_divideComplex, imaginary part

Percentage Accurate: 61.9% → 81.2%

Time: 12.0s

Alternatives: 13

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 81.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.3 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{1}{\frac{y.im}{x.im}}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1e+138)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im -3.3e-128)
     (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.im 4.6e-8)
       (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
       (/ (fma y.re (/ 1.0 (/ y.im x.im)) (- x.re)) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1e+138) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3.3e-128) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 4.6e-8) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = fma(y_46_re, (1.0 / (y_46_im / x_46_im)), -x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1e+138)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -3.3e-128)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 4.6e-8)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = Float64(fma(y_46_re, Float64(1.0 / Float64(y_46_im / x_46_im)), Float64(-x_46_re)) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1e+138], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3.3e-128], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.6e-8], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(y$46$re * N[(1.0 / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1e138

   1. Initial program 22.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 22.3%

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

      2. add-sqr-sqrt 22.3%

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

      3. times-frac 22.3%

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

      4. hypot-define 22.3%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. hypot-define 52.3%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   5. Taylor expanded in y.im around inf 87.8%

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

      1. +-commutative 87.8%

         \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}}{y.im} \]

      2. *-commutative 87.8%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.im}}{y.im} + -1 \cdot x.re}{y.im} \]

      3. associate-*r/ 90.4%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im}} + -1 \cdot x.re}{y.im} \]

      4. neg-mul-1 90.4%

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

      5. sub-neg 90.4%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im} - x.re}}{y.im} \]

   7. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}} \]

   if -1e138 < y.im < -3.3e-128

   1. Initial program 87.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   if -3.3e-128 < y.im < 4.6000000000000002e-8

   1. Initial program 70.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 91.0%

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

      1. remove-double-neg 91.0%

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

      2. mul-1-neg 91.0%

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

      3. mul-1-neg 91.0%

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

      4. distribute-neg-in 91.0%

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

      5. distribute-neg-in 91.0%

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

      6. mul-1-neg 91.0%

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

      7. remove-double-neg 91.0%

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

      8. mul-1-neg 91.0%

         \[\leadsto \frac{x.im + \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{y.re}}}{y.re} \]

      9. associate-*r/ 91.0%

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

      10. associate-*r* 91.0%

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

      11. mul-1-neg 91.0%

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

   5. Simplified 91.0%

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

   if 4.6000000000000002e-8 < y.im

   1. Initial program 57.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 78.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 78.0%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg 78.0%

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]

      3. unsub-neg 78.0%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} - \frac{x.re}{y.im}} \]

      4. unpow2 78.0%

         \[\leadsto \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]

      5. associate-/r* 80.5%

         \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re}{y.im}}{y.im}} - \frac{x.re}{y.im} \]

      6. div-sub 80.5%

         \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}} \]

      7. *-commutative 80.5%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.im}}{y.im} - x.re}{y.im} \]

      8. associate-/l* 84.5%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im}} - x.re}{y.im} \]

      9. fmm-def 84.4%

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

   5. Simplified 84.4%

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

      1. clear-num 84.5%

         \[\leadsto \frac{\mathsf{fma}\left(y.re, \color{blue}{\frac{1}{\frac{y.im}{x.im}}}, -x.re\right)}{y.im} \]

      2. inv-pow 84.5%

         \[\leadsto \frac{\mathsf{fma}\left(y.re, \color{blue}{{\left(\frac{y.im}{x.im}\right)}^{-1}}, -x.re\right)}{y.im} \]

   7. Applied egg-rr 84.5%

      \[\leadsto \frac{\mathsf{fma}\left(y.re, \color{blue}{{\left(\frac{y.im}{x.im}\right)}^{-1}}, -x.re\right)}{y.im} \]
   8. Step-by-step derivation

      1. unpow-1 84.5%

         \[\leadsto \frac{\mathsf{fma}\left(y.re, \color{blue}{\frac{1}{\frac{y.im}{x.im}}}, -x.re\right)}{y.im} \]

   9. Simplified 84.5%

      \[\leadsto \frac{\mathsf{fma}\left(y.re, \color{blue}{\frac{1}{\frac{y.im}{x.im}}}, -x.re\right)}{y.im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.3 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{1}{\frac{y.im}{x.im}}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{1}{\frac{y.im}{x.im}}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) INFINITY)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (/ (fma y.re (/ 1.0 (/ y.im x.im)) (- x.re)) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = fma(y_46_re, (1.0 / (y_46_im / x_46_im)), -x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(fma(y_46_re, Float64(1.0 / Float64(y_46_im / x_46_im)), Float64(-x_46_re)) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re * N[(1.0 / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 79.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 79.5%

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

      2. add-sqr-sqrt 79.4%

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

      3. times-frac 79.4%

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

      4. hypot-define 79.4%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. hypot-define 95.5%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Applied egg-rr 95.5%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if +inf.0 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 0.0%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 45.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 45.2%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg 45.2%

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]

      3. unsub-neg 45.2%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} - \frac{x.re}{y.im}} \]

      4. unpow2 45.2%

         \[\leadsto \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]

      5. associate-/r* 47.7%

         \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re}{y.im}}{y.im}} - \frac{x.re}{y.im} \]

      6. div-sub 47.8%

         \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}} \]

      7. *-commutative 47.8%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.im}}{y.im} - x.re}{y.im} \]

      8. associate-/l* 53.7%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im}} - x.re}{y.im} \]

      9. fmm-def 53.7%

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

   5. Simplified 53.7%

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

      1. clear-num 53.7%

         \[\leadsto \frac{\mathsf{fma}\left(y.re, \color{blue}{\frac{1}{\frac{y.im}{x.im}}}, -x.re\right)}{y.im} \]

      2. inv-pow 53.7%

         \[\leadsto \frac{\mathsf{fma}\left(y.re, \color{blue}{{\left(\frac{y.im}{x.im}\right)}^{-1}}, -x.re\right)}{y.im} \]

   7. Applied egg-rr 53.7%

      \[\leadsto \frac{\mathsf{fma}\left(y.re, \color{blue}{{\left(\frac{y.im}{x.im}\right)}^{-1}}, -x.re\right)}{y.im} \]
   8. Step-by-step derivation

      1. unpow-1 53.7%

         \[\leadsto \frac{\mathsf{fma}\left(y.re, \color{blue}{\frac{1}{\frac{y.im}{x.im}}}, -x.re\right)}{y.im} \]

   9. Simplified 53.7%

      \[\leadsto \frac{\mathsf{fma}\left(y.re, \color{blue}{\frac{1}{\frac{y.im}{x.im}}}, -x.re\right)}{y.im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))
   (if (<= y.im -2.7e+138)
     t_0
     (if (<= y.im -9.5e-129)
       (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 8.8e-11) (/ (- x.im (/ (* x.re y.im) y.re)) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -2.7e+138) {
		tmp = t_0;
	} else if (y_46_im <= -9.5e-129) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 8.8e-11) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    if (y_46im <= (-2.7d+138)) then
        tmp = t_0
    else if (y_46im <= (-9.5d-129)) then
        tmp = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 8.8d-11) then
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -2.7e+138) {
		tmp = t_0;
	} else if (y_46_im <= -9.5e-129) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 8.8e-11) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -2.7e+138:
		tmp = t_0
	elif y_46_im <= -9.5e-129:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 8.8e-11:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.7e+138)
		tmp = t_0;
	elseif (y_46_im <= -9.5e-129)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 8.8e-11)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.7e+138)
		tmp = t_0;
	elseif (y_46_im <= -9.5e-129)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 8.8e-11)
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.7e+138], t$95$0, If[LessEqual[y$46$im, -9.5e-129], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.8e-11], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.70000000000000009e138 or 8.8000000000000006e-11 < y.im

   1. Initial program 43.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 43.3%

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

      2. add-sqr-sqrt 43.3%

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

      3. times-frac 43.2%

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

      4. hypot-define 43.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. hypot-define 62.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   5. Taylor expanded in y.im around inf 83.5%

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

      1. +-commutative 83.5%

         \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}}{y.im} \]

      2. *-commutative 83.5%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.im}}{y.im} + -1 \cdot x.re}{y.im} \]

      3. associate-*r/ 86.9%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im}} + -1 \cdot x.re}{y.im} \]

      4. neg-mul-1 86.9%

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

      5. sub-neg 86.9%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im} - x.re}}{y.im} \]

   7. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}} \]

   if -2.70000000000000009e138 < y.im < -9.5000000000000006e-129

   1. Initial program 87.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   if -9.5000000000000006e-129 < y.im < 8.8000000000000006e-11

   1. Initial program 70.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 91.0%

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

      1. remove-double-neg 91.0%

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

      2. mul-1-neg 91.0%

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

      3. mul-1-neg 91.0%

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

      4. distribute-neg-in 91.0%

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

      5. distribute-neg-in 91.0%

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

      6. mul-1-neg 91.0%

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

      7. remove-double-neg 91.0%

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

      8. mul-1-neg 91.0%

         \[\leadsto \frac{x.im + \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{y.re}}}{y.re} \]

      9. associate-*r/ 91.0%

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

      10. associate-*r* 91.0%

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

      11. mul-1-neg 91.0%

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

   5. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{x.im + \frac{\left(-x.re\right) \cdot y.im}{y.re}}{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+138}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.4e+32)
   (- (/ x.im y.re) (/ (* y.im (/ x.re y.re)) y.re))
   (if (<= y.re 8.8e-11)
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
     (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.4e+32) {
		tmp = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re);
	} else if (y_46_re <= 8.8e-11) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-4.4d+32)) then
        tmp = (x_46im / y_46re) - ((y_46im * (x_46re / y_46re)) / y_46re)
    else if (y_46re <= 8.8d-11) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.4e+32) {
		tmp = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re);
	} else if (y_46_re <= 8.8e-11) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -4.4e+32:
		tmp = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re)
	elif y_46_re <= 8.8e-11:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.4e+32)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) / y_46_re));
	elseif (y_46_re <= 8.8e-11)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -4.4e+32)
		tmp = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re);
	elseif (y_46_re <= 8.8e-11)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.4e+32], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 8.8e-11], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.40000000000000002e32

   1. Initial program 56.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 75.9%

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

      1. remove-double-neg 75.9%

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

      2. mul-1-neg 75.9%

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

      3. mul-1-neg 75.9%

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

      4. distribute-neg-in 75.9%

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

      5. distribute-neg-in 75.9%

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

      6. mul-1-neg 75.9%

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

      7. remove-double-neg 75.9%

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

      8. unsub-neg 75.9%

         \[\leadsto \frac{\color{blue}{x.im - \frac{x.re \cdot y.im}{y.re}}}{y.re} \]

      9. associate-/l* 81.7%

         \[\leadsto \frac{x.im - \color{blue}{x.re \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 81.7%

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

      1. clear-num 81.7%

         \[\leadsto \frac{x.im - x.re \cdot \color{blue}{\frac{1}{\frac{y.re}{y.im}}}}{y.re} \]

      2. un-div-inv 81.7%

         \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{\frac{y.re}{y.im}}}}{y.re} \]

   7. Applied egg-rr 81.7%

      \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{\frac{y.re}{y.im}}}}{y.re} \]
   8. Step-by-step derivation

      1. div-sub 81.7%

         \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}} \]

      2. associate-/r/ 83.2%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\frac{x.re}{y.re} \cdot y.im}}{y.re} \]

      3. *-commutative 83.2%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot \frac{x.re}{y.re}}}{y.re} \]

   9. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}} \]

   if -4.40000000000000002e32 < y.re < 8.8000000000000006e-11

   1. Initial program 69.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 69.7%

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

      2. add-sqr-sqrt 69.7%

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

      3. times-frac 69.6%

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

      4. hypot-define 69.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. hypot-define 82.8%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Applied egg-rr 82.8%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   5. Taylor expanded in y.im around inf 80.9%

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

      1. +-commutative 80.9%

         \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}}{y.im} \]

      2. *-commutative 80.9%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.im}}{y.im} + -1 \cdot x.re}{y.im} \]

      3. associate-*r/ 78.9%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im}} + -1 \cdot x.re}{y.im} \]

      4. neg-mul-1 78.9%

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

      5. sub-neg 78.9%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im} - x.re}}{y.im} \]

   7. Simplified 78.9%

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

      1. clear-num 78.9%

         \[\leadsto \frac{y.re \cdot \color{blue}{\frac{1}{\frac{y.im}{x.im}}} - x.re}{y.im} \]

      2. un-div-inv 78.9%

         \[\leadsto \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.im}}} - x.re}{y.im} \]

   9. Applied egg-rr 78.9%

      \[\leadsto \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.im}}} - x.re}{y.im} \]
   10. Step-by-step derivation

       1. associate-/r/ 81.0%

          \[\leadsto \frac{\color{blue}{\frac{y.re}{y.im} \cdot x.im} - x.re}{y.im} \]

   11. Simplified 81.0%

       \[\leadsto \frac{\color{blue}{\frac{y.re}{y.im} \cdot x.im} - x.re}{y.im} \]

   if 8.8000000000000006e-11 < y.re

   1. Initial program 60.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 78.9%

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

      1. remove-double-neg 78.9%

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

      2. mul-1-neg 78.9%

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

      3. mul-1-neg 78.9%

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

      4. distribute-neg-in 78.9%

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

      5. distribute-neg-in 78.9%

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

      6. mul-1-neg 78.9%

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

      7. remove-double-neg 78.9%

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

      8. unsub-neg 78.9%

         \[\leadsto \frac{\color{blue}{x.im - \frac{x.re \cdot y.im}{y.re}}}{y.re} \]

      9. associate-/l* 81.3%

         \[\leadsto \frac{x.im - \color{blue}{x.re \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 81.3%

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

      1. clear-num 81.2%

         \[\leadsto \frac{x.im - x.re \cdot \color{blue}{\frac{1}{\frac{y.re}{y.im}}}}{y.re} \]

      2. un-div-inv 81.3%

         \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{\frac{y.re}{y.im}}}}{y.re} \]

   7. Applied egg-rr 81.3%

      \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{\frac{y.re}{y.im}}}}{y.re} \]
   8. Step-by-step derivation

      1. div-sub 81.3%

         \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}} \]

      2. associate-/r/ 81.3%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\frac{x.re}{y.re} \cdot y.im}}{y.re} \]

      3. *-commutative 81.3%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot \frac{x.re}{y.re}}}{y.re} \]

   9. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}} \]
   10. Step-by-step derivation

       1. associate-/l* 83.3%

          \[\leadsto \frac{x.im}{y.re} - \color{blue}{y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}} \]

   11. Applied egg-rr 83.3%

       \[\leadsto \frac{x.im}{y.re} - \color{blue}{y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4e+32)
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (if (<= y.re 7.8e-8)
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
     (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4e+32) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 7.8e-8) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-4d+32)) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else if (y_46re <= 7.8d-8) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4e+32) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 7.8e-8) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -4e+32:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= 7.8e-8:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4e+32)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= 7.8e-8)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -4e+32)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= 7.8e-8)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4e+32], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 7.8e-8], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.00000000000000021e32

   1. Initial program 56.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 75.9%

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

      1. remove-double-neg 75.9%

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

      2. mul-1-neg 75.9%

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

      3. mul-1-neg 75.9%

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

      4. distribute-neg-in 75.9%

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

      5. distribute-neg-in 75.9%

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

      6. mul-1-neg 75.9%

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

      7. remove-double-neg 75.9%

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

      8. unsub-neg 75.9%

         \[\leadsto \frac{\color{blue}{x.im - \frac{x.re \cdot y.im}{y.re}}}{y.re} \]

      9. associate-/l* 81.7%

         \[\leadsto \frac{x.im - \color{blue}{x.re \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 81.7%

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

      1. clear-num 81.7%

         \[\leadsto \frac{x.im - x.re \cdot \color{blue}{\frac{1}{\frac{y.re}{y.im}}}}{y.re} \]

      2. un-div-inv 81.7%

         \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{\frac{y.re}{y.im}}}}{y.re} \]

   7. Applied egg-rr 81.7%

      \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{\frac{y.re}{y.im}}}}{y.re} \]
   8. Step-by-step derivation

      1. associate-/r/ 83.1%

         \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{y.re} \cdot y.im}}{y.re} \]

   9. Simplified 83.1%

      \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{y.re} \cdot y.im}}{y.re} \]

   if -4.00000000000000021e32 < y.re < 7.7999999999999997e-8

   1. Initial program 69.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 69.7%

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

      2. add-sqr-sqrt 69.7%

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

      3. times-frac 69.6%

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

      4. hypot-define 69.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. hypot-define 82.8%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Applied egg-rr 82.8%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   5. Taylor expanded in y.im around inf 80.9%

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

      1. +-commutative 80.9%

         \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}}{y.im} \]

      2. *-commutative 80.9%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.im}}{y.im} + -1 \cdot x.re}{y.im} \]

      3. associate-*r/ 78.9%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im}} + -1 \cdot x.re}{y.im} \]

      4. neg-mul-1 78.9%

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

      5. sub-neg 78.9%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im} - x.re}}{y.im} \]

   7. Simplified 78.9%

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

      1. clear-num 78.9%

         \[\leadsto \frac{y.re \cdot \color{blue}{\frac{1}{\frac{y.im}{x.im}}} - x.re}{y.im} \]

      2. un-div-inv 78.9%

         \[\leadsto \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.im}}} - x.re}{y.im} \]

   9. Applied egg-rr 78.9%

      \[\leadsto \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.im}}} - x.re}{y.im} \]
   10. Step-by-step derivation

       1. associate-/r/ 81.0%

          \[\leadsto \frac{\color{blue}{\frac{y.re}{y.im} \cdot x.im} - x.re}{y.im} \]

   11. Simplified 81.0%

       \[\leadsto \frac{\color{blue}{\frac{y.re}{y.im} \cdot x.im} - x.re}{y.im} \]

   if 7.7999999999999997e-8 < y.re

   1. Initial program 60.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 78.9%

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

      1. remove-double-neg 78.9%

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

      2. mul-1-neg 78.9%

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

      3. mul-1-neg 78.9%

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

      4. distribute-neg-in 78.9%

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

      5. distribute-neg-in 78.9%

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

      6. mul-1-neg 78.9%

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

      7. remove-double-neg 78.9%

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

      8. unsub-neg 78.9%

         \[\leadsto \frac{\color{blue}{x.im - \frac{x.re \cdot y.im}{y.re}}}{y.re} \]

      9. associate-/l* 81.3%

         \[\leadsto \frac{x.im - \color{blue}{x.re \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 81.3%

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

      1. clear-num 81.2%

         \[\leadsto \frac{x.im - x.re \cdot \color{blue}{\frac{1}{\frac{y.re}{y.im}}}}{y.re} \]

      2. un-div-inv 81.3%

         \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{\frac{y.re}{y.im}}}}{y.re} \]

   7. Applied egg-rr 81.3%

      \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{\frac{y.re}{y.im}}}}{y.re} \]
   8. Step-by-step derivation

      1. div-sub 81.3%

         \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}} \]

      2. associate-/r/ 81.3%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\frac{x.re}{y.re} \cdot y.im}}{y.re} \]

      3. *-commutative 81.3%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot \frac{x.re}{y.re}}}{y.re} \]

   9. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}} \]
   10. Step-by-step derivation

       1. associate-/l* 83.3%

          \[\leadsto \frac{x.im}{y.re} - \color{blue}{y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}} \]

   11. Applied egg-rr 83.3%

       \[\leadsto \frac{x.im}{y.re} - \color{blue}{y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+48} \lor \neg \left(y.im \leq 2.45 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.4e+48) (not (<= y.im 2.45e-12)))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (/ (- x.im (/ (* x.re y.im) y.re)) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.4e+48) || !(y_46_im <= 2.45e-12)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-3.4d+48)) .or. (.not. (y_46im <= 2.45d-12))) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.4e+48) || !(y_46_im <= 2.45e-12)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -3.4e+48) or not (y_46_im <= 2.45e-12):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -3.4e+48) || !(y_46_im <= 2.45e-12))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -3.4e+48) || ~((y_46_im <= 2.45e-12)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -3.4e+48], N[Not[LessEqual[y$46$im, 2.45e-12]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.4000000000000003e48 or 2.44999999999999986e-12 < y.im

   1. Initial program 48.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 48.3%

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

      2. add-sqr-sqrt 48.3%

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

      3. times-frac 48.2%

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

      4. hypot-define 48.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. hypot-define 66.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Applied egg-rr 66.2%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   5. Taylor expanded in y.im around inf 81.0%

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

      1. +-commutative 81.0%

         \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}}{y.im} \]

      2. *-commutative 81.0%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.im}}{y.im} + -1 \cdot x.re}{y.im} \]

      3. associate-*r/ 84.0%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im}} + -1 \cdot x.re}{y.im} \]

      4. neg-mul-1 84.0%

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

      5. sub-neg 84.0%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im} - x.re}}{y.im} \]

   7. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}} \]

   if -3.4000000000000003e48 < y.im < 2.44999999999999986e-12

   1. Initial program 76.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 80.4%

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

      1. remove-double-neg 80.4%

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

      2. mul-1-neg 80.4%

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

      3. mul-1-neg 80.4%

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

      4. distribute-neg-in 80.4%

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

      5. distribute-neg-in 80.4%

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

      6. mul-1-neg 80.4%

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

      7. remove-double-neg 80.4%

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

      8. mul-1-neg 80.4%

         \[\leadsto \frac{x.im + \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{y.re}}}{y.re} \]

      9. associate-*r/ 80.4%

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

      10. associate-*r* 80.4%

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

      11. mul-1-neg 80.4%

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

   5. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{x.im + \frac{\left(-x.re\right) \cdot y.im}{y.re}}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+48} \lor \neg \left(y.im \leq 2.45 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+48} \lor \neg \left(y.im \leq 6.5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.4e+48) (not (<= y.im 6.5e-11)))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (/ (- x.im (/ x.re (/ y.re y.im))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.4e+48) || !(y_46_im <= 6.5e-11)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-3.4d+48)) .or. (.not. (y_46im <= 6.5d-11))) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.4e+48) || !(y_46_im <= 6.5e-11)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -3.4e+48) or not (y_46_im <= 6.5e-11):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -3.4e+48) || !(y_46_im <= 6.5e-11))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -3.4e+48) || ~((y_46_im <= 6.5e-11)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -3.4e+48], N[Not[LessEqual[y$46$im, 6.5e-11]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.4000000000000003e48 or 6.49999999999999953e-11 < y.im

   1. Initial program 48.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 48.3%

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

      2. add-sqr-sqrt 48.3%

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

      3. times-frac 48.2%

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

      4. hypot-define 48.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. hypot-define 66.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Applied egg-rr 66.2%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   5. Taylor expanded in y.im around inf 81.0%

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

      1. +-commutative 81.0%

         \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}}{y.im} \]

      2. *-commutative 81.0%

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.im}}{y.im} + -1 \cdot x.re}{y.im} \]

      3. associate-*r/ 84.0%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im}} + -1 \cdot x.re}{y.im} \]

      4. neg-mul-1 84.0%

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

      5. sub-neg 84.0%

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.im}{y.im} - x.re}}{y.im} \]

   7. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}} \]

   if -3.4000000000000003e48 < y.im < 6.49999999999999953e-11

   1. Initial program 76.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 80.4%

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

      1. remove-double-neg 80.4%

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

      2. mul-1-neg 80.4%

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

      3. mul-1-neg 80.4%

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

      4. distribute-neg-in 80.4%

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

      5. distribute-neg-in 80.4%

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

      6. mul-1-neg 80.4%

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

      7. remove-double-neg 80.4%

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

      8. unsub-neg 80.4%

         \[\leadsto \frac{\color{blue}{x.im - \frac{x.re \cdot y.im}{y.re}}}{y.re} \]

      9. associate-/l* 80.1%

         \[\leadsto \frac{x.im - \color{blue}{x.re \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 80.1%

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

      1. clear-num 80.2%

         \[\leadsto \frac{x.im - x.re \cdot \color{blue}{\frac{1}{\frac{y.re}{y.im}}}}{y.re} \]

      2. un-div-inv 80.1%

         \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{\frac{y.re}{y.im}}}}{y.re} \]

   7. Applied egg-rr 80.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+48} \lor \neg \left(y.im \leq 6.5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+48} \lor \neg \left(y.im \leq 4.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.2e+48) (not (<= y.im 4.5e-9)))
   (/ x.re (- y.im))
   (/ (- x.im (/ x.re (/ y.re y.im))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.2e+48) || !(y_46_im <= 4.5e-9)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.2d+48)) .or. (.not. (y_46im <= 4.5d-9))) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.2e+48) || !(y_46_im <= 4.5e-9)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.2e+48) or not (y_46_im <= 4.5e-9):
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.2e+48) || !(y_46_im <= 4.5e-9))
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.2e+48) || ~((y_46_im <= 4.5e-9)))
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.2e+48], N[Not[LessEqual[y$46$im, 4.5e-9]], $MachinePrecision]], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.1999999999999997e48 or 4.49999999999999976e-9 < y.im

   1. Initial program 48.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 71.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. associate-*r/ 71.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot x.re}{y.im}} \]

      2. mul-1-neg 71.5%

         \[\leadsto \frac{\color{blue}{-x.re}}{y.im} \]

   5. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{-x.re}{y.im}} \]

   if -4.1999999999999997e48 < y.im < 4.49999999999999976e-9

   1. Initial program 76.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 80.4%

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

      1. remove-double-neg 80.4%

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

      2. mul-1-neg 80.4%

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

      3. mul-1-neg 80.4%

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

      4. distribute-neg-in 80.4%

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

      5. distribute-neg-in 80.4%

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

      6. mul-1-neg 80.4%

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

      7. remove-double-neg 80.4%

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

      8. unsub-neg 80.4%

         \[\leadsto \frac{\color{blue}{x.im - \frac{x.re \cdot y.im}{y.re}}}{y.re} \]

      9. associate-/l* 80.1%

         \[\leadsto \frac{x.im - \color{blue}{x.re \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 80.1%

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

      1. clear-num 80.2%

         \[\leadsto \frac{x.im - x.re \cdot \color{blue}{\frac{1}{\frac{y.re}{y.im}}}}{y.re} \]

      2. un-div-inv 80.1%

         \[\leadsto \frac{x.im - \color{blue}{\frac{x.re}{\frac{y.re}{y.im}}}}{y.re} \]

   7. Applied egg-rr 80.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+48} \lor \neg \left(y.im \leq 4.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+53} \lor \neg \left(y.im \leq 4.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.2e+53) (not (<= y.im 4.1e-6)))
   (/ x.re (- y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.2e+53) || !(y_46_im <= 4.1e-6)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.2d+53)) .or. (.not. (y_46im <= 4.1d-6))) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.2e+53) || !(y_46_im <= 4.1e-6)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.2e+53) or not (y_46_im <= 4.1e-6):
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.2e+53) || !(y_46_im <= 4.1e-6))
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.2e+53) || ~((y_46_im <= 4.1e-6)))
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.2e+53], N[Not[LessEqual[y$46$im, 4.1e-6]], $MachinePrecision]], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.2e53 or 4.0999999999999997e-6 < y.im

   1. Initial program 48.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 71.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. associate-*r/ 71.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot x.re}{y.im}} \]

      2. mul-1-neg 71.5%

         \[\leadsto \frac{\color{blue}{-x.re}}{y.im} \]

   5. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{-x.re}{y.im}} \]

   if -1.2e53 < y.im < 4.0999999999999997e-6

   1. Initial program 76.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 80.4%

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

      1. remove-double-neg 80.4%

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

      2. mul-1-neg 80.4%

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

      3. mul-1-neg 80.4%

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

      4. distribute-neg-in 80.4%

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

      5. distribute-neg-in 80.4%

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

      6. mul-1-neg 80.4%

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

      7. remove-double-neg 80.4%

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

      8. unsub-neg 80.4%

         \[\leadsto \frac{\color{blue}{x.im - \frac{x.re \cdot y.im}{y.re}}}{y.re} \]

      9. associate-/l* 80.1%

         \[\leadsto \frac{x.im - \color{blue}{x.re \cdot \frac{y.im}{y.re}}}{y.re} \]

   5. Simplified 80.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+53} \lor \neg \left(y.im \leq 4.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.3 \cdot 10^{+32} \lor \neg \left(y.re \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.3e+32) (not (<= y.re 1.85e-6)))
   (/ x.im y.re)
   (/ x.re (- y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.3e+32) || !(y_46_re <= 1.85e-6)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-6.3d+32)) .or. (.not. (y_46re <= 1.85d-6))) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / -y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.3e+32) || !(y_46_re <= 1.85e-6)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -6.3e+32) or not (y_46_re <= 1.85e-6):
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / -y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -6.3e+32) || !(y_46_re <= 1.85e-6))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -6.3e+32) || ~((y_46_re <= 1.85e-6)))
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / -y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -6.3e+32], N[Not[LessEqual[y$46$re, 1.85e-6]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.3000000000000002e32 or 1.8500000000000001e-6 < y.re

   1. Initial program 58.6%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 68.2%

      \[\leadsto \color{blue}{\frac{x.im}{y.re}} \]

   if -6.3000000000000002e32 < y.re < 1.8500000000000001e-6

   1. Initial program 69.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 66.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. associate-*r/ 66.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot x.re}{y.im}} \]

      2. mul-1-neg 66.4%

         \[\leadsto \frac{\color{blue}{-x.re}}{y.im} \]

   5. Simplified 66.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.3 \cdot 10^{+32} \lor \neg \left(y.re \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+102} \lor \neg \left(y.im \leq 4.6 \cdot 10^{+134}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.5e+102) (not (<= y.im 4.6e+134)))
   (/ x.re y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.5e+102) || !(y_46_im <= 4.6e+134)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.5d+102)) .or. (.not. (y_46im <= 4.6d+134))) then
        tmp = x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.5e+102) || !(y_46_im <= 4.6e+134)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.5e+102) or not (y_46_im <= 4.6e+134):
		tmp = x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.5e+102) || !(y_46_im <= 4.6e+134))
		tmp = Float64(x_46_re / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.5e+102) || ~((y_46_im <= 4.6e+134)))
		tmp = x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.5e+102], N[Not[LessEqual[y$46$im, 4.6e+134]], $MachinePrecision]], N[(x$46$re / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.4999999999999999e102 or 4.5999999999999996e134 < y.im

   1. Initial program 36.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 78.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. associate-*r/ 78.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot x.re}{y.im}} \]

      2. mul-1-neg 78.0%

         \[\leadsto \frac{\color{blue}{-x.re}}{y.im} \]

   5. Simplified 78.0%

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

      1. div-inv 77.8%

         \[\leadsto \color{blue}{\left(-x.re\right) \cdot \frac{1}{y.im}} \]

      2. add-sqr-sqrt 44.8%

         \[\leadsto \color{blue}{\left(\sqrt{-x.re} \cdot \sqrt{-x.re}\right)} \cdot \frac{1}{y.im} \]

      3. sqrt-unprod 47.1%

         \[\leadsto \color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}} \cdot \frac{1}{y.im} \]

      4. sqr-neg 47.1%

         \[\leadsto \sqrt{\color{blue}{x.re \cdot x.re}} \cdot \frac{1}{y.im} \]

      5. sqrt-unprod 12.7%

         \[\leadsto \color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)} \cdot \frac{1}{y.im} \]

      6. add-sqr-sqrt 29.7%

         \[\leadsto \color{blue}{x.re} \cdot \frac{1}{y.im} \]

   7. Applied egg-rr 29.7%

      \[\leadsto \color{blue}{x.re \cdot \frac{1}{y.im}} \]
   8. Step-by-step derivation

      1. associate-*r/ 29.7%

         \[\leadsto \color{blue}{\frac{x.re \cdot 1}{y.im}} \]

      2. *-rgt-identity 29.7%

         \[\leadsto \frac{\color{blue}{x.re}}{y.im} \]

   9. Simplified 29.7%

      \[\leadsto \color{blue}{\frac{x.re}{y.im}} \]

   if -1.4999999999999999e102 < y.im < 4.5999999999999996e134

   1. Initial program 76.0%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 54.5%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+102} \lor \neg \left(y.im \leq 4.6 \cdot 10^{+134}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.42 \cdot 10^{+165}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.42e+165) (/ x.im y.im) (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.42e+165) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.42d+165)) then
        tmp = x_46im / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.42e+165) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.42e+165:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.42e+165)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.42e+165)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.42e+165], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.42e165

   1. Initial program 21.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 21.8%

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

      2. add-sqr-sqrt 21.8%

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

      3. times-frac 21.8%

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

      4. hypot-define 21.8%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. hypot-define 52.4%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   5. Taylor expanded in y.re around -inf 3.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot x.im + \frac{x.re \cdot y.im}{y.re}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 8.3%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot x.im + \color{blue}{x.re \cdot \frac{y.im}{y.re}}\right) \]

      2. +-commutative 8.3%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot \frac{y.im}{y.re} + -1 \cdot x.im\right)} \]

      3. neg-mul-1 8.3%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot \frac{y.im}{y.re} + \color{blue}{\left(-x.im\right)}\right) \]

      4. unsub-neg 8.3%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot \frac{y.im}{y.re} - x.im\right)} \]

      5. associate-*r/ 3.1%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.re \cdot y.im}{y.re}} - x.im\right) \]

      6. *-commutative 3.1%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{\color{blue}{y.im \cdot x.re}}{y.re} - x.im\right) \]

      7. associate-/l* 8.6%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{y.im \cdot \frac{x.re}{y.re}} - x.im\right) \]

   7. Simplified 8.6%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(y.im \cdot \frac{x.re}{y.re} - x.im\right)} \]

   8. Taylor expanded in y.im around -inf 3.6%

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

      1. +-commutative 3.6%

         \[\leadsto \color{blue}{\frac{x.im}{y.im} + -1 \cdot \frac{x.re}{y.re}} \]

      2. neg-mul-1 3.6%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(-\frac{x.re}{y.re}\right)} \]

      3. sub-neg 3.6%

         \[\leadsto \color{blue}{\frac{x.im}{y.im} - \frac{x.re}{y.re}} \]

   10. Simplified 3.6%

       \[\leadsto \color{blue}{\frac{x.im}{y.im} - \frac{x.re}{y.re}} \]

   11. Taylor expanded in x.im around inf 21.0%

       \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if -1.42e165 < y.im

   1. Initial program 71.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 48.3%

      \[\leadsto \color{blue}{\frac{x.im}{y.re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 9.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.im}{y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.im y.im))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_im;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = x_46im / y_46im
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_im;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return x_46_im / y_46_im

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_46_im / y_46_im)
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = x_46_im / y_46_im;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(x$46$im / y$46$im), $MachinePrecision]

Derivation

1. Initial program 64.6%

   \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 64.6%

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

   2. add-sqr-sqrt 64.6%

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

   3. times-frac 64.5%

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

   4. hypot-define 64.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

   5. hypot-define 78.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

4. Applied egg-rr 78.1%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

5. Taylor expanded in y.re around -inf 27.9%

   \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot x.im + \frac{x.re \cdot y.im}{y.re}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 29.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot x.im + \color{blue}{x.re \cdot \frac{y.im}{y.re}}\right) \]

   2. +-commutative 29.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot \frac{y.im}{y.re} + -1 \cdot x.im\right)} \]

   3. neg-mul-1 29.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot \frac{y.im}{y.re} + \color{blue}{\left(-x.im\right)}\right) \]

   4. unsub-neg 29.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot \frac{y.im}{y.re} - x.im\right)} \]

   5. associate-*r/ 27.9%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.re \cdot y.im}{y.re}} - x.im\right) \]

   6. *-commutative 27.9%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{\color{blue}{y.im \cdot x.re}}{y.re} - x.im\right) \]

   7. associate-/l* 29.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{y.im \cdot \frac{x.re}{y.re}} - x.im\right) \]

7. Simplified 29.0%

   \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(y.im \cdot \frac{x.re}{y.re} - x.im\right)} \]

8. Taylor expanded in y.im around -inf 4.9%

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

   1. +-commutative 4.9%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + -1 \cdot \frac{x.re}{y.re}} \]

   2. neg-mul-1 4.9%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(-\frac{x.re}{y.re}\right)} \]

   3. sub-neg 4.9%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} - \frac{x.re}{y.re}} \]

10. Simplified 4.9%

    \[\leadsto \color{blue}{\frac{x.im}{y.im} - \frac{x.re}{y.re}} \]

11. Taylor expanded in x.im around inf 10.1%

    \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024186 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))