_divideComplex, imaginary part

Percentage Accurate: 62.6% → 97.8%

Time: 15.6s

Alternatives: 18

Speedup: 1.8×

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: 62.6% 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: 97.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - {\left(\sqrt[3]{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}\right)}^{3}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (/ 1.0 (hypot y.re y.im))
  (-
   (/ y.re (/ (hypot y.re y.im) x.im))
   (pow (cbrt (* (/ y.im (hypot y.re y.im)) x.re)) 3.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - pow(cbrt(((y_46_im / hypot(y_46_re, y_46_im)) * x_46_re)), 3.0));
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (1.0 / Math.hypot(y_46_re, y_46_im)) * ((y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im)) - Math.pow(Math.cbrt(((y_46_im / Math.hypot(y_46_re, y_46_im)) * x_46_re)), 3.0));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im)) - (cbrt(Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * x_46_re)) ^ 3.0)))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - N[Power[N[Power[N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

   1. *-un-lft-identity 61.5%

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

   2. add-sqr-sqrt 61.5%

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

   3. times-frac 61.4%

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

   4. hypot-def 61.5%

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

   5. hypot-def 73.2%

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

3. Applied egg-rr 73.2%

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

   1. div-sub 73.2%

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

   2. *-commutative 73.2%

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

   3. *-commutative 73.2%

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

5. Applied egg-rr 73.2%

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

   1. associate-/l* 83.7%

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

   2. associate-/l* 97.3%

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

7. Simplified 97.3%

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

   1. add-cube-cbrt 96.7%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\left(\sqrt[3]{\frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}} \cdot \sqrt[3]{\frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}\right) \cdot \sqrt[3]{\frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}}\right) \]

   2. pow3 96.7%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{{\left(\sqrt[3]{\frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}\right)}^{3}}\right) \]

   3. associate-/r/ 97.6%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - {\left(\sqrt[3]{\color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}}\right)}^{3}\right) \]

9. Applied egg-rr 97.6%

   \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{{\left(\sqrt[3]{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}\right)}^{3}}\right) \]

10. Final simplification 97.6%

    \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - {\left(\sqrt[3]{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}\right)}^{3}\right) \]

Alternative 2: 88.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 5e+261)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.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 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+261) {
		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 / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_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(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+261)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(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[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $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], 5e+261], 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[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-x$46$re) / y$46$im), $MachinePrecision]), $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))) < 5.0000000000000001e261

   1. Initial program 81.7%

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

      1. *-un-lft-identity 81.7%

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

      2. add-sqr-sqrt 81.7%

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

      3. times-frac 81.6%

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

      4. hypot-def 81.7%

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

      5. hypot-def 95.9%

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

   3. Applied egg-rr 95.9%

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

   if 5.0000000000000001e261 < (/.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 12.7%

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

      1. div-sub 9.7%

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

      2. sub-neg 9.7%

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

      3. *-commutative 9.7%

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

      4. add-sqr-sqrt 9.7%

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

      5. times-frac 13.7%

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

      6. fma-def 13.7%

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

      7. hypot-def 13.7%

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

      8. hypot-def 47.0%

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

      9. associate-/l* 53.8%

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

      10. add-sqr-sqrt 53.8%

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

      11. pow2 53.8%

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

      12. hypot-def 53.8%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}}{y.im}}\right) \]

   3. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)} \]

   4. Taylor expanded in y.re around 0 67.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \]

Alternative 3: 97.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (/ 1.0 (hypot y.re y.im))
  (- (/ y.re (/ (hypot y.re y.im) x.im)) (/ y.im (/ (hypot y.re y.im) x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re)));
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (1.0 / Math.hypot(y_46_re, y_46_im)) * ((y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im)) - (y_46_im / (Math.hypot(y_46_re, y_46_im) / x_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (1.0 / math.hypot(y_46_re, y_46_im)) * ((y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im)) - (y_46_im / (math.hypot(y_46_re, y_46_im) / x_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im)) - Float64(y_46_im / Float64(hypot(y_46_re, y_46_im) / x_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(y$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

   1. *-un-lft-identity 61.5%

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

   2. add-sqr-sqrt 61.5%

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

   3. times-frac 61.4%

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

   4. hypot-def 61.5%

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

   5. hypot-def 73.2%

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

3. Applied egg-rr 73.2%

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

   1. div-sub 73.2%

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

   2. *-commutative 73.2%

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

   3. *-commutative 73.2%

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

5. Applied egg-rr 73.2%

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

   1. associate-/l* 83.7%

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

   2. associate-/l* 97.3%

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

7. Simplified 97.3%

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

8. Final simplification 97.3%

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

Alternative 4: 87.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_1}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))) (t_1 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_1 (+ (* y.re y.re) (* y.im y.im))) INFINITY)
     (* t_0 (/ t_1 (hypot y.re y.im)))
     (* t_0 (- (/ y.re (/ (hypot y.re y.im) x.im)) x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = t_0 * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_1 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = t_0 * ((y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if (t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= math.inf:
		tmp = t_0 * (t_1 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = t_0 * ((y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= Inf)
		tmp = Float64(t_0 * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(t_0 * Float64(Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Inf)
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	else
		tmp = t_0 * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $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 79.1%

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

      1. *-un-lft-identity 79.1%

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

      2. add-sqr-sqrt 79.1%

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

      3. times-frac 79.0%

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

      4. hypot-def 79.1%

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

      5. hypot-def 93.4%

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

   3. Applied egg-rr 93.4%

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

   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. Step-by-step derivation

      1. *-un-lft-identity 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 0.0%

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

      4. hypot-def 0.0%

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

      5. hypot-def 2.7%

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

   3. Applied egg-rr 2.7%

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

      1. div-sub 2.7%

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

      2. *-commutative 2.7%

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

      3. *-commutative 2.7%

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

   5. Applied egg-rr 2.7%

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

      1. associate-/l* 48.7%

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

      2. associate-/l* 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in y.im around inf 63.8%

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

4. Final simplification 86.8%

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

Alternative 5: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{y.im}{\frac{y.re}{x.re}} \cdot \frac{-1}{y.re}\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-160}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 y.im) (- (/ x.im (/ y.im y.re)) x.re))))
   (if (<= y.re -3.6e+59)
     (+ (/ x.im y.re) (* (/ y.im (/ y.re x.re)) (/ -1.0 y.re)))
     (if (<= y.re -8.6e-51)
       t_0
       (if (<= y.re -1.45e-160)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 7.2e-122)
           t_0
           (*
            (/ 1.0 (hypot y.re y.im))
            (- x.im (/ y.im (/ (hypot y.re y.im) x.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	double tmp;
	if (y_46_re <= -3.6e+59) {
		tmp = (x_46_im / y_46_re) + ((y_46_im / (y_46_re / x_46_re)) * (-1.0 / y_46_re));
	} else if (y_46_re <= -8.6e-51) {
		tmp = t_0;
	} else if (y_46_re <= -1.45e-160) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 7.2e-122) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	double tmp;
	if (y_46_re <= -3.6e+59) {
		tmp = (x_46_im / y_46_re) + ((y_46_im / (y_46_re / x_46_re)) * (-1.0 / y_46_re));
	} else if (y_46_re <= -8.6e-51) {
		tmp = t_0;
	} else if (y_46_re <= -1.45e-160) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 7.2e-122) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im - (y_46_im / (Math.hypot(y_46_re, y_46_im) / x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re)
	tmp = 0
	if y_46_re <= -3.6e+59:
		tmp = (x_46_im / y_46_re) + ((y_46_im / (y_46_re / x_46_re)) * (-1.0 / y_46_re))
	elif y_46_re <= -8.6e-51:
		tmp = t_0
	elif y_46_re <= -1.45e-160:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 7.2e-122:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im - (y_46_im / (math.hypot(y_46_re, y_46_im) / x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re))
	tmp = 0.0
	if (y_46_re <= -3.6e+59)
		tmp = Float64(Float64(x_46_im / y_46_re) + Float64(Float64(y_46_im / Float64(y_46_re / x_46_re)) * Float64(-1.0 / y_46_re)));
	elseif (y_46_re <= -8.6e-51)
		tmp = t_0;
	elseif (y_46_re <= -1.45e-160)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 7.2e-122)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im - Float64(y_46_im / Float64(hypot(y_46_re, y_46_im) / x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	tmp = 0.0;
	if (y_46_re <= -3.6e+59)
		tmp = (x_46_im / y_46_re) + ((y_46_im / (y_46_re / x_46_re)) * (-1.0 / y_46_re));
	elseif (y_46_re <= -8.6e-51)
		tmp = t_0;
	elseif (y_46_re <= -1.45e-160)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 7.2e-122)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re)));
	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[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.6e+59], N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -8.6e-51], t$95$0, If[LessEqual[y$46$re, -1.45e-160], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2e-122], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im - N[(y$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.5999999999999999e59

   1. Initial program 45.8%

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

   2. Taylor expanded in y.re around inf 76.7%

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

      1. *-un-lft-identity 76.7%

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

      2. pow2 76.7%

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

      3. times-frac 76.9%

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

      4. *-commutative 76.9%

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

   4. Applied egg-rr 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-/l* 84.0%

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

   6. Simplified 84.0%

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

   if -3.5999999999999999e59 < y.re < -8.5999999999999995e-51 or -1.45e-160 < y.re < 7.19999999999999989e-122

   1. Initial program 67.3%

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

      1. *-un-lft-identity 67.3%

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

      2. add-sqr-sqrt 67.3%

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

      3. times-frac 67.2%

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

      4. hypot-def 67.2%

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

      5. hypot-def 77.7%

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

   3. Applied egg-rr 77.7%

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

   4. Taylor expanded in y.re around 0 50.0%

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

      1. +-commutative 50.0%

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

      2. mul-1-neg 50.0%

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

      3. unsub-neg 50.0%

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

      4. associate-/l* 50.4%

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

   6. Simplified 50.4%

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

   7. Taylor expanded in y.re around 0 90.1%

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

   if -8.5999999999999995e-51 < y.re < -1.45e-160

   1. Initial program 91.5%

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

   if 7.19999999999999989e-122 < y.re

   1. Initial program 56.5%

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

      1. *-un-lft-identity 56.5%

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

      2. add-sqr-sqrt 56.5%

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

      3. times-frac 56.5%

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

      4. hypot-def 56.5%

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

      5. hypot-def 73.2%

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

   3. Applied egg-rr 73.2%

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

      1. div-sub 73.1%

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

      2. *-commutative 73.1%

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

      3. *-commutative 73.1%

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

   5. Applied egg-rr 73.1%

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

      1. associate-/l* 88.4%

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

      2. associate-/l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in y.re around inf 82.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{y.im}{\frac{y.re}{x.re}} \cdot \frac{-1}{y.re}\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-160}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)\\ \end{array} \]

Alternative 6: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -15000:\\ \;\;\;\;t_0 \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{y.im \cdot x.re}{y.re} \cdot \frac{-1}{y.re} + \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-53}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))))
   (if (<= y.im -15000.0)
     (* t_0 (- x.re (/ x.im (/ y.im y.re))))
     (if (<= y.im 8.2e-141)
       (+ (* (/ (* y.im x.re) y.re) (/ -1.0 y.re)) (/ x.im y.re))
       (if (<= y.im 2.05e-53)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.im 1.16e-15)
           (/ x.im y.re)
           (* t_0 (- (/ y.re (/ (hypot y.re y.im) x.im)) x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -15000.0) {
		tmp = t_0 * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	} else if (y_46_im <= 8.2e-141) {
		tmp = (((y_46_im * x_46_re) / y_46_re) * (-1.0 / y_46_re)) + (x_46_im / y_46_re);
	} else if (y_46_im <= 2.05e-53) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.16e-15) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0 * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -15000.0) {
		tmp = t_0 * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	} else if (y_46_im <= 8.2e-141) {
		tmp = (((y_46_im * x_46_re) / y_46_re) * (-1.0 / y_46_re)) + (x_46_im / y_46_re);
	} else if (y_46_im <= 2.05e-53) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.16e-15) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0 * ((y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	tmp = 0
	if y_46_im <= -15000.0:
		tmp = t_0 * (x_46_re - (x_46_im / (y_46_im / y_46_re)))
	elif y_46_im <= 8.2e-141:
		tmp = (((y_46_im * x_46_re) / y_46_re) * (-1.0 / y_46_re)) + (x_46_im / y_46_re)
	elif y_46_im <= 2.05e-53:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 1.16e-15:
		tmp = x_46_im / y_46_re
	else:
		tmp = t_0 * ((y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_im <= -15000.0)
		tmp = Float64(t_0 * Float64(x_46_re - Float64(x_46_im / Float64(y_46_im / y_46_re))));
	elseif (y_46_im <= 8.2e-141)
		tmp = Float64(Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) * Float64(-1.0 / y_46_re)) + Float64(x_46_im / y_46_re));
	elseif (y_46_im <= 2.05e-53)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.16e-15)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(t_0 * Float64(Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	tmp = 0.0;
	if (y_46_im <= -15000.0)
		tmp = t_0 * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	elseif (y_46_im <= 8.2e-141)
		tmp = (((y_46_im * x_46_re) / y_46_re) * (-1.0 / y_46_re)) + (x_46_im / y_46_re);
	elseif (y_46_im <= 2.05e-53)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 1.16e-15)
		tmp = x_46_im / y_46_re;
	else
		tmp = t_0 * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -15000.0], N[(t$95$0 * N[(x$46$re - N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.2e-141], N[(N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.05e-53], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $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, 1.16e-15], N[(x$46$im / y$46$re), $MachinePrecision], N[(t$95$0 * N[(N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -15000

   1. Initial program 52.1%

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

      1. *-un-lft-identity 52.1%

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

      2. add-sqr-sqrt 52.1%

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

      3. times-frac 52.1%

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

      4. hypot-def 52.1%

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

      5. hypot-def 64.7%

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

   3. Applied egg-rr 64.7%

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

   4. Taylor expanded in y.im around -inf 75.8%

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

      1. mul-1-neg 75.8%

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

      2. unsub-neg 75.8%

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

      3. associate-/l* 82.8%

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

   6. Simplified 82.8%

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

   if -15000 < y.im < 8.20000000000000005e-141

   1. Initial program 69.8%

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

   2. Taylor expanded in y.re around inf 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. pow2 76.3%

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

      3. times-frac 83.6%

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

      4. *-commutative 83.6%

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

   4. Applied egg-rr 83.6%

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

   if 8.20000000000000005e-141 < y.im < 2.05e-53

   1. Initial program 93.4%

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

   if 2.05e-53 < y.im < 1.1599999999999999e-15

   1. Initial program 42.5%

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

   2. Taylor expanded in y.re around inf 100.0%

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

   if 1.1599999999999999e-15 < y.im

   1. Initial program 53.2%

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

      1. *-un-lft-identity 53.2%

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

      2. add-sqr-sqrt 53.2%

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

      3. times-frac 53.2%

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

      4. hypot-def 53.2%

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

      5. hypot-def 65.5%

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

   3. Applied egg-rr 65.5%

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

      1. div-sub 65.5%

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

      2. *-commutative 65.5%

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

      3. *-commutative 65.5%

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

   5. Applied egg-rr 65.5%

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

      1. associate-/l* 75.9%

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

      2. associate-/l* 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in y.im around inf 93.8%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{x.re}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -15000:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{y.im \cdot x.re}{y.re} \cdot \frac{-1}{y.re} + \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-53}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - x.re\right)\\ \end{array} \]

Alternative 7: 80.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ t_1 := \frac{x.im}{y.re} + \frac{y.im}{\frac{y.re}{x.re}} \cdot \frac{-1}{y.re}\\ t_2 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -5.1 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 y.im) (- (/ x.im (/ y.im y.re)) x.re)))
        (t_1 (+ (/ x.im y.re) (* (/ y.im (/ y.re x.re)) (/ -1.0 y.re))))
        (t_2
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -5.1e+61)
     t_1
     (if (<= y.re -7.8e-51)
       t_0
       (if (<= y.re -3.8e-160)
         t_2
         (if (<= y.re 4.6e-141) t_0 (if (<= y.re 7.2e+73) t_2 t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	double t_1 = (x_46_im / y_46_re) + ((y_46_im / (y_46_re / x_46_re)) * (-1.0 / y_46_re));
	double t_2 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -5.1e+61) {
		tmp = t_1;
	} else if (y_46_re <= -7.8e-51) {
		tmp = t_0;
	} else if (y_46_re <= -3.8e-160) {
		tmp = t_2;
	} else if (y_46_re <= 4.6e-141) {
		tmp = t_0;
	} else if (y_46_re <= 7.2e+73) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (1.0d0 / y_46im) * ((x_46im / (y_46im / y_46re)) - x_46re)
    t_1 = (x_46im / y_46re) + ((y_46im / (y_46re / x_46re)) * ((-1.0d0) / y_46re))
    t_2 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-5.1d+61)) then
        tmp = t_1
    else if (y_46re <= (-7.8d-51)) then
        tmp = t_0
    else if (y_46re <= (-3.8d-160)) then
        tmp = t_2
    else if (y_46re <= 4.6d-141) then
        tmp = t_0
    else if (y_46re <= 7.2d+73) then
        tmp = t_2
    else
        tmp = t_1
    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 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	double t_1 = (x_46_im / y_46_re) + ((y_46_im / (y_46_re / x_46_re)) * (-1.0 / y_46_re));
	double t_2 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -5.1e+61) {
		tmp = t_1;
	} else if (y_46_re <= -7.8e-51) {
		tmp = t_0;
	} else if (y_46_re <= -3.8e-160) {
		tmp = t_2;
	} else if (y_46_re <= 4.6e-141) {
		tmp = t_0;
	} else if (y_46_re <= 7.2e+73) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re)
	t_1 = (x_46_im / y_46_re) + ((y_46_im / (y_46_re / x_46_re)) * (-1.0 / y_46_re))
	t_2 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -5.1e+61:
		tmp = t_1
	elif y_46_re <= -7.8e-51:
		tmp = t_0
	elif y_46_re <= -3.8e-160:
		tmp = t_2
	elif y_46_re <= 4.6e-141:
		tmp = t_0
	elif y_46_re <= 7.2e+73:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re))
	t_1 = Float64(Float64(x_46_im / y_46_re) + Float64(Float64(y_46_im / Float64(y_46_re / x_46_re)) * Float64(-1.0 / y_46_re)))
	t_2 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -5.1e+61)
		tmp = t_1;
	elseif (y_46_re <= -7.8e-51)
		tmp = t_0;
	elseif (y_46_re <= -3.8e-160)
		tmp = t_2;
	elseif (y_46_re <= 4.6e-141)
		tmp = t_0;
	elseif (y_46_re <= 7.2e+73)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	t_1 = (x_46_im / y_46_re) + ((y_46_im / (y_46_re / x_46_re)) * (-1.0 / y_46_re));
	t_2 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -5.1e+61)
		tmp = t_1;
	elseif (y_46_re <= -7.8e-51)
		tmp = t_0;
	elseif (y_46_re <= -3.8e-160)
		tmp = t_2;
	elseif (y_46_re <= 4.6e-141)
		tmp = t_0;
	elseif (y_46_re <= 7.2e+73)
		tmp = t_2;
	else
		tmp = t_1;
	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[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.1e+61], t$95$1, If[LessEqual[y$46$re, -7.8e-51], t$95$0, If[LessEqual[y$46$re, -3.8e-160], t$95$2, If[LessEqual[y$46$re, 4.6e-141], t$95$0, If[LessEqual[y$46$re, 7.2e+73], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.1000000000000001e61 or 7.1999999999999998e73 < y.re

   1. Initial program 40.1%

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

   2. Taylor expanded in y.re around inf 73.4%

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

      1. *-un-lft-identity 73.4%

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

      2. pow2 73.4%

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

      3. times-frac 73.6%

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

      4. *-commutative 73.6%

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

   4. Applied egg-rr 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-/l* 81.9%

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

   6. Simplified 81.9%

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

   if -5.1000000000000001e61 < y.re < -7.7999999999999995e-51 or -3.7999999999999998e-160 < y.re < 4.5999999999999999e-141

   1. Initial program 66.6%

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

      1. *-un-lft-identity 66.6%

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

      2. add-sqr-sqrt 66.6%

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

      3. times-frac 66.6%

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

      4. hypot-def 66.6%

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

      5. hypot-def 77.2%

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

   3. Applied egg-rr 77.2%

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

   4. Taylor expanded in y.re around 0 49.5%

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

      1. +-commutative 49.5%

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

      2. mul-1-neg 49.5%

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

      3. unsub-neg 49.5%

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

      4. associate-/l* 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in y.re around 0 89.9%

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

   if -7.7999999999999995e-51 < y.re < -3.7999999999999998e-160 or 4.5999999999999999e-141 < y.re < 7.1999999999999998e73

   1. Initial program 86.1%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.1 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{y.im}{\frac{y.re}{x.re}} \cdot \frac{-1}{y.re}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{y.im}{\frac{y.re}{x.re}} \cdot \frac{-1}{y.re}\\ \end{array} \]

Alternative 8: 70.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-53}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 0.1:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 y.im) (- (/ x.im (/ y.im y.re)) x.re))))
   (if (<= y.im -4.1e+29)
     t_0
     (if (<= y.im -3.6e-54)
       (/ (* y.im (- x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 1.3e-121)
         (/ x.im y.re)
         (if (<= y.im 1.45e-53)
           (- (/ x.im (* y.im (/ y.im y.re))) (/ x.re y.im))
           (if (<= y.im 0.1) (/ x.im 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 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	double tmp;
	if (y_46_im <= -4.1e+29) {
		tmp = t_0;
	} else if (y_46_im <= -3.6e-54) {
		tmp = (y_46_im * -x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.3e-121) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.45e-53) {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	} else if (y_46_im <= 0.1) {
		tmp = x_46_im / 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 = (1.0d0 / y_46im) * ((x_46im / (y_46im / y_46re)) - x_46re)
    if (y_46im <= (-4.1d+29)) then
        tmp = t_0
    else if (y_46im <= (-3.6d-54)) then
        tmp = (y_46im * -x_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 1.3d-121) then
        tmp = x_46im / y_46re
    else if (y_46im <= 1.45d-53) then
        tmp = (x_46im / (y_46im * (y_46im / y_46re))) - (x_46re / y_46im)
    else if (y_46im <= 0.1d0) then
        tmp = x_46im / 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 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	double tmp;
	if (y_46_im <= -4.1e+29) {
		tmp = t_0;
	} else if (y_46_im <= -3.6e-54) {
		tmp = (y_46_im * -x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.3e-121) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.45e-53) {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	} else if (y_46_im <= 0.1) {
		tmp = x_46_im / 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 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re)
	tmp = 0
	if y_46_im <= -4.1e+29:
		tmp = t_0
	elif y_46_im <= -3.6e-54:
		tmp = (y_46_im * -x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 1.3e-121:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 1.45e-53:
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im)
	elif y_46_im <= 0.1:
		tmp = x_46_im / 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(1.0 / y_46_im) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re))
	tmp = 0.0
	if (y_46_im <= -4.1e+29)
		tmp = t_0;
	elseif (y_46_im <= -3.6e-54)
		tmp = Float64(Float64(y_46_im * Float64(-x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.3e-121)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1.45e-53)
		tmp = Float64(Float64(x_46_im / Float64(y_46_im * Float64(y_46_im / y_46_re))) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= 0.1)
		tmp = Float64(x_46_im / 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 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	tmp = 0.0;
	if (y_46_im <= -4.1e+29)
		tmp = t_0;
	elseif (y_46_im <= -3.6e-54)
		tmp = (y_46_im * -x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 1.3e-121)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 1.45e-53)
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	elseif (y_46_im <= 0.1)
		tmp = x_46_im / 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[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.1e+29], t$95$0, If[LessEqual[y$46$im, -3.6e-54], N[(N[(y$46$im * (-x$46$re)), $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, 1.3e-121], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.45e-53], N[(N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 0.1], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.1000000000000003e29 or 0.10000000000000001 < y.im

   1. Initial program 50.5%

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

      1. *-un-lft-identity 50.5%

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

      2. add-sqr-sqrt 50.5%

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

      3. times-frac 50.4%

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

      4. hypot-def 50.4%

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

      5. hypot-def 63.5%

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

   3. Applied egg-rr 63.5%

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

   4. Taylor expanded in y.re around 0 47.9%

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

      1. +-commutative 47.9%

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

      2. mul-1-neg 47.9%

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

      3. unsub-neg 47.9%

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

      4. associate-/l* 49.6%

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

   6. Simplified 49.6%

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

   7. Taylor expanded in y.re around 0 80.8%

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

   if -4.1000000000000003e29 < y.im < -3.59999999999999976e-54

   1. Initial program 90.5%

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

   2. Taylor expanded in x.im around 0 74.2%

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

      1. associate-*r* 74.2%

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

      2. neg-mul-1 74.2%

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

      3. *-commutative 74.2%

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

   4. Simplified 74.2%

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

   if -3.59999999999999976e-54 < y.im < 1.29999999999999993e-121 or 1.4499999999999999e-53 < y.im < 0.10000000000000001

   1. Initial program 66.3%

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

   2. Taylor expanded in y.re around inf 76.7%

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

   if 1.29999999999999993e-121 < y.im < 1.4499999999999999e-53

   1. Initial program 91.7%

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

   2. Taylor expanded in y.re around 0 66.6%

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

      1. +-commutative 66.6%

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

      2. mul-1-neg 66.6%

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

      3. unsub-neg 66.6%

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

      4. associate-/l* 66.6%

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

   4. Simplified 66.6%

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

      1. div-inv 66.6%

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

      2. unpow2 66.6%

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

      3. associate-*l* 66.6%

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

      4. div-inv 66.7%

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

   6. Applied egg-rr 66.7%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{elif}\;y.im \leq -3.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-53}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 0.1:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]

Alternative 9: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1500 \lor \neg \left(y.im \leq 8.4 \cdot 10^{-122} \lor \neg \left(y.im \leq 3 \cdot 10^{-57}\right) \land y.im \leq 0.215\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1500.0)
         (not
          (or (<= y.im 8.4e-122) (and (not (<= y.im 3e-57)) (<= y.im 0.215)))))
   (* (/ 1.0 y.im) (- (/ x.im (/ y.im y.re)) x.re))
   (/ 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 <= -1500.0) || !((y_46_im <= 8.4e-122) || (!(y_46_im <= 3e-57) && (y_46_im <= 0.215)))) {
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	} 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 <= (-1500.0d0)) .or. (.not. (y_46im <= 8.4d-122) .or. (.not. (y_46im <= 3d-57)) .and. (y_46im <= 0.215d0))) then
        tmp = (1.0d0 / y_46im) * ((x_46im / (y_46im / y_46re)) - x_46re)
    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 <= -1500.0) || !((y_46_im <= 8.4e-122) || (!(y_46_im <= 3e-57) && (y_46_im <= 0.215)))) {
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	} 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 <= -1500.0) or not ((y_46_im <= 8.4e-122) or (not (y_46_im <= 3e-57) and (y_46_im <= 0.215))):
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re)
	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 <= -1500.0) || !((y_46_im <= 8.4e-122) || (!(y_46_im <= 3e-57) && (y_46_im <= 0.215))))
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re));
	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 <= -1500.0) || ~(((y_46_im <= 8.4e-122) || (~((y_46_im <= 3e-57)) && (y_46_im <= 0.215)))))
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	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, -1500.0], N[Not[Or[LessEqual[y$46$im, 8.4e-122], And[N[Not[LessEqual[y$46$im, 3e-57]], $MachinePrecision], LessEqual[y$46$im, 0.215]]]], $MachinePrecision]], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1500 or 8.39999999999999969e-122 < y.im < 3.00000000000000001e-57 or 0.214999999999999997 < y.im

   1. Initial program 55.5%

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

      1. *-un-lft-identity 55.5%

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

      2. add-sqr-sqrt 55.5%

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

      3. times-frac 55.5%

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

      4. hypot-def 55.5%

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

      5. hypot-def 67.0%

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

   3. Applied egg-rr 67.0%

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

   4. Taylor expanded in y.re around 0 48.0%

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

      1. +-commutative 48.0%

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

      2. mul-1-neg 48.0%

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

      3. unsub-neg 48.0%

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

      4. associate-/l* 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in y.re around 0 79.6%

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

   if -1500 < y.im < 8.39999999999999969e-122 or 3.00000000000000001e-57 < y.im < 0.214999999999999997

   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. Taylor expanded in y.re around inf 71.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1500 \lor \neg \left(y.im \leq 8.4 \cdot 10^{-122} \lor \neg \left(y.im \leq 3 \cdot 10^{-57}\right) \land y.im \leq 0.215\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10: 69.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{if}\;y.im \leq -1600:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2100:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 y.im) (- (/ x.im (/ y.im y.re)) x.re))))
   (if (<= y.im -1600.0)
     t_0
     (if (<= y.im 1e-121)
       (/ x.im y.re)
       (if (<= y.im 3.5e-57)
         (- (/ x.im (* y.im (/ y.im y.re))) (/ x.re y.im))
         (if (<= y.im 2100.0) (/ x.im 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 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	double tmp;
	if (y_46_im <= -1600.0) {
		tmp = t_0;
	} else if (y_46_im <= 1e-121) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.5e-57) {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	} else if (y_46_im <= 2100.0) {
		tmp = x_46_im / 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 = (1.0d0 / y_46im) * ((x_46im / (y_46im / y_46re)) - x_46re)
    if (y_46im <= (-1600.0d0)) then
        tmp = t_0
    else if (y_46im <= 1d-121) then
        tmp = x_46im / y_46re
    else if (y_46im <= 3.5d-57) then
        tmp = (x_46im / (y_46im * (y_46im / y_46re))) - (x_46re / y_46im)
    else if (y_46im <= 2100.0d0) then
        tmp = x_46im / 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 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	double tmp;
	if (y_46_im <= -1600.0) {
		tmp = t_0;
	} else if (y_46_im <= 1e-121) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.5e-57) {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	} else if (y_46_im <= 2100.0) {
		tmp = x_46_im / 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 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re)
	tmp = 0
	if y_46_im <= -1600.0:
		tmp = t_0
	elif y_46_im <= 1e-121:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 3.5e-57:
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im)
	elif y_46_im <= 2100.0:
		tmp = x_46_im / 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(1.0 / y_46_im) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re))
	tmp = 0.0
	if (y_46_im <= -1600.0)
		tmp = t_0;
	elseif (y_46_im <= 1e-121)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.5e-57)
		tmp = Float64(Float64(x_46_im / Float64(y_46_im * Float64(y_46_im / y_46_re))) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= 2100.0)
		tmp = Float64(x_46_im / 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 = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	tmp = 0.0;
	if (y_46_im <= -1600.0)
		tmp = t_0;
	elseif (y_46_im <= 1e-121)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 3.5e-57)
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	elseif (y_46_im <= 2100.0)
		tmp = x_46_im / 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[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1600.0], t$95$0, If[LessEqual[y$46$im, 1e-121], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.5e-57], N[(N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2100.0], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1600 or 2100 < y.im

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. add-sqr-sqrt 52.3%

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

      3. times-frac 52.3%

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

      4. hypot-def 52.3%

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

      5. hypot-def 64.8%

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

   3. Applied egg-rr 64.8%

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

   4. Taylor expanded in y.re around 0 46.3%

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

      1. +-commutative 46.3%

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

      2. mul-1-neg 46.3%

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

      3. unsub-neg 46.3%

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

      4. associate-/l* 47.9%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in y.re around 0 80.8%

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

   if -1600 < y.im < 9.9999999999999998e-122 or 3.49999999999999991e-57 < y.im < 2100

   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. Taylor expanded in y.re around inf 71.4%

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

   if 9.9999999999999998e-122 < y.im < 3.49999999999999991e-57

   1. Initial program 91.7%

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

   2. Taylor expanded in y.re around 0 66.6%

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

      1. +-commutative 66.6%

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

      2. mul-1-neg 66.6%

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

      3. unsub-neg 66.6%

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

      4. associate-/l* 66.6%

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

   4. Simplified 66.6%

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

      1. div-inv 66.6%

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

      2. unpow2 66.6%

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

      3. associate-*l* 66.6%

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

      4. div-inv 66.7%

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

   6. Applied egg-rr 66.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1600:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{elif}\;y.im \leq 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2100:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]

Alternative 11: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6800000 \lor \neg \left(y.im \leq 20000\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im \cdot x.re}{y.re} \cdot \frac{-1}{y.re} + \frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -6800000.0) (not (<= y.im 20000.0)))
   (* (/ 1.0 y.im) (- (/ x.im (/ y.im y.re)) x.re))
   (+ (* (/ (* y.im x.re) y.re) (/ -1.0 y.re)) (/ 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 <= -6800000.0) || !(y_46_im <= 20000.0)) {
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	} else {
		tmp = (((y_46_im * x_46_re) / y_46_re) * (-1.0 / y_46_re)) + (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 <= (-6800000.0d0)) .or. (.not. (y_46im <= 20000.0d0))) then
        tmp = (1.0d0 / y_46im) * ((x_46im / (y_46im / y_46re)) - x_46re)
    else
        tmp = (((y_46im * x_46re) / y_46re) * ((-1.0d0) / y_46re)) + (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 <= -6800000.0) || !(y_46_im <= 20000.0)) {
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	} else {
		tmp = (((y_46_im * x_46_re) / y_46_re) * (-1.0 / y_46_re)) + (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 <= -6800000.0) or not (y_46_im <= 20000.0):
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re)
	else:
		tmp = (((y_46_im * x_46_re) / y_46_re) * (-1.0 / y_46_re)) + (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 <= -6800000.0) || !(y_46_im <= 20000.0))
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re));
	else
		tmp = Float64(Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) * Float64(-1.0 / y_46_re)) + 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 <= -6800000.0) || ~((y_46_im <= 20000.0)))
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	else
		tmp = (((y_46_im * x_46_re) / y_46_re) * (-1.0 / y_46_re)) + (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, -6800000.0], N[Not[LessEqual[y$46$im, 20000.0]], $MachinePrecision]], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.8e6 or 2e4 < y.im

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. add-sqr-sqrt 52.3%

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

      3. times-frac 52.3%

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

      4. hypot-def 52.3%

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

      5. hypot-def 64.8%

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

   3. Applied egg-rr 64.8%

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

   4. Taylor expanded in y.re around 0 46.3%

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

      1. +-commutative 46.3%

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

      2. mul-1-neg 46.3%

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

      3. unsub-neg 46.3%

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

      4. associate-/l* 47.9%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in y.re around 0 80.8%

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

   if -6.8e6 < y.im < 2e4

   1. Initial program 71.9%

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

   2. Taylor expanded in y.re around inf 74.8%

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

      1. *-un-lft-identity 74.8%

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

      2. pow2 74.8%

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

      3. times-frac 80.9%

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

      4. *-commutative 80.9%

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

   4. Applied egg-rr 80.9%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{1}{y.re} \cdot \frac{y.im \cdot x.re}{y.re}\right)} + \frac{x.im}{y.re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6800000 \lor \neg \left(y.im \leq 20000\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im \cdot x.re}{y.re} \cdot \frac{-1}{y.re} + \frac{x.im}{y.re}\\ \end{array} \]

Alternative 12: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3350 \lor \neg \left(y.im \leq 0.086\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3350.0) (not (<= y.im 0.086)))
   (* (/ 1.0 y.im) (- (/ x.im (/ y.im y.re)) x.re))
   (- (/ x.im y.re) (* (/ y.im y.re) (/ x.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 <= -3350.0) || !(y_46_im <= 0.086)) {
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	} else {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_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 <= (-3350.0d0)) .or. (.not. (y_46im <= 0.086d0))) then
        tmp = (1.0d0 / y_46im) * ((x_46im / (y_46im / y_46re)) - x_46re)
    else
        tmp = (x_46im / y_46re) - ((y_46im / y_46re) * (x_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 <= -3350.0) || !(y_46_im <= 0.086)) {
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	} else {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_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 <= -3350.0) or not (y_46_im <= 0.086):
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re)
	else:
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_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 <= -3350.0) || !(y_46_im <= 0.086))
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) * Float64(x_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 <= -3350.0) || ~((y_46_im <= 0.086)))
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	else
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_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, -3350.0], N[Not[LessEqual[y$46$im, 0.086]], $MachinePrecision]], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3350 or 0.085999999999999993 < y.im

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. add-sqr-sqrt 52.3%

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

      3. times-frac 52.3%

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

      4. hypot-def 52.3%

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

      5. hypot-def 64.8%

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

   3. Applied egg-rr 64.8%

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

   4. Taylor expanded in y.re around 0 46.3%

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

      1. +-commutative 46.3%

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

      2. mul-1-neg 46.3%

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

      3. unsub-neg 46.3%

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

      4. associate-/l* 47.9%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in y.re around 0 80.8%

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

   if -3350 < y.im < 0.085999999999999993

   1. Initial program 71.9%

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

   2. Taylor expanded in y.re around inf 74.8%

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

      1. *-commutative 74.8%

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

      2. pow2 74.8%

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

      3. times-frac 78.5%

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

   4. Applied egg-rr 78.5%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\right)} + \frac{x.im}{y.re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3350 \lor \neg \left(y.im \leq 0.086\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternative 13: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -34000000 \lor \neg \left(y.im \leq 10200\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -34000000.0) (not (<= y.im 10200.0)))
   (* (/ 1.0 y.im) (- (/ x.im (/ y.im y.re)) x.re))
   (- (/ x.im y.re) (/ x.re (* y.re (/ y.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 <= -34000000.0) || !(y_46_im <= 10200.0)) {
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_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_46im <= (-34000000.0d0)) .or. (.not. (y_46im <= 10200.0d0))) then
        tmp = (1.0d0 / y_46im) * ((x_46im / (y_46im / y_46re)) - x_46re)
    else
        tmp = (x_46im / y_46re) - (x_46re / (y_46re * (y_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_im <= -34000000.0) || !(y_46_im <= 10200.0)) {
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_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_im <= -34000000.0) or not (y_46_im <= 10200.0):
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re)
	else:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_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 <= -34000000.0) || !(y_46_im <= 10200.0))
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re / 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_im <= -34000000.0) || ~((y_46_im <= 10200.0)))
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	else
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_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$46$im, -34000000.0], N[Not[LessEqual[y$46$im, 10200.0]], $MachinePrecision]], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.4e7 or 10200 < y.im

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. add-sqr-sqrt 52.3%

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

      3. times-frac 52.3%

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

      4. hypot-def 52.3%

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

      5. hypot-def 64.8%

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

   3. Applied egg-rr 64.8%

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

   4. Taylor expanded in y.re around 0 46.3%

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

      1. +-commutative 46.3%

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

      2. mul-1-neg 46.3%

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

      3. unsub-neg 46.3%

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

      4. associate-/l* 47.9%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in y.re around 0 80.8%

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

   if -3.4e7 < y.im < 10200

   1. Initial program 71.9%

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

   2. Taylor expanded in y.re around inf 74.8%

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

      1. *-commutative 74.8%

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

      2. pow2 74.8%

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

      3. times-frac 78.5%

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

   4. Applied egg-rr 78.5%

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

      1. clear-num 78.5%

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

      2. frac-times 79.3%

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

      3. *-un-lft-identity 79.3%

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

   6. Applied egg-rr 79.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -34000000 \lor \neg \left(y.im \leq 10200\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]

Alternative 14: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5500 \lor \neg \left(y.im \leq 18000000000000\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5500.0) (not (<= y.im 18000000000000.0)))
   (* (/ 1.0 y.im) (- (/ x.im (/ y.im y.re)) x.re))
   (- (/ x.im y.re) (/ (/ 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 <= -5500.0) || !(y_46_im <= 18000000000000.0)) {
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	} else {
		tmp = (x_46_im / y_46_re) - ((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 <= (-5500.0d0)) .or. (.not. (y_46im <= 18000000000000.0d0))) then
        tmp = (1.0d0 / y_46im) * ((x_46im / (y_46im / y_46re)) - x_46re)
    else
        tmp = (x_46im / y_46re) - ((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 <= -5500.0) || !(y_46_im <= 18000000000000.0)) {
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	} else {
		tmp = (x_46_im / y_46_re) - ((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 <= -5500.0) or not (y_46_im <= 18000000000000.0):
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re)
	else:
		tmp = (x_46_im / y_46_re) - ((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 <= -5500.0) || !(y_46_im <= 18000000000000.0))
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(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 <= -5500.0) || ~((y_46_im <= 18000000000000.0)))
		tmp = (1.0 / y_46_im) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	else
		tmp = (x_46_im / y_46_re) - ((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, -5500.0], N[Not[LessEqual[y$46$im, 18000000000000.0]], $MachinePrecision]], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5500 or 1.8e13 < y.im

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. add-sqr-sqrt 52.3%

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

      3. times-frac 52.3%

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

      4. hypot-def 52.3%

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

      5. hypot-def 64.8%

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

   3. Applied egg-rr 64.8%

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

   4. Taylor expanded in y.re around 0 46.3%

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

      1. +-commutative 46.3%

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

      2. mul-1-neg 46.3%

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

      3. unsub-neg 46.3%

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

      4. associate-/l* 47.9%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in y.re around 0 80.8%

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

   if -5500 < y.im < 1.8e13

   1. Initial program 71.9%

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

   2. Taylor expanded in y.re around inf 74.8%

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

      1. *-un-lft-identity 74.8%

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

      2. pow2 74.8%

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

      3. times-frac 80.9%

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

      4. *-commutative 80.9%

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

   4. Applied egg-rr 80.9%

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

      1. associate-*l/ 80.9%

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

      2. *-commutative 80.9%

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

      3. *-lft-identity 80.9%

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

      4. associate-/l* 80.1%

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

   6. Simplified 80.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5500 \lor \neg \left(y.im \leq 18000000000000\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]

Alternative 15: 64.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9000000000 \lor \neg \left(y.im \leq 1.5 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9000000000.0) (not (<= y.im 1.5e+26)))
   (/ (- 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 <= -9000000000.0) || !(y_46_im <= 1.5e+26)) {
		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 <= (-9000000000.0d0)) .or. (.not. (y_46im <= 1.5d+26))) 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 <= -9000000000.0) || !(y_46_im <= 1.5e+26)) {
		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 <= -9000000000.0) or not (y_46_im <= 1.5e+26):
		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 <= -9000000000.0) || !(y_46_im <= 1.5e+26))
		tmp = Float64(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 <= -9000000000.0) || ~((y_46_im <= 1.5e+26)))
		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, -9000000000.0], N[Not[LessEqual[y$46$im, 1.5e+26]], $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 < -9e9 or 1.49999999999999999e26 < y.im

   1. Initial program 51.9%

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

   2. Taylor expanded in y.re around 0 67.7%

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

      1. associate-*r/ 67.7%

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

      2. neg-mul-1 67.7%

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

   4. Simplified 67.7%

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

   if -9e9 < y.im < 1.49999999999999999e26

   1. Initial program 72.1%

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

   2. Taylor expanded in y.re around inf 66.6%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9000000000 \lor \neg \left(y.im \leq 1.5 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 16: 46.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+237} \lor \neg \left(y.im \leq 9.2 \cdot 10^{+191}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.2e+237) (not (<= y.im 9.2e+191)))
   (/ 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 <= -2.2e+237) || !(y_46_im <= 9.2e+191)) {
		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 <= (-2.2d+237)) .or. (.not. (y_46im <= 9.2d+191))) 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 <= -2.2e+237) || !(y_46_im <= 9.2e+191)) {
		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 <= -2.2e+237) or not (y_46_im <= 9.2e+191):
		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 <= -2.2e+237) || !(y_46_im <= 9.2e+191))
		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 <= -2.2e+237) || ~((y_46_im <= 9.2e+191)))
		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, -2.2e+237], N[Not[LessEqual[y$46$im, 9.2e+191]], $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 < -2.2e237 or 9.1999999999999997e191 < y.im

   1. Initial program 42.2%

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

      1. *-un-lft-identity 42.2%

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

      2. add-sqr-sqrt 42.2%

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

      3. times-frac 42.2%

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

      4. hypot-def 42.2%

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

      5. hypot-def 59.1%

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

   3. Applied egg-rr 59.1%

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

   4. Taylor expanded in y.re around 0 64.9%

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

      1. +-commutative 64.9%

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

      2. mul-1-neg 64.9%

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

      3. unsub-neg 64.9%

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

      4. associate-/l* 69.4%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in y.im around -inf 43.2%

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

   if -2.2e237 < y.im < 9.1999999999999997e191

   1. Initial program 65.9%

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

   2. Taylor expanded in y.re around inf 47.5%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+237} \lor \neg \left(y.im \leq 9.2 \cdot 10^{+191}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 17: 9.5% 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 61.5%

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

   1. *-un-lft-identity 61.5%

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

   2. add-sqr-sqrt 61.5%

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

   3. times-frac 61.4%

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

   4. hypot-def 61.5%

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

   5. hypot-def 73.2%

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

3. Applied egg-rr 73.2%

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

4. Taylor expanded in y.re around -inf 27.0%

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

   1. neg-mul-1 27.0%

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

6. Simplified 27.0%

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

7. Taylor expanded in y.im around -inf 10.7%

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

8. Final simplification 10.7%

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

Alternative 18: 42.4% accurate, 5.0× speedup

Math

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

FPCore

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

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;
}

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
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;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

2. Taylor expanded in y.re around inf 40.8%

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

3. Final simplification 40.8%

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

Reproduce

herbie shell --seed 2023308 
(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))))