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

Percentage Accurate: 95.4% → 98.6%

Time: 8.8s

Alternatives: 7

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: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -350:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-50}:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{e^{z}} \cdot -0.8862269254527579\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -350.0)
   (+ x (/ -1.0 x))
   (if (<= z 5.2e-50)
     (+ x (/ 1.0 (- (/ 1.1283791670955126 y) x)))
     (- x (* (/ y (exp z)) -0.8862269254527579)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -350.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 5.2e-50) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x - ((y / exp(z)) * -0.8862269254527579);
	}
	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 <= (-350.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 5.2d-50) then
        tmp = x + (1.0d0 / ((1.1283791670955126d0 / y) - x))
    else
        tmp = x - ((y / exp(z)) * (-0.8862269254527579d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -350.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 5.2e-50) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x - ((y / Math.exp(z)) * -0.8862269254527579);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -350.0:
		tmp = x + (-1.0 / x)
	elif z <= 5.2e-50:
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x))
	else:
		tmp = x - ((y / math.exp(z)) * -0.8862269254527579)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -350.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 5.2e-50)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(1.1283791670955126 / y) - x)));
	else
		tmp = Float64(x - Float64(Float64(y / exp(z)) * -0.8862269254527579));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -350.0)
		tmp = x + (-1.0 / x);
	elseif (z <= 5.2e-50)
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	else
		tmp = x - ((y / exp(z)) * -0.8862269254527579);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -350.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.2e-50], N[(x + N[(1.0 / N[(N[(1.1283791670955126 / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y / N[Exp[z], $MachinePrecision]), $MachinePrecision] * -0.8862269254527579), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -350

   1. Initial program 87.5%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -350 < z < 5.2000000000000003e-50

   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 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-neg 99.9%

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

      3. metadata-eval 99.9%

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

   7. Simplified 99.9%

      \[\leadsto x - \color{blue}{\frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}} \]
   8. Step-by-step derivation

      1. metadata-eval 99.9%

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

      2. fma-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. clear-num 99.9%

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

      5. inv-pow 99.9%

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

      6. div-sub 99.9%

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

      7. add-sqr-sqrt 53.0%

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

      8. sqrt-unprod 95.6%

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

      9. sqr-neg 95.6%

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

      10. sqrt-unprod 43.3%

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

      11. add-sqr-sqrt 85.7%

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

      12. associate-/l* 85.7%

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

      13. *-inverses 85.7%

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

      14. *-commutative 85.7%

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

      15. add-sqr-sqrt 43.3%

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

      16. sqrt-unprod 95.6%

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

      17. sqr-neg 95.6%

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

      18. sqrt-unprod 53.1%

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

      19. *-un-lft-identity 53.1%

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

      20. add-sqr-sqrt 99.9%

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

   9. Applied egg-rr 99.9%

      \[\leadsto x - \color{blue}{{\left(x - \frac{1.1283791670955126}{y}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 99.9%

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

   11. Simplified 99.9%

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

   if 5.2000000000000003e-50 < z

   1. Initial program 91.4%

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

      1. remove-double-neg 91.4%

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

      2. distribute-frac-neg 91.4%

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

      3. unsub-neg 91.4%

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

      4. distribute-frac-neg 91.4%

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

      5. distribute-neg-frac2 91.4%

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

      6. neg-sub0 91.4%

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

      7. associate--r- 91.4%

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

      8. neg-sub0 91.4%

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

      9. +-commutative 91.4%

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

      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. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -350:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-50}:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{e^{z}} \cdot -0.8862269254527579\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 1.8 \cdot 10^{-307}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 1.8e-307)
   (+ 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) <= 1.8e-307) {
		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) <= 1.8e-307)
		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], 1.8e-307], 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) < 1.80000000000000003e-307

   1. Initial program 87.5%

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

   3. Taylor expanded in y around inf 100.0%

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

   if 1.80000000000000003e-307 < (exp.f64 z)

   1. Initial program 96.7%

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

      1. remove-double-neg 96.7%

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

      2. distribute-frac-neg 96.7%

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

      3. unsub-neg 96.7%

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

      4. distribute-frac-neg 96.7%

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

      5. distribute-neg-frac2 96.7%

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

      6. neg-sub0 96.7%

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

      7. associate--r- 96.7%

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

      8. neg-sub0 96.7%

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

      9. +-commutative 96.7%

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

      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. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 1.8 \cdot 10^{-307}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x - y \cdot -0.8862269254527579\\ \mathbf{if}\;z \leq -0.00078:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-300}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))) (t_1 (- x (* y -0.8862269254527579))))
   (if (<= z -0.00078)
     t_0
     (if (<= z -2.9e-300)
       t_1
       (if (<= z 2.3e-101) t_0 (if (<= z 3.1e-6) t_1 x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x - (y * -0.8862269254527579);
	double tmp;
	if (z <= -0.00078) {
		tmp = t_0;
	} else if (z <= -2.9e-300) {
		tmp = t_1;
	} else if (z <= 2.3e-101) {
		tmp = t_0;
	} else if (z <= 3.1e-6) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x - (y * (-0.8862269254527579d0))
    if (z <= (-0.00078d0)) then
        tmp = t_0
    else if (z <= (-2.9d-300)) then
        tmp = t_1
    else if (z <= 2.3d-101) then
        tmp = t_0
    else if (z <= 3.1d-6) then
        tmp = t_1
    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 t_1 = x - (y * -0.8862269254527579);
	double tmp;
	if (z <= -0.00078) {
		tmp = t_0;
	} else if (z <= -2.9e-300) {
		tmp = t_1;
	} else if (z <= 2.3e-101) {
		tmp = t_0;
	} else if (z <= 3.1e-6) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x - (y * -0.8862269254527579)
	tmp = 0
	if z <= -0.00078:
		tmp = t_0
	elif z <= -2.9e-300:
		tmp = t_1
	elif z <= 2.3e-101:
		tmp = t_0
	elif z <= 3.1e-6:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x - Float64(y * -0.8862269254527579))
	tmp = 0.0
	if (z <= -0.00078)
		tmp = t_0;
	elseif (z <= -2.9e-300)
		tmp = t_1;
	elseif (z <= 2.3e-101)
		tmp = t_0;
	elseif (z <= 3.1e-6)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x - (y * -0.8862269254527579);
	tmp = 0.0;
	if (z <= -0.00078)
		tmp = t_0;
	elseif (z <= -2.9e-300)
		tmp = t_1;
	elseif (z <= 2.3e-101)
		tmp = t_0;
	elseif (z <= 3.1e-6)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.00078], t$95$0, If[LessEqual[z, -2.9e-300], t$95$1, If[LessEqual[z, 2.3e-101], t$95$0, If[LessEqual[z, 3.1e-6], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.79999999999999986e-4 or -2.89999999999999992e-300 < z < 2.2999999999999999e-101

   1. Initial program 92.1%

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

   3. Taylor expanded in y around inf 93.2%

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

   if -7.79999999999999986e-4 < z < -2.89999999999999992e-300 or 2.2999999999999999e-101 < z < 3.1e-6

   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 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-neg 99.9%

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

      3. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 81.4%

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

   if 3.1e-6 < z

   1. Initial program 90.3%

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

   3. Taylor expanded in x around inf 98.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00078:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-300}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-101}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x - \frac{y}{-1.1283791670955126}\\ \mathbf{if}\;z \leq -4 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-299}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))) (t_1 (- x (/ y -1.1283791670955126))))
   (if (<= z -4e-5)
     t_0
     (if (<= z -2.3e-299)
       t_1
       (if (<= z 1e-133) t_0 (if (<= z 1.7e-6) t_1 x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x - (y / -1.1283791670955126);
	double tmp;
	if (z <= -4e-5) {
		tmp = t_0;
	} else if (z <= -2.3e-299) {
		tmp = t_1;
	} else if (z <= 1e-133) {
		tmp = t_0;
	} else if (z <= 1.7e-6) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x - (y / (-1.1283791670955126d0))
    if (z <= (-4d-5)) then
        tmp = t_0
    else if (z <= (-2.3d-299)) then
        tmp = t_1
    else if (z <= 1d-133) then
        tmp = t_0
    else if (z <= 1.7d-6) then
        tmp = t_1
    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 t_1 = x - (y / -1.1283791670955126);
	double tmp;
	if (z <= -4e-5) {
		tmp = t_0;
	} else if (z <= -2.3e-299) {
		tmp = t_1;
	} else if (z <= 1e-133) {
		tmp = t_0;
	} else if (z <= 1.7e-6) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x - (y / -1.1283791670955126)
	tmp = 0
	if z <= -4e-5:
		tmp = t_0
	elif z <= -2.3e-299:
		tmp = t_1
	elif z <= 1e-133:
		tmp = t_0
	elif z <= 1.7e-6:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x - Float64(y / -1.1283791670955126))
	tmp = 0.0
	if (z <= -4e-5)
		tmp = t_0;
	elseif (z <= -2.3e-299)
		tmp = t_1;
	elseif (z <= 1e-133)
		tmp = t_0;
	elseif (z <= 1.7e-6)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x - (y / -1.1283791670955126);
	tmp = 0.0;
	if (z <= -4e-5)
		tmp = t_0;
	elseif (z <= -2.3e-299)
		tmp = t_1;
	elseif (z <= 1e-133)
		tmp = t_0;
	elseif (z <= 1.7e-6)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4e-5], t$95$0, If[LessEqual[z, -2.3e-299], t$95$1, If[LessEqual[z, 1e-133], t$95$0, If[LessEqual[z, 1.7e-6], t$95$1, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.00000000000000033e-5 or -2.3000000000000001e-299 < z < 1.0000000000000001e-133

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf 94.6%

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

   if -4.00000000000000033e-5 < z < -2.3000000000000001e-299 or 1.0000000000000001e-133 < z < 1.70000000000000003e-6

   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 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-neg 99.9%

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

      3. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 80.3%

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

   if 1.70000000000000003e-6 < z

   1. Initial program 90.3%

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

   3. Taylor expanded in x around inf 98.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-299}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{elif}\;z \leq 10^{-133}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -300:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 55:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -300.0)
   (+ x (/ -1.0 x))
   (if (<= z 55.0) (+ x (/ 1.0 (- (/ 1.1283791670955126 y) x))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -300.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 55.0) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-300.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 55.0d0) then
        tmp = x + (1.0d0 / ((1.1283791670955126d0 / y) - x))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -300.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 55.0) {
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -300.0:
		tmp = x + (-1.0 / x)
	elif z <= 55.0:
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x))
	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 <= 55.0)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(1.1283791670955126 / y) - x)));
	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 <= 55.0)
		tmp = x + (1.0 / ((1.1283791670955126 / y) - x));
	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, 55.0], N[(x + N[(1.0 / N[(N[(1.1283791670955126 / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -300

   1. Initial program 87.5%

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

   3. Taylor expanded in y around inf 100.0%

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

   if -300 < z < 55

   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 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-neg 99.9%

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

      3. metadata-eval 99.9%

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

   7. Simplified 99.9%

      \[\leadsto x - \color{blue}{\frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}} \]
   8. Step-by-step derivation

      1. metadata-eval 99.9%

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

      2. fma-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. clear-num 99.9%

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

      5. inv-pow 99.9%

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

      6. div-sub 99.9%

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

      7. add-sqr-sqrt 53.6%

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

      8. sqrt-unprod 95.8%

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

      9. sqr-neg 95.8%

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

      10. sqrt-unprod 42.9%

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

      11. add-sqr-sqrt 86.6%

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

      12. associate-/l* 86.6%

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

      13. *-inverses 86.6%

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

      14. *-commutative 86.6%

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

      15. add-sqr-sqrt 42.9%

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

      16. sqrt-unprod 95.9%

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

      17. sqr-neg 95.9%

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

      18. sqrt-unprod 53.7%

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

      19. *-un-lft-identity 53.7%

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

      20. add-sqr-sqrt 99.9%

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

   9. Applied egg-rr 99.9%

      \[\leadsto x - \color{blue}{{\left(x - \frac{1.1283791670955126}{y}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 99.9%

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

   11. Simplified 99.9%

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

   if 55 < z

   1. Initial program 90.3%

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

   3. Taylor expanded in x around inf 98.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -300:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 55:\\ \;\;\;\;x + \frac{1}{\frac{1.1283791670955126}{y} - x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.3e-16) x (if (<= z 4.2e-5) (- x (* y -0.8862269254527579)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e-16) {
		tmp = x;
	} else if (z <= 4.2e-5) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.3d-16)) then
        tmp = x
    else if (z <= 4.2d-5) then
        tmp = x - (y * (-0.8862269254527579d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e-16) {
		tmp = x;
	} else if (z <= 4.2e-5) {
		tmp = x - (y * -0.8862269254527579);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.3e-16:
		tmp = x
	elif z <= 4.2e-5:
		tmp = x - (y * -0.8862269254527579)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.3e-16)
		tmp = x;
	elseif (z <= 4.2e-5)
		tmp = Float64(x - Float64(y * -0.8862269254527579));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.3e-16)
		tmp = x;
	elseif (z <= 4.2e-5)
		tmp = x - (y * -0.8862269254527579);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.3e-16], x, If[LessEqual[z, 4.2e-5], N[(x - N[(y * -0.8862269254527579), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2999999999999999e-16 or 4.19999999999999977e-5 < z

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 76.7%

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

   if -1.2999999999999999e-16 < z < 4.19999999999999977e-5

   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 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-neg 99.9%

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

      3. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 76.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;x - y \cdot -0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.2% 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 94.1%

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

3. Taylor expanded in x around inf 72.2%

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

4. Final simplification 72.2%

   \[\leadsto x \]
5. Add Preprocessing

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 2024067 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

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