95.1%

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

      18. /-rgt-identity 95.1%

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

      19. *-commutative 95.1%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 69.8%

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

   5. Taylor expanded in x around inf 82.5%

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

   if -3.7999999999999997e-58 < z < 1.05e-81

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.8%

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

   3. Taylor expanded in x around 0 78.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-81}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 73.8% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-82}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.4e-56) x (if (<= z 1.2e-82) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.4e-56) {
		tmp = x;
	} else if (z <= 1.2e-82) {
		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 <= (-2.4d-56)) then
        tmp = x
    else if (z <= 1.2d-82) 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 <= -2.4e-56) {
		tmp = x;
	} else if (z <= 1.2e-82) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.4e-56:
		tmp = x
	elif z <= 1.2e-82:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.4e-56)
		tmp = x;
	elseif (z <= 1.2e-82)
		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 <= -2.4e-56)
		tmp = x;
	elseif (z <= 1.2e-82)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.4e-56], x, If[LessEqual[z, 1.2e-82], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -2.40000000000000001e-56 or 1.20000000000000004e-82 < z

   1. Initial program 95.0%

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

      1. remove-double-neg 95.0%

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

      2. neg-mul-1 95.0%

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

      3. associate-/l* 95.0%

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

      4. neg-mul-1 95.0%

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

      5. associate-/r* 95.0%

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

      6. div-sub 95.1%

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

      7. metadata-eval 95.1%

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

      8. associate-/l* 95.1%

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

      9. *-commutative 95.1%

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

      10. neg-mul-1 95.1%

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

      11. distribute-lft-neg-out 95.1%

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

      12. /-rgt-identity 95.1%

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

      13. div-sub 95.1%

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

      14. associate-/r* 95.1%

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

      15. neg-mul-1 95.1%

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

      16. *-rgt-identity 95.1%

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

      17. times-frac 95.1%

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

      18. /-rgt-identity 95.1%

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

      19. *-commutative 95.1%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 69.8%

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

   5. Taylor expanded in x around inf 82.5%

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

   if -2.40000000000000001e-56 < z < 1.20000000000000004e-82

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      20. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.8%

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

   5. Taylor expanded in x around 0 78.2%

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

      1. *-commutative 78.2%

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

   7. Simplified 78.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-82}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 85.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-105}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.2e-105)
   (+ x (/ -1.0 x))
   (if (<= z 5e-83) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.2e-105) {
		tmp = x + (-1.0 / x);
	} else if (z <= 5e-83) {
		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 <= (-4.2d-105)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 5d-83) 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 <= -4.2e-105) {
		tmp = x + (-1.0 / x);
	} else if (z <= 5e-83) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.2e-105:
		tmp = x + (-1.0 / x)
	elif z <= 5e-83:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.2e-105)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 5e-83)
		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 <= -4.2e-105)
		tmp = x + (-1.0 / x);
	elseif (z <= 5e-83)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.2e-105], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e-83], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2e-105

   1. Initial program 92.6%

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

      1. remove-double-neg 92.6%

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

      2. neg-mul-1 92.6%

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

      3. associate-/l* 92.6%

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

      4. neg-mul-1 92.6%

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

      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. neg-mul-1 92.7%

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

      11. distribute-lft-neg-out 92.7%

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

      12. /-rgt-identity 92.7%

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

      13. div-sub 92.7%

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

      14. associate-/r* 92.7%

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

      15. neg-mul-1 92.7%

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

      16. *-rgt-identity 92.7%

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

      17. times-frac 92.7%

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

      18. /-rgt-identity 92.7%

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

      19. *-commutative 92.7%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 94.8%

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

   if -4.2e-105 < z < 5e-83

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

      4. neg-mul-1 99.8%

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

      5. associate-/r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. *-commutative 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-out 99.8%

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

      12. /-rgt-identity 99.8%

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

      13. div-sub 99.8%

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

      14. associate-/r* 99.8%

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

      15. neg-mul-1 99.8%

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

      16. *-rgt-identity 99.8%

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

      17. times-frac 99.8%

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

      18. /-rgt-identity 99.8%

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

      19. *-commutative 99.8%

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

      20. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.8%

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

   5. Taylor expanded in x around 0 80.1%

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

      1. *-commutative 80.1%

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

   7. Simplified 80.1%

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

   if 5e-83 < z

   1. Initial program 97.6%

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

      1. remove-double-neg 97.6%

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

      2. neg-mul-1 97.6%

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

      3. associate-/l* 97.6%

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

      4. neg-mul-1 97.6%

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

      5. associate-/r* 97.6%

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

      6. div-sub 97.6%

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

      7. metadata-eval 97.6%

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

      8. associate-/l* 97.6%

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

      9. *-commutative 97.6%

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

      10. neg-mul-1 97.6%

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

      11. distribute-lft-neg-out 97.6%

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

      12. /-rgt-identity 97.6%

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

      13. div-sub 97.6%

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

      14. associate-/r* 97.6%

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

      15. neg-mul-1 97.6%

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

      16. *-rgt-identity 97.6%

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

      17. times-frac 97.6%

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

      18. /-rgt-identity 97.6%

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

      19. *-commutative 97.6%

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

      20. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 48.4%

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

   5. Taylor expanded in x around inf 95.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-105}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 69.6% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 96.9%

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

   1. remove-double-neg 96.9%

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

   2. neg-mul-1 96.9%

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

   3. associate-/l* 96.9%

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

   4. neg-mul-1 96.9%

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

   5. associate-/r* 96.9%

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

   6. div-sub 97.0%

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

   7. metadata-eval 97.0%

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

   8. associate-/l* 97.0%

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

   9. *-commutative 97.0%

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

   10. neg-mul-1 97.0%

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

   11. distribute-lft-neg-out 97.0%

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

   12. /-rgt-identity 97.0%

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

   13. div-sub 97.0%

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

   14. associate-/r* 97.0%

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

   15. neg-mul-1 97.0%

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

   16. *-rgt-identity 97.0%

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

   17. times-frac 97.0%

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

   18. /-rgt-identity 97.0%

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

   19. *-commutative 97.0%

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

   20. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 67.2%

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

5. Taylor expanded in x around inf 76.2%

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

6. Final simplification 76.2%

   \[\leadsto x \]

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / (((1.1283791670955126d0 / y) * exp(z)) - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023275 
(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)))))