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

Percentage Accurate: 95.4% → 99.9%

Time: 12.9s

Alternatives: 14

Speedup: 5.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.4% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.7%

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

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.3%

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

      1. remove-double-neg 97.3%

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

      2. distribute-frac-neg 97.3%

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

      3. unsub-neg 97.3%

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

      4. distribute-frac-neg 97.3%

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

      5. distribute-neg-frac2 97.3%

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

      6. neg-sub0 97.3%

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

      7. associate--r- 97.3%

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

      8. neg-sub0 97.3%

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

      9. +-commutative 97.3%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.00000002:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot \left(1.1283791670955126 - z \cdot \left(z \cdot -0.18806319451591877 - 0.5641895835477563\right)\right) - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot -0.8862269254527579}{e^{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (- x (/ 1.0 x))
   (if (<= (exp z) 1.00000002)
     (+
      x
      (/
       y
       (+
        1.1283791670955126
        (-
         (*
          z
          (-
           1.1283791670955126
           (* z (- (* z -0.18806319451591877) 0.5641895835477563))))
         (* x y)))))
     (- x (/ (* y -0.8862269254527579) (exp z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (exp(z) <= 1.00000002) {
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * ((z * -0.18806319451591877) - 0.5641895835477563)))) - (x * y))));
	} else {
		tmp = x - ((y * -0.8862269254527579) / exp(z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x - (1.0d0 / x)
    else if (exp(z) <= 1.00000002d0) then
        tmp = x + (y / (1.1283791670955126d0 + ((z * (1.1283791670955126d0 - (z * ((z * (-0.18806319451591877d0)) - 0.5641895835477563d0)))) - (x * y))))
    else
        tmp = x - ((y * (-0.8862269254527579d0)) / exp(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (Math.exp(z) <= 1.00000002) {
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * ((z * -0.18806319451591877) - 0.5641895835477563)))) - (x * y))));
	} else {
		tmp = x - ((y * -0.8862269254527579) / Math.exp(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x - (1.0 / x)
	elif math.exp(z) <= 1.00000002:
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * ((z * -0.18806319451591877) - 0.5641895835477563)))) - (x * y))))
	else:
		tmp = x - ((y * -0.8862269254527579) / math.exp(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (exp(z) <= 1.00000002)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 + Float64(Float64(z * Float64(1.1283791670955126 - Float64(z * Float64(Float64(z * -0.18806319451591877) - 0.5641895835477563)))) - Float64(x * y)))));
	else
		tmp = Float64(x - Float64(Float64(y * -0.8862269254527579) / exp(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x - (1.0 / x);
	elseif (exp(z) <= 1.00000002)
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * ((z * -0.18806319451591877) - 0.5641895835477563)))) - (x * y))));
	else
		tmp = x - ((y * -0.8862269254527579) / exp(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.00000002], N[(x + N[(y / N[(1.1283791670955126 + N[(N[(z * N[(1.1283791670955126 - N[(z * N[(N[(z * -0.18806319451591877), $MachinePrecision] - 0.5641895835477563), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y * -0.8862269254527579), $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.7%

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

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1.0000000200000001

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

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

      7. associate--r- 99.8%

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

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.8%

      \[\leadsto x - \frac{y}{\color{blue}{\left(x \cdot y + z \cdot \left(z \cdot \left(-0.18806319451591877 \cdot z - 0.5641895835477563\right) - 1.1283791670955126\right)\right) - 1.1283791670955126}} \]

   if 1.0000000200000001 < (exp.f64 z)

   1. Initial program 91.2%

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

      1. remove-double-neg 91.2%

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

      2. distribute-frac-neg 91.2%

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

      3. unsub-neg 91.2%

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

      4. distribute-frac-neg 91.2%

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

      5. distribute-neg-frac2 91.2%

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

      6. neg-sub0 91.2%

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

      7. associate--r- 91.2%

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

      8. neg-sub0 91.2%

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

      9. +-commutative 91.2%

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

      10. fma-define 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

         \[\leadsto x - \color{blue}{\frac{-0.8862269254527579 \cdot y}{e^{z}}} \]

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.00000002:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot \left(1.1283791670955126 - z \cdot \left(z \cdot -0.18806319451591877 - 0.5641895835477563\right)\right) - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot -0.8862269254527579}{e^{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot \left(1.1283791670955126 - z \cdot \left(z \cdot -0.18806319451591877 - 0.5641895835477563\right)\right) - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (- x (/ 1.0 x))
   (if (<= (exp z) 1.2)
     (+
      x
      (/
       y
       (+
        1.1283791670955126
        (-
         (*
          z
          (-
           1.1283791670955126
           (* z (- (* z -0.18806319451591877) 0.5641895835477563))))
         (* x y)))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (exp(z) <= 1.2) {
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * ((z * -0.18806319451591877) - 0.5641895835477563)))) - (x * y))));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x - (1.0d0 / x)
    else if (exp(z) <= 1.2d0) then
        tmp = x + (y / (1.1283791670955126d0 + ((z * (1.1283791670955126d0 - (z * ((z * (-0.18806319451591877d0)) - 0.5641895835477563d0)))) - (x * y))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x - (1.0 / x);
	} else if (Math.exp(z) <= 1.2) {
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * ((z * -0.18806319451591877) - 0.5641895835477563)))) - (x * y))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x - (1.0 / x)
	elif math.exp(z) <= 1.2:
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * ((z * -0.18806319451591877) - 0.5641895835477563)))) - (x * y))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (exp(z) <= 1.2)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 + Float64(Float64(z * Float64(1.1283791670955126 - Float64(z * Float64(Float64(z * -0.18806319451591877) - 0.5641895835477563)))) - Float64(x * y)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x - (1.0 / x);
	elseif (exp(z) <= 1.2)
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * ((z * -0.18806319451591877) - 0.5641895835477563)))) - (x * y))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.2], N[(x + N[(y / N[(1.1283791670955126 + N[(N[(z * N[(1.1283791670955126 - N[(z * N[(N[(z * -0.18806319451591877), $MachinePrecision] - 0.5641895835477563), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.7%

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

   3. Taylor expanded in y around inf 100.0%

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

   if 0.0 < (exp.f64 z) < 1.19999999999999996

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

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

      7. associate--r- 99.8%

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

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.4%

      \[\leadsto x - \frac{y}{\color{blue}{\left(x \cdot y + z \cdot \left(z \cdot \left(-0.18806319451591877 \cdot z - 0.5641895835477563\right) - 1.1283791670955126\right)\right) - 1.1283791670955126}} \]

   if 1.19999999999999996 < (exp.f64 z)

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot \left(1.1283791670955126 - z \cdot \left(z \cdot -0.18806319451591877 - 0.5641895835477563\right)\right) - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_0 \leq 10^{+292}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_0 1e+292) 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+292) {
		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+292) 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+292) {
		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+292:
		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+292)
		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+292)
		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+292], 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 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1e292

   1. Initial program 98.3%

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

   if 1e292 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 28.7%

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

   3. Taylor expanded in y around inf 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 10^{+292}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3600000000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.27:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot \left(1.1283791670955126 - z \cdot -0.5641895835477563\right) - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3600000000000.0)
   (- x (/ 1.0 x))
   (if (<= z 0.27)
     (+
      x
      (/
       y
       (+
        1.1283791670955126
        (- (* z (- 1.1283791670955126 (* z -0.5641895835477563))) (* x y)))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3600000000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.27) {
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * -0.5641895835477563))) - (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 <= (-3600000000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 0.27d0) then
        tmp = x + (y / (1.1283791670955126d0 + ((z * (1.1283791670955126d0 - (z * (-0.5641895835477563d0)))) - (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 <= -3600000000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.27) {
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * -0.5641895835477563))) - (x * y))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3600000000000.0:
		tmp = x - (1.0 / x)
	elif z <= 0.27:
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * -0.5641895835477563))) - (x * y))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3600000000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 0.27)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 + Float64(Float64(z * Float64(1.1283791670955126 - Float64(z * -0.5641895835477563))) - Float64(x * y)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3600000000000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 0.27)
		tmp = x + (y / (1.1283791670955126 + ((z * (1.1283791670955126 - (z * -0.5641895835477563))) - (x * y))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3600000000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.27], N[(x + N[(y / N[(1.1283791670955126 + N[(N[(z * N[(1.1283791670955126 - N[(z * -0.5641895835477563), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -3.6e12

   1. Initial program 89.2%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -3.6e12 < z < 0.27000000000000002

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. distribute-neg-frac2 99.9%

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

      6. neg-sub0 99.9%

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

      7. associate--r- 99.9%

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

      8. neg-sub0 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.2%

      \[\leadsto x - \frac{y}{\color{blue}{\left(x \cdot y + z \cdot \left(-0.5641895835477563 \cdot z - 1.1283791670955126\right)\right) - 1.1283791670955126}} \]

   if 0.27000000000000002 < z

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3600000000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.27:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + \left(z \cdot \left(1.1283791670955126 - z \cdot -0.5641895835477563\right) - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-227}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.8e-5)
   x
   (if (<= x -9.6e-155)
     (/ -1.0 x)
     (if (<= x 1.6e-227)
       (- x (* y -0.8862269254527579))
       (if (<= x 1.3e-18) (/ -1.0 x) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.8e-5) {
		tmp = x;
	} else if (x <= -9.6e-155) {
		tmp = -1.0 / x;
	} else if (x <= 1.6e-227) {
		tmp = x - (y * -0.8862269254527579);
	} else if (x <= 1.3e-18) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.8d-5)) then
        tmp = x
    else if (x <= (-9.6d-155)) then
        tmp = (-1.0d0) / x
    else if (x <= 1.6d-227) then
        tmp = x - (y * (-0.8862269254527579d0))
    else if (x <= 1.3d-18) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.8e-5) {
		tmp = x;
	} else if (x <= -9.6e-155) {
		tmp = -1.0 / x;
	} else if (x <= 1.6e-227) {
		tmp = x - (y * -0.8862269254527579);
	} else if (x <= 1.3e-18) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.8e-5:
		tmp = x
	elif x <= -9.6e-155:
		tmp = -1.0 / x
	elif x <= 1.6e-227:
		tmp = x - (y * -0.8862269254527579)
	elif x <= 1.3e-18:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.8e-5)
		tmp = x;
	elseif (x <= -9.6e-155)
		tmp = Float64(-1.0 / x);
	elseif (x <= 1.6e-227)
		tmp = Float64(x - Float64(y * -0.8862269254527579));
	elseif (x <= 1.3e-18)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.8e-5)
		tmp = x;
	elseif (x <= -9.6e-155)
		tmp = -1.0 / x;
	elseif (x <= 1.6e-227)
		tmp = x - (y * -0.8862269254527579);
	elseif (x <= 1.3e-18)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.8e-5], x, If[LessEqual[x, -9.6e-155], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 1.6e-227], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e-18], N[(-1.0 / x), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.79999999999999996e-5 or 1.3e-18 < x

   1. Initial program 96.4%

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

   3. Taylor expanded in x around inf 97.3%

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

   if -2.79999999999999996e-5 < x < -9.600000000000001e-155 or 1.60000000000000005e-227 < x < 1.3e-18

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf 61.1%

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

   4. Taylor expanded in x around 0 60.7%

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

   if -9.600000000000001e-155 < x < 1.60000000000000005e-227

   1. Initial program 91.6%

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

      1. remove-double-neg 91.6%

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

      2. distribute-frac-neg 91.6%

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

      3. unsub-neg 91.6%

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

      4. distribute-frac-neg 91.6%

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

      5. distribute-neg-frac2 91.6%

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

      6. neg-sub0 91.5%

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

      7. associate--r- 91.5%

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

      8. neg-sub0 91.9%

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

      9. +-commutative 91.9%

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

      10. fma-define 91.9%

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

      11. *-commutative 91.9%

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

      12. distribute-rgt-neg-in 91.9%

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

      13. metadata-eval 91.9%

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

   3. Simplified 91.9%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 51.7%

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

   6. Taylor expanded in y around 0 43.5%

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

      1. *-commutative 43.5%

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

   8. Simplified 43.5%

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

Alternative 7: 71.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-155}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-261}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.5e-5)
   x
   (if (<= x -1.95e-155)
     (/ -1.0 x)
     (if (<= x 1.1e-261)
       (/ y 1.1283791670955126)
       (if (<= x 2e-21) (/ -1.0 x) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e-5) {
		tmp = x;
	} else if (x <= -1.95e-155) {
		tmp = -1.0 / x;
	} else if (x <= 1.1e-261) {
		tmp = y / 1.1283791670955126;
	} else if (x <= 2e-21) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.5d-5)) then
        tmp = x
    else if (x <= (-1.95d-155)) then
        tmp = (-1.0d0) / x
    else if (x <= 1.1d-261) then
        tmp = y / 1.1283791670955126d0
    else if (x <= 2d-21) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e-5) {
		tmp = x;
	} else if (x <= -1.95e-155) {
		tmp = -1.0 / x;
	} else if (x <= 1.1e-261) {
		tmp = y / 1.1283791670955126;
	} else if (x <= 2e-21) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.5e-5:
		tmp = x
	elif x <= -1.95e-155:
		tmp = -1.0 / x
	elif x <= 1.1e-261:
		tmp = y / 1.1283791670955126
	elif x <= 2e-21:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.5e-5)
		tmp = x;
	elseif (x <= -1.95e-155)
		tmp = Float64(-1.0 / x);
	elseif (x <= 1.1e-261)
		tmp = Float64(y / 1.1283791670955126);
	elseif (x <= 2e-21)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.5e-5)
		tmp = x;
	elseif (x <= -1.95e-155)
		tmp = -1.0 / x;
	elseif (x <= 1.1e-261)
		tmp = y / 1.1283791670955126;
	elseif (x <= 2e-21)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.5e-5], x, If[LessEqual[x, -1.95e-155], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 1.1e-261], N[(y / 1.1283791670955126), $MachinePrecision], If[LessEqual[x, 2e-21], N[(-1.0 / x), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000012e-5 or 1.99999999999999982e-21 < x

   1. Initial program 96.4%

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

   3. Taylor expanded in x around inf 97.3%

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

   if -2.50000000000000012e-5 < x < -1.9500000000000002e-155 or 1.1000000000000001e-261 < x < 1.99999999999999982e-21

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf 60.0%

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

   4. Taylor expanded in x around 0 59.6%

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

   if -1.9500000000000002e-155 < x < 1.1000000000000001e-261

   1. Initial program 93.3%

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

      1. remove-double-neg 93.3%

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

      2. distribute-frac-neg 93.3%

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

      3. unsub-neg 93.3%

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

      4. distribute-frac-neg 93.3%

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

      5. distribute-neg-frac2 93.3%

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

      6. neg-sub0 93.1%

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

      7. associate--r- 93.1%

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

      8. neg-sub0 93.5%

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

      9. +-commutative 93.5%

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

      10. fma-define 93.5%

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

      11. *-commutative 93.5%

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

      12. distribute-rgt-neg-in 93.5%

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

      13. metadata-eval 93.5%

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

   3. Simplified 93.5%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 52.8%

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

   6. Taylor expanded in x around 0 42.6%

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

      1. metadata-eval 42.6%

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

      2. distribute-lft-neg-in 42.6%

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

      3. *-commutative 42.6%

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

      4. metadata-eval 43.0%

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

      5. div-inv 43.0%

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

      6. distribute-neg-frac2 43.0%

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

      7. metadata-eval 43.0%

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

   8. Applied egg-rr 43.0%

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

Alternative 8: 71.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.82 \cdot 10^{-155}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-262}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.6e-5)
   x
   (if (<= x -1.82e-155)
     (/ -1.0 x)
     (if (<= x 2.7e-262)
       (* y 0.8862269254527579)
       (if (<= x 5e-25) (/ -1.0 x) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e-5) {
		tmp = x;
	} else if (x <= -1.82e-155) {
		tmp = -1.0 / x;
	} else if (x <= 2.7e-262) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 5e-25) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.6d-5)) then
        tmp = x
    else if (x <= (-1.82d-155)) then
        tmp = (-1.0d0) / x
    else if (x <= 2.7d-262) then
        tmp = y * 0.8862269254527579d0
    else if (x <= 5d-25) then
        tmp = (-1.0d0) / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e-5) {
		tmp = x;
	} else if (x <= -1.82e-155) {
		tmp = -1.0 / x;
	} else if (x <= 2.7e-262) {
		tmp = y * 0.8862269254527579;
	} else if (x <= 5e-25) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.6e-5:
		tmp = x
	elif x <= -1.82e-155:
		tmp = -1.0 / x
	elif x <= 2.7e-262:
		tmp = y * 0.8862269254527579
	elif x <= 5e-25:
		tmp = -1.0 / x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.6e-5)
		tmp = x;
	elseif (x <= -1.82e-155)
		tmp = Float64(-1.0 / x);
	elseif (x <= 2.7e-262)
		tmp = Float64(y * 0.8862269254527579);
	elseif (x <= 5e-25)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.6e-5)
		tmp = x;
	elseif (x <= -1.82e-155)
		tmp = -1.0 / x;
	elseif (x <= 2.7e-262)
		tmp = y * 0.8862269254527579;
	elseif (x <= 5e-25)
		tmp = -1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.6e-5], x, If[LessEqual[x, -1.82e-155], N[(-1.0 / x), $MachinePrecision], If[LessEqual[x, 2.7e-262], N[(y * 0.8862269254527579), $MachinePrecision], If[LessEqual[x, 5e-25], N[(-1.0 / x), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.60000000000000009e-5 or 4.99999999999999962e-25 < x

   1. Initial program 96.4%

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

   3. Taylor expanded in x around inf 97.3%

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

   if -3.60000000000000009e-5 < x < -1.82000000000000006e-155 or 2.7000000000000001e-262 < x < 4.99999999999999962e-25

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf 60.0%

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

   4. Taylor expanded in x around 0 59.6%

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

   if -1.82000000000000006e-155 < x < 2.7000000000000001e-262

   1. Initial program 93.3%

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

      1. remove-double-neg 93.3%

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

      2. distribute-frac-neg 93.3%

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

      3. unsub-neg 93.3%

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

      4. distribute-frac-neg 93.3%

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

      5. distribute-neg-frac2 93.3%

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

      6. neg-sub0 93.1%

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

      7. associate--r- 93.1%

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

      8. neg-sub0 93.5%

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

      9. +-commutative 93.5%

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

      10. fma-define 93.5%

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

      11. *-commutative 93.5%

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

      12. distribute-rgt-neg-in 93.5%

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

      13. metadata-eval 93.5%

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

   3. Simplified 93.5%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 52.8%

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

   6. Taylor expanded in x around 0 42.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.82 \cdot 10^{-155}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-262}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3600000000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.27:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - \left(x \cdot y + z \cdot -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3600000000000.0)
   (- x (/ 1.0 x))
   (if (<= z 0.27)
     (+ x (/ y (- 1.1283791670955126 (+ (* x y) (* z -1.1283791670955126)))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3600000000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.27) {
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (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 <= (-3600000000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 0.27d0) then
        tmp = x + (y / (1.1283791670955126d0 - ((x * y) + (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 <= -3600000000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.27) {
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * -1.1283791670955126))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3600000000000.0:
		tmp = x - (1.0 / x)
	elif z <= 0.27:
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * -1.1283791670955126))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3600000000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 0.27)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(Float64(x * y) + Float64(z * -1.1283791670955126)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3600000000000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 0.27)
		tmp = x + (y / (1.1283791670955126 - ((x * y) + (z * -1.1283791670955126))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3600000000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.27], N[(x + N[(y / N[(1.1283791670955126 - N[(N[(x * y), $MachinePrecision] + N[(z * -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -3.6e12

   1. Initial program 89.2%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -3.6e12 < z < 0.27000000000000002

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. distribute-neg-frac2 99.9%

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

      6. neg-sub0 99.9%

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

      7. associate--r- 99.9%

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

      8. neg-sub0 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.0%

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

   if 0.27000000000000002 < z

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3600000000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.27:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - \left(x \cdot y + z \cdot -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3600000000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.27:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3600000000000.0)
   (- x (/ 1.0 x))
   (if (<= z 0.27) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3600000000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.27) {
		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 <= (-3600000000000.0d0)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 0.27d0) 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 <= -3600000000000.0) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.27) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3600000000000.0:
		tmp = x - (1.0 / x)
	elif z <= 0.27:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3600000000000.0)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 0.27)
		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 <= -3600000000000.0)
		tmp = x - (1.0 / x);
	elseif (z <= 0.27)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3600000000000.0], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.27], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -3.6e12

   1. Initial program 89.2%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -3.6e12 < z < 0.27000000000000002

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. distribute-neg-frac2 99.9%

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

      6. neg-sub0 99.9%

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

      7. associate--r- 99.9%

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

      8. neg-sub0 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-define 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.7%

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

   if 0.27000000000000002 < z

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3600000000000:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.27:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-192}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.082:\\ \;\;\;\;x - \left(y \cdot 0.8862269254527579\right) \cdot \left(z + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.45e-192)
   (- x (/ 1.0 x))
   (if (<= z 0.082) (- x (* (* y 0.8862269254527579) (+ z -1.0))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.45e-192) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.082) {
		tmp = x - ((y * 0.8862269254527579) * (z + -1.0));
	} 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) :: tmp
    if (z <= (-1.45d-192)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 0.082d0) then
        tmp = x - ((y * 0.8862269254527579d0) * (z + (-1.0d0)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.45e-192) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.082) {
		tmp = x - ((y * 0.8862269254527579) * (z + -1.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.45e-192:
		tmp = x - (1.0 / x)
	elif z <= 0.082:
		tmp = x - ((y * 0.8862269254527579) * (z + -1.0))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.45e-192)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 0.082)
		tmp = Float64(x - Float64(Float64(y * 0.8862269254527579) * Float64(z + -1.0)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.45e-192)
		tmp = x - (1.0 / x);
	elseif (z <= 0.082)
		tmp = x - ((y * 0.8862269254527579) * (z + -1.0));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.45e-192], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.082], N[(x - N[(N[(y * 0.8862269254527579), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -1.45000000000000008e-192

   1. Initial program 93.8%

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

   3. Taylor expanded in y around inf 94.8%

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

   if -1.45000000000000008e-192 < z < 0.0820000000000000034

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

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

      7. associate--r- 99.8%

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

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.2%

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

      1. associate-*r/ 77.2%

         \[\leadsto x - \color{blue}{\frac{-0.8862269254527579 \cdot y}{e^{z}}} \]

   7. Simplified 77.2%

      \[\leadsto x - \color{blue}{\frac{-0.8862269254527579 \cdot y}{e^{z}}} \]

   8. Taylor expanded in z around 0 76.5%

      \[\leadsto x - \frac{-0.8862269254527579 \cdot y}{\color{blue}{1 + z \cdot \left(1 + 0.5 \cdot z\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 76.5%

         \[\leadsto x - \frac{-0.8862269254527579 \cdot y}{1 + z \cdot \left(1 + \color{blue}{z \cdot 0.5}\right)} \]

   10. Simplified 76.5%

       \[\leadsto x - \frac{-0.8862269254527579 \cdot y}{\color{blue}{1 + z \cdot \left(1 + z \cdot 0.5\right)}} \]

   11. Taylor expanded in z around 0 76.4%

       \[\leadsto x - \color{blue}{\left(-0.8862269254527579 \cdot y + 0.8862269254527579 \cdot \left(y \cdot z\right)\right)} \]
   12. Step-by-step derivation

       1. metadata-eval 76.4%

          \[\leadsto x - \left(\color{blue}{\left(-1 \cdot 0.8862269254527579\right)} \cdot y + 0.8862269254527579 \cdot \left(y \cdot z\right)\right) \]

       2. associate-*r* 76.4%

          \[\leadsto x - \left(\color{blue}{-1 \cdot \left(0.8862269254527579 \cdot y\right)} + 0.8862269254527579 \cdot \left(y \cdot z\right)\right) \]

       3. *-commutative 76.4%

          \[\leadsto x - \left(\color{blue}{\left(0.8862269254527579 \cdot y\right) \cdot -1} + 0.8862269254527579 \cdot \left(y \cdot z\right)\right) \]

       4. associate-*r* 76.4%

          \[\leadsto x - \left(\left(0.8862269254527579 \cdot y\right) \cdot -1 + \color{blue}{\left(0.8862269254527579 \cdot y\right) \cdot z}\right) \]

       5. distribute-lft-out 76.4%

          \[\leadsto x - \color{blue}{\left(0.8862269254527579 \cdot y\right) \cdot \left(-1 + z\right)} \]

       6. *-commutative 76.4%

          \[\leadsto x - \color{blue}{\left(y \cdot 0.8862269254527579\right)} \cdot \left(-1 + z\right) \]

   13. Simplified 76.4%

       \[\leadsto x - \color{blue}{\left(y \cdot 0.8862269254527579\right) \cdot \left(-1 + z\right)} \]

   if 0.0820000000000000034 < z

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-192}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.082:\\ \;\;\;\;x - \left(y \cdot 0.8862269254527579\right) \cdot \left(z + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-190}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.055:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.05e-190)
   (- x (/ 1.0 x))
   (if (<= z 0.055) (- x (/ y -1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.05e-190) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.055) {
		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 <= (-2.05d-190)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 0.055d0) 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 <= -2.05e-190) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.055) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.05e-190:
		tmp = x - (1.0 / x)
	elif z <= 0.055:
		tmp = x - (y / -1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.05e-190)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 0.055)
		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 <= -2.05e-190)
		tmp = x - (1.0 / x);
	elseif (z <= 0.055)
		tmp = x - (y / -1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.05e-190], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.055], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -2.0500000000000001e-190

   1. Initial program 93.8%

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

   3. Taylor expanded in y around inf 94.8%

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

   if -2.0500000000000001e-190 < z < 0.0550000000000000003

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

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

      7. associate--r- 99.8%

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

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.8%

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

   6. Taylor expanded in x around 0 76.3%

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

   if 0.0550000000000000003 < z

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf 100.0%

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

Alternative 13: 85.5% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-184}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 0.05:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.4e-184)
   (- x (/ 1.0 x))
   (if (<= z 0.05) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.4e-184) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.05) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.4d-184)) then
        tmp = x - (1.0d0 / x)
    else if (z <= 0.05d0) then
        tmp = x - (y * (-0.8862269254527579d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.4e-184) {
		tmp = x - (1.0 / x);
	} else if (z <= 0.05) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.4e-184:
		tmp = x - (1.0 / x)
	elif z <= 0.05:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.4e-184)
		tmp = Float64(x - Float64(1.0 / x));
	elseif (z <= 0.05)
		tmp = Float64(x - Float64(y * -0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.4e-184)
		tmp = x - (1.0 / x);
	elseif (z <= 0.05)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.4e-184], N[(x - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.05], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -2.40000000000000024e-184

   1. Initial program 93.8%

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

   3. Taylor expanded in y around inf 94.8%

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

   if -2.40000000000000024e-184 < z < 0.050000000000000003

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. neg-sub0 99.8%

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

      7. associate--r- 99.8%

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

      8. neg-sub0 99.8%

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

      9. +-commutative 99.8%

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

      10. fma-define 99.8%

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

      11. *-commutative 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.8%

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

   6. Taylor expanded in y around 0 76.1%

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

      1. *-commutative 76.1%

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

   8. Simplified 76.1%

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

   if 0.050000000000000003 < z

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf 100.0%

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

Alternative 14: 69.6% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in x around inf 67.3%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Developer Target 1: 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 2024165 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ 1 (- (* (/ 5641895835477563/5000000000000000 y) (exp z)) x))))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))