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

Percentage Accurate: 95.8% → 99.4%

Time: 9.2s

Alternatives: 11

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: 95.8% 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: 99.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot 1.1283791670955126 - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 2.5e-5)
     (+ x (/ y (+ 1.1283791670955126 (- (* z 1.1283791670955126) (* x y)))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 2.5e-5) {
		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 <= (-7000000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 2.5d-5) 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 <= -7000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 2.5e-5) {
		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 <= -7000000.0:
		tmp = x + (-1.0 / x)
	elif z <= 2.5e-5:
		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 <= -7000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 2.5e-5)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 + Float64(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 <= -7000000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 2.5e-5)
		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, -7000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5e-5], N[(x + N[(y / N[(1.1283791670955126 + N[(N[(z * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -7e6

   1. Initial program 93.9%

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

      1. *-lft-identity 93.9%

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

      2. metadata-eval 93.9%

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

      3. times-frac 93.9%

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

      4. neg-mul-1 93.9%

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

      5. sub0-neg 93.7%

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

      6. associate-+l- 93.7%

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

      7. neg-sub0 93.9%

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

      8. +-commutative 93.9%

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

      9. sub-neg 93.9%

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

      10. associate-/l* 94.1%

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

      11. div-sub 94.1%

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

      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 x around inf 100.0%

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

   if -7e6 < z < 2.50000000000000012e-5

   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%

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

      1. associate--l+ 99.9%

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

      2. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   if 2.50000000000000012e-5 < z

   1. Initial program 93.0%

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

      1. *-lft-identity 93.0%

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

      2. metadata-eval 93.0%

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

      3. times-frac 93.0%

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

      4. neg-mul-1 93.0%

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

      5. sub0-neg 93.0%

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

      6. associate-+l- 93.0%

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

      7. neg-sub0 93.0%

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

      8. +-commutative 93.0%

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

      9. sub-neg 93.0%

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

      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 x around inf 50.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot 1.1283791670955126 - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{+252}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_0 4e+252) 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 <= 4e+252) {
		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 <= 4d+252) 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 <= 4e+252) {
		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 <= 4e+252:
		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 <= 4e+252)
		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 <= 4e+252)
		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, 4e+252], 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)))) < 4.0000000000000004e252

   1. Initial program 99.5%

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

   if 4.0000000000000004e252 < (+.f64 x (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))

   1. Initial program 68.5%

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

      1. *-lft-identity 68.5%

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

      2. metadata-eval 68.5%

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

      3. times-frac 68.5%

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

      4. neg-mul-1 68.5%

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

      5. sub0-neg 68.4%

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

      6. associate-+l- 68.4%

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

      7. neg-sub0 68.6%

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

      8. +-commutative 68.6%

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

      9. sub-neg 68.6%

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

      10. associate-/l* 68.6%

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

      11. div-sub 68.6%

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

      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 x around inf 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 4 \cdot 10^{+252}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]

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 96.6%

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

   1. *-lft-identity 96.6%

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

   2. metadata-eval 96.6%

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

   3. times-frac 96.6%

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

   4. neg-mul-1 96.6%

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

   5. sub0-neg 96.5%

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

   6. associate-+l- 96.5%

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

   7. neg-sub0 96.6%

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

   8. +-commutative 96.6%

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

   9. sub-neg 96.6%

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

   10. associate-/l* 96.6%

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

   11. div-sub 96.6%

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

   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: 86.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-42}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.2e-42)
   (+ x (/ -1.0 x))
   (if (<= z 2.6e-6)
     (+ x (/ y (+ 1.1283791670955126 (* z 1.1283791670955126))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.2e-42) {
		tmp = x + (-1.0 / x);
	} else if (z <= 2.6e-6) {
		tmp = x + (y / (1.1283791670955126 + (z * 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 <= (-8.2d-42)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 2.6d-6) then
        tmp = x + (y / (1.1283791670955126d0 + (z * 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 <= -8.2e-42) {
		tmp = x + (-1.0 / x);
	} else if (z <= 2.6e-6) {
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.2e-42:
		tmp = x + (-1.0 / x)
	elif z <= 2.6e-6:
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.2e-42)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 2.6e-6)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 + Float64(z * 1.1283791670955126))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8.2e-42)
		tmp = x + (-1.0 / x);
	elseif (z <= 2.6e-6)
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.2e-42], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e-6], N[(x + N[(y / N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -8.2000000000000003e-42

   1. Initial program 94.8%

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

      1. *-lft-identity 94.8%

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

      2. metadata-eval 94.8%

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

      3. times-frac 94.8%

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

      4. neg-mul-1 94.8%

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

      5. sub0-neg 94.6%

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

      6. associate-+l- 94.6%

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

      7. neg-sub0 94.9%

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

      8. +-commutative 94.9%

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

      9. sub-neg 94.9%

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

      10. associate-/l* 95.0%

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

      11. div-sub 95.0%

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

      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 x around inf 94.8%

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

   if -8.2000000000000003e-42 < z < 2.60000000000000009e-6

   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%

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

      1. associate--l+ 99.9%

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

      2. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around 0 79.5%

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

   if 2.60000000000000009e-6 < z

   1. Initial program 93.0%

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

      1. *-lft-identity 93.0%

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

      2. metadata-eval 93.0%

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

      3. times-frac 93.0%

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

      4. neg-mul-1 93.0%

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

      5. sub0-neg 93.0%

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

      6. associate-+l- 93.0%

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

      7. neg-sub0 93.0%

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

      8. +-commutative 93.0%

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

      9. sub-neg 93.0%

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

      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 x around inf 50.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-42}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 99.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 2.5e-5) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 2.5e-5) {
		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 <= (-7000000.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 2.5d-5) 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 <= -7000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 2.5e-5) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7000000.0:
		tmp = x + (-1.0 / x)
	elif z <= 2.5e-5:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 2.5e-5)
		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 <= -7000000.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 2.5e-5)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5e-5], 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 < -7e6

   1. Initial program 93.9%

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

      1. *-lft-identity 93.9%

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

      2. metadata-eval 93.9%

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

      3. times-frac 93.9%

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

      4. neg-mul-1 93.9%

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

      5. sub0-neg 93.7%

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

      6. associate-+l- 93.7%

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

      7. neg-sub0 93.9%

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

      8. +-commutative 93.9%

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

      9. sub-neg 93.9%

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

      10. associate-/l* 94.1%

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

      11. div-sub 94.1%

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

      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 x around inf 100.0%

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

   if -7e6 < z < 2.50000000000000012e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.1%

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

   if 2.50000000000000012e-5 < z

   1. Initial program 93.0%

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

      1. *-lft-identity 93.0%

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

      2. metadata-eval 93.0%

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

      3. times-frac 93.0%

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

      4. neg-mul-1 93.0%

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

      5. sub0-neg 93.0%

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

      6. associate-+l- 93.0%

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

      7. neg-sub0 93.0%

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

      8. +-commutative 93.0%

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

      9. sub-neg 93.0%

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

      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 x around inf 50.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 74.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e-41) x (if (<= z 2.4e-5) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-41) {
		tmp = x;
	} else if (z <= 2.4e-5) {
		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 <= (-2d-41)) then
        tmp = x
    else if (z <= 2.4d-5) 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 <= -2e-41) {
		tmp = x;
	} else if (z <= 2.4e-5) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2e-41:
		tmp = x
	elif z <= 2.4e-5:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e-41)
		tmp = x;
	elseif (z <= 2.4e-5)
		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 <= -2e-41)
		tmp = x;
	elseif (z <= 2.4e-5)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e-41], x, If[LessEqual[z, 2.4e-5], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000001e-41 or 2.4000000000000001e-5 < z

   1. Initial program 93.9%

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

      1. *-lft-identity 93.9%

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

      2. metadata-eval 93.9%

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

      3. times-frac 93.9%

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

      4. neg-mul-1 93.9%

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

      5. sub0-neg 93.8%

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

      6. associate-+l- 93.8%

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

      7. neg-sub0 93.9%

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

      8. +-commutative 93.9%

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

      9. sub-neg 93.9%

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

      10. associate-/l* 94.0%

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

      11. div-sub 94.0%

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

      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 x around inf 72.5%

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

   5. Taylor expanded in x around inf 72.4%

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

   if -2.00000000000000001e-41 < z < 2.4000000000000001e-5

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      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 99.7%

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

   5. Taylor expanded in x around 0 79.2%

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

      1. *-commutative 79.2%

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

   7. Simplified 79.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 86.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.78 \cdot 10^{-40}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.78e-40)
   (+ x (/ -1.0 x))
   (if (<= z 2.35e-7) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.78e-40) {
		tmp = x + (-1.0 / x);
	} else if (z <= 2.35e-7) {
		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 <= (-1.78d-40)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 2.35d-7) 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 <= -1.78e-40) {
		tmp = x + (-1.0 / x);
	} else if (z <= 2.35e-7) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.78e-40:
		tmp = x + (-1.0 / x)
	elif z <= 2.35e-7:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.78e-40)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 2.35e-7)
		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 <= -1.78e-40)
		tmp = x + (-1.0 / x);
	elseif (z <= 2.35e-7)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.78e-40], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.35e-7], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.78000000000000001e-40

   1. Initial program 94.8%

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

      1. *-lft-identity 94.8%

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

      2. metadata-eval 94.8%

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

      3. times-frac 94.8%

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

      4. neg-mul-1 94.8%

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

      5. sub0-neg 94.6%

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

      6. associate-+l- 94.6%

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

      7. neg-sub0 94.9%

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

      8. +-commutative 94.9%

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

      9. sub-neg 94.9%

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

      10. associate-/l* 95.0%

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

      11. div-sub 95.0%

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

      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 x around inf 94.8%

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

   if -1.78000000000000001e-40 < z < 2.35e-7

   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 99.7%

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

   5. Taylor expanded in x around 0 79.2%

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

      1. *-commutative 79.2%

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

   7. Simplified 79.2%

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

   if 2.35e-7 < z

   1. Initial program 93.0%

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

      1. *-lft-identity 93.0%

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

      2. metadata-eval 93.0%

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

      3. times-frac 93.0%

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

      4. neg-mul-1 93.0%

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

      5. sub0-neg 93.0%

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

      6. associate-+l- 93.0%

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

      7. neg-sub0 93.0%

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

      8. +-commutative 93.0%

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

      9. sub-neg 93.0%

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

      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 x around inf 50.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.78 \cdot 10^{-40}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 86.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.78 \cdot 10^{-40}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.78e-40)
   (+ x (/ -1.0 x))
   (if (<= z 3.15e-9) (- x (/ y -1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.78e-40) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.15e-9) {
		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 <= (-1.78d-40)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 3.15d-9) 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 <= -1.78e-40) {
		tmp = x + (-1.0 / x);
	} else if (z <= 3.15e-9) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.78e-40:
		tmp = x + (-1.0 / x)
	elif z <= 3.15e-9:
		tmp = x - (y / -1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.78e-40)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 3.15e-9)
		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 <= -1.78e-40)
		tmp = x + (-1.0 / x);
	elseif (z <= 3.15e-9)
		tmp = x - (y / -1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.78e-40], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.15e-9], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.78000000000000001e-40

   1. Initial program 94.8%

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

      1. *-lft-identity 94.8%

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

      2. metadata-eval 94.8%

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

      3. times-frac 94.8%

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

      4. neg-mul-1 94.8%

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

      5. sub0-neg 94.6%

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

      6. associate-+l- 94.6%

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

      7. neg-sub0 94.9%

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

      8. +-commutative 94.9%

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

      9. sub-neg 94.9%

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

      10. associate-/l* 95.0%

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

      11. div-sub 95.0%

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

      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 x around inf 94.8%

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

   if -1.78000000000000001e-40 < z < 3.1500000000000001e-9

   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 99.7%

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

   5. Taylor expanded in x around 0 79.2%

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

      1. *-commutative 79.2%

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

   7. Simplified 79.2%

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

      1. metadata-eval 79.2%

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

      2. div-inv 79.3%

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

   9. Applied egg-rr 79.3%

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

   if 3.1500000000000001e-9 < z

   1. Initial program 93.0%

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

      1. *-lft-identity 93.0%

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

      2. metadata-eval 93.0%

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

      3. times-frac 93.0%

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

      4. neg-mul-1 93.0%

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

      5. sub0-neg 93.0%

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

      6. associate-+l- 93.0%

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

      7. neg-sub0 93.0%

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

      8. +-commutative 93.0%

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

      9. sub-neg 93.0%

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

      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 x around inf 50.5%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.78 \cdot 10^{-40}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 66.7% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-294}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{-y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.7e-294) x (if (<= x 2e-22) (/ (- y) -1.1283791670955126) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.7e-294) {
		tmp = x;
	} else if (x <= 2e-22) {
		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 <= (-5.7d-294)) then
        tmp = x
    else if (x <= 2d-22) 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 <= -5.7e-294) {
		tmp = x;
	} else if (x <= 2e-22) {
		tmp = -y / -1.1283791670955126;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.7e-294:
		tmp = x
	elif x <= 2e-22:
		tmp = -y / -1.1283791670955126
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.7e-294)
		tmp = x;
	elseif (x <= 2e-22)
		tmp = Float64(Float64(-y) / -1.1283791670955126);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.7e-294)
		tmp = x;
	elseif (x <= 2e-22)
		tmp = -y / -1.1283791670955126;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.7e-294], x, If[LessEqual[x, 2e-22], N[((-y) / -1.1283791670955126), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.70000000000000032e-294 or 2.0000000000000001e-22 < 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 96.6%

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

      6. associate-+l- 96.6%

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

      7. neg-sub0 96.7%

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

      8. +-commutative 96.7%

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

      9. sub-neg 96.7%

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

      10. associate-/l* 96.7%

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

      11. div-sub 96.7%

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

      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 x around inf 75.7%

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

   5. Taylor expanded in x around inf 78.7%

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

   if -5.70000000000000032e-294 < x < 2.0000000000000001e-22

   1. Initial program 96.4%

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

      1. *-lft-identity 96.4%

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

      2. metadata-eval 96.4%

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

      3. times-frac 96.4%

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

      4. neg-mul-1 96.4%

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

      5. sub0-neg 96.2%

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

      6. associate-+l- 96.2%

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

      7. neg-sub0 96.4%

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

      8. +-commutative 96.4%

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

      9. sub-neg 96.4%

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

      10. associate-/l* 96.4%

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

      11. div-sub 96.4%

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

      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 63.3%

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

   5. Taylor expanded in x around 0 53.0%

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

      1. *-commutative 53.0%

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

   7. Simplified 53.0%

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

   8. Taylor expanded in x around 0 45.1%

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

      1. *-commutative 45.1%

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

   10. Simplified 45.1%

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

       1. metadata-eval 45.1%

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

       2. metadata-eval 45.1%

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

       3. distribute-rgt-neg-in 45.1%

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

       4. div-inv 45.3%

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

       5. distribute-neg-frac 45.3%

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

   12. Applied egg-rr 45.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-294}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{-y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 66.4% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{-307}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-22}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.02e-307) x (if (<= x 2e-22) (* y 0.8862269254527579) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.02e-307) {
		tmp = x;
	} else if (x <= 2e-22) {
		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 <= 1.02d-307) then
        tmp = x
    else if (x <= 2d-22) 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 <= 1.02e-307) {
		tmp = x;
	} else if (x <= 2e-22) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.02e-307:
		tmp = x
	elif x <= 2e-22:
		tmp = y * 0.8862269254527579
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.02e-307)
		tmp = x;
	elseif (x <= 2e-22)
		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 <= 1.02e-307)
		tmp = x;
	elseif (x <= 2e-22)
		tmp = y * 0.8862269254527579;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.02e-307], x, If[LessEqual[x, 2e-22], N[(y * 0.8862269254527579), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < 1.02000000000000005e-307 or 2.0000000000000001e-22 < 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 96.7%

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

      6. associate-+l- 96.7%

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

      7. neg-sub0 96.7%

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

      8. +-commutative 96.7%

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

      9. sub-neg 96.7%

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

      10. associate-/l* 96.7%

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

      11. div-sub 96.7%

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

      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 x around inf 75.1%

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

   5. Taylor expanded in x around inf 78.1%

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

   if 1.02000000000000005e-307 < x < 2.0000000000000001e-22

   1. Initial program 96.2%

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

      1. *-lft-identity 96.2%

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

      2. metadata-eval 96.2%

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

      3. times-frac 96.2%

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

      4. neg-mul-1 96.2%

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

      5. sub0-neg 96.0%

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

      6. associate-+l- 96.0%

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

      7. neg-sub0 96.2%

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

      8. +-commutative 96.2%

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

      9. sub-neg 96.2%

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

      10. associate-/l* 96.2%

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

      11. div-sub 96.2%

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

      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 64.9%

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

   5. Taylor expanded in x around 0 54.0%

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

      1. *-commutative 54.0%

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

   7. Simplified 54.0%

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

   8. Taylor expanded in x around 0 45.7%

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

      1. *-commutative 45.7%

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

   10. Simplified 45.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{-307}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-22}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 69.8% 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 96.6%

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

   1. *-lft-identity 96.6%

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

   2. metadata-eval 96.6%

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

   3. times-frac 96.6%

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

   4. neg-mul-1 96.6%

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

   5. sub0-neg 96.5%

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

   6. associate-+l- 96.5%

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

   7. neg-sub0 96.6%

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

   8. +-commutative 96.6%

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

   9. sub-neg 96.6%

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

   10. associate-/l* 96.6%

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

   11. div-sub 96.6%

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

   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 x around inf 66.1%

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

5. Taylor expanded in x around inf 67.1%

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

6. Final simplification 67.1%

   \[\leadsto x \]

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 2023230 
(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)))))