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

Percentage Accurate: 95.7% → 99.9%

Time: 7.9s

Alternatives: 8

Speedup: 8.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. *-lft-identity 97.9%

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

   2. metadata-eval 97.9%

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

   3. times-frac 97.9%

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

   4. neg-mul-1 97.9%

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

   5. sub0-neg 97.9%

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

   6. associate-+l- 97.9%

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

   7. neg-sub0 98.0%

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

   8. +-commutative 98.0%

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

   9. sub-neg 98.0%

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

   10. associate-/l* 98.0%

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

   11. div-sub 98.0%

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

   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. Final simplification 99.9%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t_0 \leq 10^{+225}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_0 1e+225) 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 <= 1e+225) {
		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 <= 1d+225) 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 <= 1e+225) {
		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 <= 1e+225:
		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 <= 1e+225)
		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 <= 1e+225)
		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, 1e+225], 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)))) < 9.99999999999999928e224

   1. Initial program 99.9%

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

   if 9.99999999999999928e224 < (+.f64 x (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))

   1. Initial program 69.2%

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

      1. *-lft-identity 69.2%

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

      2. metadata-eval 69.2%

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

      3. times-frac 69.2%

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

      4. neg-mul-1 69.2%

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

      5. sub0-neg 69.1%

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

      6. associate-+l- 69.1%

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

      7. neg-sub0 69.5%

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

      8. +-commutative 69.5%

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

      9. sub-neg 69.5%

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

      10. associate-/l* 69.5%

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

      11. div-sub 69.5%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 10^{+225}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 2.0d0) then
        tmp = x - (1.0d0 / (x + ((-1.1283791670955126d0) / y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 2.0:
		tmp = x - (1.0 / (x + (-1.1283791670955126 / y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 2.0)
		tmp = Float64(x - Float64(1.0 / Float64(x + Float64(-1.1283791670955126 / y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 2.0)
		tmp = x - (1.0 / (x + (-1.1283791670955126 / y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(x - N[(1.0 / N[(x + N[(-1.1283791670955126 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   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 96.6%

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

      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 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. metadata-eval 99.8%

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

      3. times-frac 99.8%

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

      4. neg-mul-1 99.8%

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

      5. sub0-neg 99.8%

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

      6. associate-+l- 99.8%

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

      7. neg-sub0 99.8%

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

      8. +-commutative 99.8%

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

      9. sub-neg 99.8%

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

      10. associate-/l* 99.8%

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

      11. div-sub 99.8%

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

      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 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.1%

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

      4. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   if 2 < (exp.f64 z)

   1. Initial program 95.4%

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

      1. *-lft-identity 95.4%

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

      2. metadata-eval 95.4%

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

      3. times-frac 95.4%

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

      4. neg-mul-1 95.4%

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

      5. sub0-neg 95.4%

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

      6. associate-+l- 95.4%

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

      7. neg-sub0 95.4%

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

      8. +-commutative 95.4%

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

      9. sub-neg 95.4%

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

      10. associate-/l* 95.4%

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

      11. div-sub 95.4%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.5%

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

Alternative 4: 73.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+232}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{+187}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-134}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.9e+232)
   x
   (if (<= z -2.85e+187)
     (/ -1.0 x)
     (if (<= z -3.8e-19)
       x
       (if (<= z 6.6e-134) (+ x (/ y 1.1283791670955126)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9e+232) {
		tmp = x;
	} else if (z <= -2.85e+187) {
		tmp = -1.0 / x;
	} else if (z <= -3.8e-19) {
		tmp = x;
	} else if (z <= 6.6e-134) {
		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 <= (-2.9d+232)) then
        tmp = x
    else if (z <= (-2.85d+187)) then
        tmp = (-1.0d0) / x
    else if (z <= (-3.8d-19)) then
        tmp = x
    else if (z <= 6.6d-134) 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 <= -2.9e+232) {
		tmp = x;
	} else if (z <= -2.85e+187) {
		tmp = -1.0 / x;
	} else if (z <= -3.8e-19) {
		tmp = x;
	} else if (z <= 6.6e-134) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.9e+232:
		tmp = x
	elif z <= -2.85e+187:
		tmp = -1.0 / x
	elif z <= -3.8e-19:
		tmp = x
	elif z <= 6.6e-134:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.9e+232)
		tmp = x;
	elseif (z <= -2.85e+187)
		tmp = Float64(-1.0 / x);
	elseif (z <= -3.8e-19)
		tmp = x;
	elseif (z <= 6.6e-134)
		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 <= -2.9e+232)
		tmp = x;
	elseif (z <= -2.85e+187)
		tmp = -1.0 / x;
	elseif (z <= -3.8e-19)
		tmp = x;
	elseif (z <= 6.6e-134)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.9e+232], x, If[LessEqual[z, -2.85e+187], N[(-1.0 / x), $MachinePrecision], If[LessEqual[z, -3.8e-19], x, If[LessEqual[z, 6.6e-134], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.90000000000000023e232 or -2.8500000000000002e187 < z < -3.8e-19 or 6.60000000000000038e-134 < z

   1. Initial program 97.4%

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

      1. *-lft-identity 97.4%

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

      2. metadata-eval 97.4%

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

      3. times-frac 97.4%

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

      4. neg-mul-1 97.4%

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

      5. sub0-neg 97.4%

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

      6. associate-+l- 97.4%

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

      7. neg-sub0 97.4%

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

      8. +-commutative 97.4%

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

      9. sub-neg 97.4%

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

      10. associate-/l* 97.5%

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

      11. div-sub 97.5%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 77.9%

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

   if -2.90000000000000023e232 < z < -2.8500000000000002e187

   1. Initial program 88.7%

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

      1. *-lft-identity 88.7%

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

      2. metadata-eval 88.7%

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

      3. times-frac 88.7%

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

      4. neg-mul-1 88.7%

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

      5. sub0-neg 89.3%

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

      6. associate-+l- 89.3%

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

      7. neg-sub0 89.3%

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

      8. +-commutative 89.3%

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

      9. sub-neg 89.3%

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

      10. associate-/l* 89.1%

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

      11. div-sub 89.1%

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

      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%

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

   5. Taylor expanded in x around 0 78.1%

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

   if -3.8e-19 < z < 6.60000000000000038e-134

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.8%

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

   3. Taylor expanded in y around 0 83.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+232}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{+187}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-134}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 86.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -1.38 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-94}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{elif}\;z \leq 0.00049:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z -1.38e-14)
     t_0
     (if (<= z 1.9e-94)
       (+ x (/ y (+ 1.1283791670955126 (* z 1.1283791670955126))))
       (if (<= z 0.00049) t_0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -1.38e-14) {
		tmp = t_0;
	} else if (z <= 1.9e-94) {
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	} else if (z <= 0.00049) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    if (z <= (-1.38d-14)) then
        tmp = t_0
    else if (z <= 1.9d-94) then
        tmp = x + (y / (1.1283791670955126d0 + (z * 1.1283791670955126d0)))
    else if (z <= 0.00049d0) then
        tmp = t_0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -1.38e-14) {
		tmp = t_0;
	} else if (z <= 1.9e-94) {
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	} else if (z <= 0.00049) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if z <= -1.38e-14:
		tmp = t_0
	elif z <= 1.9e-94:
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)))
	elif z <= 0.00049:
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if (z <= -1.38e-14)
		tmp = t_0;
	elseif (z <= 1.9e-94)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 + Float64(z * 1.1283791670955126))));
	elseif (z <= 0.00049)
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	tmp = 0.0;
	if (z <= -1.38e-14)
		tmp = t_0;
	elseif (z <= 1.9e-94)
		tmp = x + (y / (1.1283791670955126 + (z * 1.1283791670955126)));
	elseif (z <= 0.00049)
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.38e-14], t$95$0, If[LessEqual[z, 1.9e-94], N[(x + N[(y / N[(1.1283791670955126 + N[(z * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.00049], t$95$0, x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.38000000000000002e-14 or 1.9e-94 < z < 4.8999999999999998e-4

   1. Initial program 97.6%

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

      1. *-lft-identity 97.6%

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

      2. metadata-eval 97.6%

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

      3. times-frac 97.6%

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

      4. neg-mul-1 97.6%

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

      5. sub0-neg 97.5%

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

      6. associate-+l- 97.5%

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

      7. neg-sub0 97.6%

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

      8. +-commutative 97.6%

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

      9. sub-neg 97.6%

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

      10. associate-/l* 97.7%

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

      11. div-sub 97.7%

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

      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 96.5%

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

   if -1.38000000000000002e-14 < z < 1.9e-94

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.8%

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

   3. Taylor expanded in y around 0 81.4%

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

   if 4.8999999999999998e-4 < z

   1. Initial program 95.4%

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

      1. *-lft-identity 95.4%

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

      2. metadata-eval 95.4%

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

      3. times-frac 95.4%

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

      4. neg-mul-1 95.4%

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

      5. sub0-neg 95.4%

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

      6. associate-+l- 95.4%

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

      7. neg-sub0 95.4%

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

      8. +-commutative 95.4%

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

      9. sub-neg 95.4%

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

      10. associate-/l* 95.4%

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

      11. div-sub 95.4%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{-14}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-94}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{elif}\;z \leq 0.00049:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 99.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -300:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 305:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -300.0)
   (+ x (/ -1.0 x))
   (if (<= z 305.0) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -300.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 305.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-300.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 305.0d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -300.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 305.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -300.0:
		tmp = x + (-1.0 / x)
	elif z <= 305.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -300.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 305.0)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -300.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 305.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -300.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 305.0], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -300

   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 96.6%

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

      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 100.0%

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

   if -300 < z < 305

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.1%

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

   if 305 < z

   1. Initial program 95.4%

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

      1. *-lft-identity 95.4%

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

      2. metadata-eval 95.4%

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

      3. times-frac 95.4%

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

      4. neg-mul-1 95.4%

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

      5. sub0-neg 95.4%

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

      6. associate-+l- 95.4%

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

      7. neg-sub0 95.4%

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

      8. +-commutative 95.4%

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

      9. sub-neg 95.4%

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

      10. associate-/l* 95.4%

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

      11. div-sub 95.4%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -300:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 305:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 86.3% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;z \leq 0.0078:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (<= z -1.5e-14)
     t_0
     (if (<= z 2.8e-93)
       (+ x (/ y 1.1283791670955126))
       (if (<= z 0.0078) t_0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -1.5e-14) {
		tmp = t_0;
	} else if (z <= 2.8e-93) {
		tmp = x + (y / 1.1283791670955126);
	} else if (z <= 0.0078) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    if (z <= (-1.5d-14)) then
        tmp = t_0
    else if (z <= 2.8d-93) then
        tmp = x + (y / 1.1283791670955126d0)
    else if (z <= 0.0078d0) then
        tmp = t_0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if (z <= -1.5e-14) {
		tmp = t_0;
	} else if (z <= 2.8e-93) {
		tmp = x + (y / 1.1283791670955126);
	} else if (z <= 0.0078) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if z <= -1.5e-14:
		tmp = t_0
	elif z <= 2.8e-93:
		tmp = x + (y / 1.1283791670955126)
	elif z <= 0.0078:
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if (z <= -1.5e-14)
		tmp = t_0;
	elseif (z <= 2.8e-93)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	elseif (z <= 0.0078)
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	tmp = 0.0;
	if (z <= -1.5e-14)
		tmp = t_0;
	elseif (z <= 2.8e-93)
		tmp = x + (y / 1.1283791670955126);
	elseif (z <= 0.0078)
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.5e-14], t$95$0, If[LessEqual[z, 2.8e-93], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.0078], t$95$0, x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4999999999999999e-14 or 2.79999999999999998e-93 < z < 0.0077999999999999996

   1. Initial program 97.6%

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

      1. *-lft-identity 97.6%

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

      2. metadata-eval 97.6%

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

      3. times-frac 97.6%

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

      4. neg-mul-1 97.6%

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

      5. sub0-neg 97.5%

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

      6. associate-+l- 97.5%

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

      7. neg-sub0 97.6%

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

      8. +-commutative 97.6%

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

      9. sub-neg 97.6%

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

      10. associate-/l* 97.7%

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

      11. div-sub 97.7%

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

      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 96.5%

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

   if -1.4999999999999999e-14 < z < 2.79999999999999998e-93

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.7%

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

   3. Taylor expanded in y around 0 81.4%

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

   if 0.0077999999999999996 < z

   1. Initial program 95.4%

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

      1. *-lft-identity 95.4%

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

      2. metadata-eval 95.4%

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

      3. times-frac 95.4%

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

      4. neg-mul-1 95.4%

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

      5. sub0-neg 95.4%

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

      6. associate-+l- 95.4%

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

      7. neg-sub0 95.4%

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

      8. +-commutative 95.4%

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

      9. sub-neg 95.4%

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

      10. associate-/l* 95.4%

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

      11. div-sub 95.4%

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

      12. associate-*r/ 100.0%

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

      13. *-inverses 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. cancel-sign-sub-inv 100.0%

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

      17. distribute-lft-neg-in 100.0%

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

      18. distribute-rgt-neg-in 100.0%

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

      19. associate-*l/ 100.0%

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

      20. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-14}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{elif}\;z \leq 0.0078:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 69.0% 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 97.9%

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

   1. *-lft-identity 97.9%

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

   2. metadata-eval 97.9%

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

   3. times-frac 97.9%

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

   4. neg-mul-1 97.9%

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

   5. sub0-neg 97.9%

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

   6. associate-+l- 97.9%

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

   7. neg-sub0 98.0%

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

   8. +-commutative 98.0%

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

   9. sub-neg 98.0%

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

   10. associate-/l* 98.0%

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

   11. div-sub 98.0%

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

   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 68.4%

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

5. Final simplification 68.4%

   \[\leadsto x \]

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023171 
(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)))))