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

Percentage Accurate: 95.6% → 99.6%

Time: 7.7s

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: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 2.0)
     (+ x (/ y (- (+ 1.1283791670955126 (* z 1.1283791670955126)) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 2.0) {
		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 (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 2.0d0) 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 (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 2.0) {
		tmp = x + (y / ((1.1283791670955126 + (z * 1.1283791670955126)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 2.0:
		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 (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 2.0)
		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 (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 2.0)
		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[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], 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 (exp.f64 z) < 0.0

   1. Initial program 87.8%

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

      1. *-lft-identity 87.8%

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

      2. metadata-eval 87.8%

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

      3. times-frac 87.8%

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

      4. neg-mul-1 87.8%

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

      5. sub0-neg 87.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 87.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 88.1%

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

      8. +-commutative 88.1%

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

      9. sub-neg 88.1%

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

      10. associate-/l* 88.4%

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

      11. div-sub 88.4%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.9%

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

   if 2 < (exp.f64 z)

   1. Initial program 94.5%

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

      1. *-lft-identity 94.5%

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

      2. metadata-eval 94.5%

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

      3. times-frac 94.5%

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

      4. neg-mul-1 94.5%

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

      5. sub0-neg 94.5%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 94.5%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 94.5%

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

      8. +-commutative 94.5%

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

      9. sub-neg 94.5%

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

      10. associate-/l* 94.5%

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

      11. div-sub 94.5%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 z around 0 60.7%

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

      1. associate-*r/ 60.7%

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

      2. metadata-eval 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in y around 0 51.6%

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

   8. Taylor expanded in y around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+261}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_0 5e+261) 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 <= 5e+261) {
		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 <= 5d+261) 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 <= 5e+261) {
		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 <= 5e+261:
		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 <= 5e+261)
		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 <= 5e+261)
		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, 5e+261], 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)))) < 5.0000000000000001e261

   1. Initial program 98.8%

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

   if 5.0000000000000001e261 < (+.f64 x (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))

   1. Initial program 62.1%

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

      1. *-lft-identity 62.1%

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

      2. metadata-eval 62.1%

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

      3. times-frac 62.1%

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

      4. neg-mul-1 62.1%

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

      5. sub0-neg 62.6%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 62.6%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 62.9%

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

      8. +-commutative 62.9%

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

      9. sub-neg 62.9%

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

      10. associate-/l* 63.0%

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

      11. div-sub 63.0%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 y around inf 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 5 \cdot 10^{+261}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 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 95.8%

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

   1. *-lft-identity 95.8%

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

   2. metadata-eval 95.8%

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

   3. times-frac 95.8%

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

   4. neg-mul-1 95.8%

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

   5. sub0-neg 95.8%

      \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

   6. associate-+l- 95.8%

      \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

   7. neg-sub0 95.8%

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

   8. +-commutative 95.8%

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

   9. sub-neg 95.8%

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

   10. associate-/l* 95.9%

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

   11. div-sub 95.9%

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

   12. associate-*r/ 99.9%

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

   13. *-inverses 99.9%

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

   14. *-rgt-identity 99.9%

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

   15. associate-*l/ 99.9%

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

   16. cancel-sign-sub-inv 99.9%

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

   17. distribute-lft-neg-in 99.9%

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

   18. distribute-rgt-neg-in 99.9%

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

   19. associate-*l/ 99.9%

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

   20. distribute-rgt-neg-in 99.9%

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

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 4: 99.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+15}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 13:\\ \;\;\;\;x + \frac{-1}{x - \frac{1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.1e+15)
   (+ x (/ -1.0 x))
   (if (<= z 13.0) (+ x (/ -1.0 (- x (/ 1.1283791670955126 y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e+15) {
		tmp = x + (-1.0 / x);
	} else if (z <= 13.0) {
		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 <= (-3.1d+15)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 13.0d0) 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 <= -3.1e+15) {
		tmp = x + (-1.0 / x);
	} else if (z <= 13.0) {
		tmp = x + (-1.0 / (x - (1.1283791670955126 / y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.1e+15:
		tmp = x + (-1.0 / x)
	elif z <= 13.0:
		tmp = x + (-1.0 / (x - (1.1283791670955126 / y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.1e+15)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 13.0)
		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 <= -3.1e+15)
		tmp = x + (-1.0 / x);
	elseif (z <= 13.0)
		tmp = x + (-1.0 / (x - (1.1283791670955126 / y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.1e+15], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 13.0], 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 < -3.1e15

   1. Initial program 87.3%

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

      1. *-lft-identity 87.3%

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

      2. metadata-eval 87.3%

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

      3. times-frac 87.3%

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

      4. neg-mul-1 87.3%

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

      5. sub0-neg 87.3%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 87.3%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 87.7%

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

      8. +-commutative 87.7%

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

      9. sub-neg 87.7%

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

      10. associate-/l* 87.9%

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

      11. div-sub 87.9%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 y around inf 100.0%

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

   if -3.1e15 < z < 13

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. times-frac 99.9%

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

      4. neg-mul-1 99.9%

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

      5. sub0-neg 99.9%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 99.9%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 99.9%

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

      8. +-commutative 99.9%

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

      9. sub-neg 99.9%

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

      10. associate-/l* 99.9%

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

      11. div-sub 99.9%

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

      12. associate-*r/ 99.9%

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

      13. *-inverses 99.9%

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

      14. *-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      16. cancel-sign-sub-inv 99.9%

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

      17. distribute-lft-neg-in 99.9%

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

      18. distribute-rgt-neg-in 99.9%

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

      19. associate-*l/ 99.9%

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

      20. distribute-rgt-neg-in 99.9%

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

   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 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   if 13 < z

   1. Initial program 94.5%

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

      1. *-lft-identity 94.5%

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

      2. metadata-eval 94.5%

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

      3. times-frac 94.5%

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

      4. neg-mul-1 94.5%

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

      5. sub0-neg 94.5%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 94.5%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 94.5%

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

      8. +-commutative 94.5%

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

      9. sub-neg 94.5%

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

      10. associate-/l* 94.5%

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

      11. div-sub 94.5%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 z around 0 60.7%

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

      1. associate-*r/ 60.7%

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

      2. metadata-eval 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in y around 0 51.6%

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

   8. Taylor expanded in y around 0 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+15}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 13:\\ \;\;\;\;x + \frac{-1}{x - \frac{1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 99.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+15}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 13:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.1e+15)
   (+ x (/ -1.0 x))
   (if (<= z 13.0) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e+15) {
		tmp = x + (-1.0 / x);
	} else if (z <= 13.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3.1d+15)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 13.0d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e+15) {
		tmp = x + (-1.0 / x);
	} else if (z <= 13.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.1e+15:
		tmp = x + (-1.0 / x)
	elif z <= 13.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.1e+15)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 13.0)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.1e+15)
		tmp = x + (-1.0 / x);
	elseif (z <= 13.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.1e+15], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 13.0], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1e15

   1. Initial program 87.3%

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

      1. *-lft-identity 87.3%

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

      2. metadata-eval 87.3%

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

      3. times-frac 87.3%

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

      4. neg-mul-1 87.3%

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

      5. sub0-neg 87.3%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 87.3%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 87.7%

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

      8. +-commutative 87.7%

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

      9. sub-neg 87.7%

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

      10. associate-/l* 87.9%

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

      11. div-sub 87.9%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 y around inf 100.0%

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

   if -3.1e15 < z < 13

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.7%

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

   if 13 < z

   1. Initial program 94.5%

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

      1. *-lft-identity 94.5%

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

      2. metadata-eval 94.5%

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

      3. times-frac 94.5%

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

      4. neg-mul-1 94.5%

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

      5. sub0-neg 94.5%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 94.5%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 94.5%

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

      8. +-commutative 94.5%

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

      9. sub-neg 94.5%

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

      10. associate-/l* 94.5%

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

      11. div-sub 94.5%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 z around 0 60.7%

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

      1. associate-*r/ 60.7%

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

      2. metadata-eval 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in y around 0 51.6%

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

   8. Taylor expanded in y around 0 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+15}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 13:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 81.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-41} \lor \neg \left(z \leq -2.1 \cdot 10^{-195}\right) \land z \leq 1.15 \cdot 10^{-208}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.55e-41) (and (not (<= z -2.1e-195)) (<= z 1.15e-208)))
   (+ x (/ -1.0 x))
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.55e-41) || (!(z <= -2.1e-195) && (z <= 1.15e-208))) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.55d-41)) .or. (.not. (z <= (-2.1d-195))) .and. (z <= 1.15d-208)) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.55e-41) || (!(z <= -2.1e-195) && (z <= 1.15e-208))) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.55e-41) or (not (z <= -2.1e-195) and (z <= 1.15e-208)):
		tmp = x + (-1.0 / x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.55e-41) || (!(z <= -2.1e-195) && (z <= 1.15e-208)))
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.55e-41) || (~((z <= -2.1e-195)) && (z <= 1.15e-208)))
		tmp = x + (-1.0 / x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.55e-41], And[N[Not[LessEqual[z, -2.1e-195]], $MachinePrecision], LessEqual[z, 1.15e-208]]], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.55e-41 or -2.1e-195 < z < 1.14999999999999998e-208

   1. Initial program 93.4%

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

      1. *-lft-identity 93.4%

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

      2. metadata-eval 93.4%

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

      3. times-frac 93.4%

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

      4. neg-mul-1 93.4%

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

      5. sub0-neg 93.4%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 93.4%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 93.6%

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

      8. +-commutative 93.6%

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

      9. sub-neg 93.6%

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

      10. associate-/l* 93.7%

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

      11. div-sub 93.7%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 y around inf 84.8%

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

   if -1.55e-41 < z < -2.1e-195 or 1.14999999999999998e-208 < z

   1. Initial program 97.4%

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

      1. *-lft-identity 97.4%

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

      2. metadata-eval 97.4%

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

      3. times-frac 97.4%

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

      4. neg-mul-1 97.4%

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

      5. sub0-neg 97.4%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 97.4%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 97.4%

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

      8. +-commutative 97.4%

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

      9. sub-neg 97.4%

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

      10. associate-/l* 97.3%

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

      11. div-sub 97.3%

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

      12. associate-*r/ 99.9%

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

      13. *-inverses 99.9%

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

      14. *-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      16. cancel-sign-sub-inv 99.9%

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

      17. distribute-lft-neg-in 99.9%

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

      18. distribute-rgt-neg-in 99.9%

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

      19. associate-*l/ 99.9%

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

      20. distribute-rgt-neg-in 99.9%

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

   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 81.3%

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

      1. associate-*r/ 81.3%

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

      2. metadata-eval 81.3%

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

   6. Simplified 81.3%

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

   7. Taylor expanded in y around 0 63.3%

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

   8. Taylor expanded in y around 0 80.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-41} \lor \neg \left(z \leq -2.1 \cdot 10^{-195}\right) \land z \leq 1.15 \cdot 10^{-208}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 85.2% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-172}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.1e-172)
   (+ x (/ -1.0 x))
   (if (<= z 1.05e-29) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e-172) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.05e-29) {
		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 <= (-3.1d-172)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 1.05d-29) 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 <= -3.1e-172) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.05e-29) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.1e-172:
		tmp = x + (-1.0 / x)
	elif z <= 1.05e-29:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.1e-172)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.05e-29)
		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 <= -3.1e-172)
		tmp = x + (-1.0 / x);
	elseif (z <= 1.05e-29)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.1e-172], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e-29], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1000000000000003e-172

   1. Initial program 92.7%

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

      1. *-lft-identity 92.7%

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

      2. metadata-eval 92.7%

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

      3. times-frac 92.7%

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

      4. neg-mul-1 92.7%

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

      5. sub0-neg 92.7%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 92.7%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 92.9%

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

      8. +-commutative 92.9%

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

      9. sub-neg 92.9%

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

      10. associate-/l* 93.0%

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

      11. div-sub 93.0%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 y around inf 89.4%

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

   if -3.1000000000000003e-172 < z < 1.04999999999999995e-29

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.9%

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

   3. Taylor expanded in y around 0 77.4%

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

   if 1.04999999999999995e-29 < z

   1. Initial program 95.2%

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

      1. *-lft-identity 95.2%

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

      2. metadata-eval 95.2%

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

      3. times-frac 95.2%

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

      4. neg-mul-1 95.2%

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

      5. sub0-neg 95.2%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 95.2%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 95.2%

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

      8. +-commutative 95.2%

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

      9. sub-neg 95.2%

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

      10. associate-/l* 95.2%

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

      11. div-sub 95.2%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 z around 0 65.4%

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

      1. associate-*r/ 65.4%

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

      2. metadata-eval 65.4%

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

   6. Simplified 65.4%

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

   7. Taylor expanded in y around 0 50.5%

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

   8. Taylor expanded in y around 0 94.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-172}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 70.5% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-184}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-188}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.2e-184) x (if (<= x 2.65e-188) (* y 0.8862269254527579) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.2e-184) {
		tmp = x;
	} else if (x <= 2.65e-188) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.2d-184)) then
        tmp = x
    else if (x <= 2.65d-188) then
        tmp = y * 0.8862269254527579d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.2e-184) {
		tmp = x;
	} else if (x <= 2.65e-188) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.2e-184:
		tmp = x
	elif x <= 2.65e-188:
		tmp = y * 0.8862269254527579
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.2e-184)
		tmp = x;
	elseif (x <= 2.65e-188)
		tmp = Float64(y * 0.8862269254527579);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.2e-184)
		tmp = x;
	elseif (x <= 2.65e-188)
		tmp = y * 0.8862269254527579;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.2e-184], x, If[LessEqual[x, 2.65e-188], N[(y * 0.8862269254527579), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999998e-184 or 2.65000000000000007e-188 < x

   1. Initial program 96.7%

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

      1. *-lft-identity 96.7%

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

      2. metadata-eval 96.7%

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

      3. times-frac 96.7%

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

      4. neg-mul-1 96.7%

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

      5. sub0-neg 96.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 96.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 96.8%

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

      8. +-commutative 96.8%

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

      9. sub-neg 96.8%

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

      10. associate-/l* 96.8%

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

      11. div-sub 96.8%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 z around 0 88.0%

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

      1. associate-*r/ 88.0%

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

      2. metadata-eval 88.0%

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

   6. Simplified 88.0%

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

   7. Taylor expanded in y around 0 57.2%

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

   8. Taylor expanded in y around 0 78.0%

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

   if -4.1999999999999998e-184 < x < 2.65000000000000007e-188

   1. Initial program 92.7%

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

      1. *-lft-identity 92.7%

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

      2. metadata-eval 92.7%

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

      3. times-frac 92.7%

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

      4. neg-mul-1 92.7%

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

      5. sub0-neg 92.6%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 92.6%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 92.9%

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

      8. +-commutative 92.9%

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

      9. sub-neg 92.9%

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

      10. associate-/l* 93.0%

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

      11. div-sub 93.0%

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

      12. associate-*r/ 99.8%

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

      13. *-inverses 99.8%

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

      14. *-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. cancel-sign-sub-inv 99.8%

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

      17. distribute-lft-neg-in 99.8%

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

      18. distribute-rgt-neg-in 99.8%

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

      19. associate-*l/ 99.8%

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

      20. distribute-rgt-neg-in 99.8%

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

   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 52.4%

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

      1. associate-*r/ 52.4%

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

      2. metadata-eval 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in y around 0 43.3%

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

   8. Taylor expanded in y around inf 41.5%

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

      1. *-commutative 41.5%

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

   10. Simplified 41.5%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-184}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-188}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 70.5% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-183}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-191}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e-183) x (if (<= x 1.35e-191) (/ y 1.1283791670955126) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-183) {
		tmp = x;
	} else if (x <= 1.35e-191) {
		tmp = y / 1.1283791670955126;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.4d-183)) then
        tmp = x
    else if (x <= 1.35d-191) then
        tmp = y / 1.1283791670955126d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-183) {
		tmp = x;
	} else if (x <= 1.35e-191) {
		tmp = y / 1.1283791670955126;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.4e-183:
		tmp = x
	elif x <= 1.35e-191:
		tmp = y / 1.1283791670955126
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e-183)
		tmp = x;
	elseif (x <= 1.35e-191)
		tmp = Float64(y / 1.1283791670955126);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4e-183)
		tmp = x;
	elseif (x <= 1.35e-191)
		tmp = y / 1.1283791670955126;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4e-183], x, If[LessEqual[x, 1.35e-191], N[(y / 1.1283791670955126), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.39999999999999992e-183 or 1.34999999999999999e-191 < x

   1. Initial program 96.7%

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

      1. *-lft-identity 96.7%

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

      2. metadata-eval 96.7%

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

      3. times-frac 96.7%

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

      4. neg-mul-1 96.7%

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

      5. sub0-neg 96.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

      6. associate-+l- 96.8%

         \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

      7. neg-sub0 96.8%

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

      8. +-commutative 96.8%

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

      9. sub-neg 96.8%

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

      10. associate-/l* 96.8%

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

      11. div-sub 96.8%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   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 z around 0 88.0%

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

      1. associate-*r/ 88.0%

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

      2. metadata-eval 88.0%

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

   6. Simplified 88.0%

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

   7. Taylor expanded in y around 0 57.2%

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

   8. Taylor expanded in y around 0 78.0%

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

   if -1.39999999999999992e-183 < x < 1.34999999999999999e-191

   1. Initial program 92.7%

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

   2. Taylor expanded in z around 0 52.5%

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

      1. div-inv 52.4%

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

      2. add-sqr-sqrt 34.3%

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

      3. sqrt-unprod 21.8%

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

      4. sqr-neg 21.8%

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

      5. sqrt-unprod 2.3%

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

      6. add-sqr-sqrt 3.4%

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

      7. cancel-sign-sub-inv 3.4%

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

      8. div-inv 3.4%

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

      9. cancel-sign-sub-inv 3.4%

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

      10. add-sqr-sqrt 2.3%

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

      11. sqrt-unprod 16.1%

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

      12. sqr-neg 16.1%

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

      13. sqrt-unprod 7.4%

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

      14. add-sqr-sqrt 12.7%

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

   4. Applied egg-rr 12.7%

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

   5. Taylor expanded in x around 0 2.7%

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

      1. *-commutative 2.7%

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

   7. Simplified 2.7%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 20.7%

         \[\leadsto \color{blue}{\sqrt{\left(y \cdot -0.8862269254527579\right) \cdot \left(y \cdot -0.8862269254527579\right)}} \]

      3. swap-sqr 20.7%

         \[\leadsto \sqrt{\color{blue}{\left(y \cdot y\right) \cdot \left(-0.8862269254527579 \cdot -0.8862269254527579\right)}} \]

      4. metadata-eval 20.7%

         \[\leadsto \sqrt{\left(y \cdot y\right) \cdot \color{blue}{0.7853981633974483}} \]

      5. metadata-eval 20.7%

         \[\leadsto \sqrt{\left(y \cdot y\right) \cdot \color{blue}{\left(0.8862269254527579 \cdot 0.8862269254527579\right)}} \]

      6. swap-sqr 20.7%

         \[\leadsto \sqrt{\color{blue}{\left(y \cdot 0.8862269254527579\right) \cdot \left(y \cdot 0.8862269254527579\right)}} \]

      7. metadata-eval 20.7%

         \[\leadsto \sqrt{\left(y \cdot \color{blue}{\frac{1}{1.1283791670955126}}\right) \cdot \left(y \cdot 0.8862269254527579\right)} \]

      8. div-inv 20.7%

         \[\leadsto \sqrt{\color{blue}{\frac{y}{1.1283791670955126}} \cdot \left(y \cdot 0.8862269254527579\right)} \]

      9. metadata-eval 20.7%

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

      10. div-inv 20.7%

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

      11. sqrt-unprod 27.8%

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

      12. add-sqr-sqrt 41.6%

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

   9. Applied egg-rr 41.6%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-183}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-191}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 68.9% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 95.8%

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

   1. *-lft-identity 95.8%

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

   2. metadata-eval 95.8%

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

   3. times-frac 95.8%

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

   4. neg-mul-1 95.8%

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

   5. sub0-neg 95.8%

      \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{0 - \left(1.1283791670955126 \cdot e^{z} - x \cdot y\right)}} \]

   6. associate-+l- 95.8%

      \[\leadsto x + \frac{-1 \cdot y}{\color{blue}{\left(0 - 1.1283791670955126 \cdot e^{z}\right) + x \cdot y}} \]

   7. neg-sub0 95.8%

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

   8. +-commutative 95.8%

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

   9. sub-neg 95.8%

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

   10. associate-/l* 95.9%

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

   11. div-sub 95.9%

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

   12. associate-*r/ 99.9%

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

   13. *-inverses 99.9%

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

   14. *-rgt-identity 99.9%

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

   15. associate-*l/ 99.9%

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

   16. cancel-sign-sub-inv 99.9%

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

   17. distribute-lft-neg-in 99.9%

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

   18. distribute-rgt-neg-in 99.9%

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

   19. associate-*l/ 99.9%

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

   20. distribute-rgt-neg-in 99.9%

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

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 79.4%

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

   1. associate-*r/ 79.4%

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

   2. metadata-eval 79.4%

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

6. Simplified 79.4%

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

7. Taylor expanded in y around 0 53.8%

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

8. Taylor expanded in y around 0 65.4%

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

9. Final simplification 65.4%

   \[\leadsto x \]

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 2023238 
(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)))))