_divideComplex, real part

Percentage Accurate: 61.2% → 84.7%

Time: 13.6s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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: 84.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + x.im \cdot y.im\\ t_1 := \frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* x.im y.im)))
        (t_1 (/ t_0 (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_1 2e+300)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (if (<= t_1 INFINITY)
       (/ (fma y.re (/ x.re y.im) x.im) y.im)
       (+ (/ x.re y.re) (* y.im (/ (/ x.im y.re) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_1 <= 2e+300) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	} else {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im))
	t_1 = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (t_1 <= 2e+300)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	elseif (t_1 <= Inf)
		tmp = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_im / y_46_re) / y_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+300], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.0000000000000001e300

   1. Initial program 84.3%

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

      1. *-un-lft-identity 84.3%

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

      2. associate-*r/ 84.3%

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

      3. fma-def 84.3%

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

      4. add-sqr-sqrt 84.3%

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

      5. times-frac 84.3%

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

      6. fma-def 84.3%

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

      7. hypot-def 84.3%

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

      8. fma-def 84.3%

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

      9. fma-def 84.3%

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

      10. hypot-def 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. associate-*l/ 98.8%

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

      2. *-un-lft-identity 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. fma-def 98.8%

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

   8. Applied egg-rr 98.8%

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

   if 2.0000000000000001e300 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 32.5%

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

      1. *-un-lft-identity 32.5%

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

      2. associate-*r/ 32.5%

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

      3. fma-def 32.5%

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

      4. add-sqr-sqrt 32.5%

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

      5. times-frac 32.5%

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

      6. fma-def 32.5%

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

      7. hypot-def 32.5%

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

      8. fma-def 32.5%

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

      9. fma-def 32.5%

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

      10. hypot-def 59.4%

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

   4. Applied egg-rr 59.4%

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

   5. Taylor expanded in y.re around 0 38.6%

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

   6. Taylor expanded in y.re around 0 77.2%

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

      1. distribute-rgt-in 76.8%

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

      2. un-div-inv 77.0%

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

      3. div-inv 77.0%

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

      4. associate-*l* 58.5%

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

      5. inv-pow 58.5%

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

      6. inv-pow 58.5%

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

      7. pow-prod-up 58.5%

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

      8. metadata-eval 58.5%

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

   8. Applied egg-rr 58.5%

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

      1. *-lft-identity 58.5%

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

      2. associate-*l/ 58.4%

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

      3. *-commutative 58.4%

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

      4. metadata-eval 58.4%

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

      5. pow-sqr 58.4%

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

      6. unpow-1 58.4%

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

      7. unpow-1 58.4%

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

      8. associate-*l* 76.8%

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

      9. associate-*l/ 76.8%

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

      10. *-lft-identity 76.8%

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

      11. associate-*r/ 81.3%

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

      12. distribute-lft-in 81.7%

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

      13. +-commutative 81.7%

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

      14. fma-udef 81.7%

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

      15. associate-*l/ 81.8%

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

      16. *-lft-identity 81.8%

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

      17. fma-udef 81.8%

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

      18. *-commutative 81.8%

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

      19. associate-*l/ 77.3%

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

      20. associate-*r/ 81.8%

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

      21. fma-def 81.8%

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

   10. Simplified 81.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y.re around inf 48.4%

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

      1. associate-/l* 53.7%

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

      2. associate-/r/ 53.7%

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

   5. Simplified 53.7%

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

      1. *-un-lft-identity 53.7%

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

      2. pow2 53.7%

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

      3. times-frac 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. associate-*l/ 62.2%

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

      2. *-lft-identity 62.2%

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

   9. Simplified 62.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + x.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{-x.im}{\frac{y.re}{y.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-138}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+81}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* x.im y.im))))
   (if (<= y.re -1.35e+78)
     (/ (- (/ (- x.im) (/ y.re y.im)) x.re) (hypot y.re y.im))
     (if (<= y.re -9e-138)
       (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 5.2e-152)
         (* (/ 1.0 y.im) (+ x.im (/ (* x.re y.re) y.im)))
         (if (<= y.re 2.3e+81)
           (/ t_0 (fma y.re y.re (* y.im y.im)))
           (/ (+ x.re (/ x.im (/ y.re y.im))) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -1.35e+78) {
		tmp = ((-x_46_im / (y_46_re / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -9e-138) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 5.2e-152) {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 2.3e+81) {
		tmp = t_0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.35e+78)
		tmp = Float64(Float64(Float64(Float64(-x_46_im) / Float64(y_46_re / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -9e-138)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 5.2e-152)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)));
	elseif (y_46_re <= 2.3e+81)
		tmp = Float64(t_0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.35e+78], N[(N[(N[((-x$46$im) / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -9e-138], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.2e-152], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.3e+81], N[(t$95$0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.35000000000000002e78

   1. Initial program 36.4%

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

      1. *-un-lft-identity 36.4%

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

      2. associate-*r/ 36.4%

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

      3. fma-def 36.4%

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

      4. add-sqr-sqrt 36.4%

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

      5. times-frac 36.4%

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

      6. fma-def 36.4%

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

      7. hypot-def 36.4%

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

      8. fma-def 36.4%

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

      9. fma-def 36.4%

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

      10. hypot-def 46.4%

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

   4. Applied egg-rr 46.4%

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

      1. associate-*l/ 46.4%

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

      2. *-un-lft-identity 46.4%

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

   6. Applied egg-rr 46.4%

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

   7. Taylor expanded in y.re around -inf 85.3%

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

      1. neg-mul-1 85.3%

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

      2. +-commutative 85.3%

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

      3. unsub-neg 85.3%

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

      4. mul-1-neg 85.3%

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

      5. associate-/l* 90.5%

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

      6. distribute-neg-frac 90.5%

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

   9. Simplified 90.5%

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

   if -1.35000000000000002e78 < y.re < -9.00000000000000016e-138

   1. Initial program 90.0%

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

   if -9.00000000000000016e-138 < y.re < 5.20000000000000026e-152

   1. Initial program 65.3%

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

      1. *-un-lft-identity 65.3%

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

      2. associate-*r/ 65.3%

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

      3. fma-def 65.3%

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

      4. add-sqr-sqrt 65.3%

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

      5. times-frac 65.3%

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

      6. fma-def 65.3%

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

      7. hypot-def 65.3%

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

      8. fma-def 65.3%

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

      9. fma-def 65.3%

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

      10. hypot-def 84.4%

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

   4. Applied egg-rr 84.4%

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

   5. Taylor expanded in y.re around 0 50.3%

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

   6. Taylor expanded in y.re around 0 93.5%

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

   if 5.20000000000000026e-152 < y.re < 2.2999999999999999e81

   1. Initial program 82.7%

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

      1. fma-def 82.7%

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

      2. fma-def 82.7%

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

   3. Simplified 82.7%

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

      1. fma-def 90.3%

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

   6. Applied egg-rr 82.7%

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

   if 2.2999999999999999e81 < y.re

   1. Initial program 40.1%

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

      1. *-un-lft-identity 40.1%

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

      2. associate-*r/ 40.1%

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

      3. fma-def 40.1%

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

      4. add-sqr-sqrt 40.1%

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

      5. times-frac 40.0%

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

      6. fma-def 40.0%

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

      7. hypot-def 40.0%

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

      8. fma-def 40.0%

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

      9. fma-def 40.0%

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

      10. hypot-def 59.4%

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

   4. Applied egg-rr 59.4%

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

      1. associate-*l/ 59.5%

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

      2. *-un-lft-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

   7. Taylor expanded in y.re around inf 76.0%

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

      1. associate-/l* 88.0%

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

   9. Simplified 88.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{-x.im}{\frac{y.re}{y.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6.4 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -8.2 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -6.4e+78)
     (+ (/ x.re y.re) (* y.im (/ (/ x.im y.re) y.re)))
     (if (<= y.re -8.2e-138)
       t_0
       (if (<= y.re 3.1e-158)
         (* (/ 1.0 y.im) (+ x.im (/ (* x.re y.re) y.im)))
         (if (<= y.re 1.2e+81)
           t_0
           (/ (+ x.re (/ x.im (/ y.re y.im))) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -6.4e+78) {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	} else if (y_46_re <= -8.2e-138) {
		tmp = t_0;
	} else if (y_46_re <= 3.1e-158) {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.2e+81) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	}
	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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -6.4e+78) {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	} else if (y_46_re <= -8.2e-138) {
		tmp = t_0;
	} else if (y_46_re <= 3.1e-158) {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.2e+81) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -6.4e+78:
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re))
	elif y_46_re <= -8.2e-138:
		tmp = t_0
	elif y_46_re <= 3.1e-158:
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im))
	elif y_46_re <= 1.2e+81:
		tmp = t_0
	else:
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / math.hypot(y_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(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -6.4e+78)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_im / y_46_re) / y_46_re)));
	elseif (y_46_re <= -8.2e-138)
		tmp = t_0;
	elseif (y_46_re <= 3.1e-158)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)));
	elseif (y_46_re <= 1.2e+81)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) / hypot(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)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -6.4e+78)
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	elseif (y_46_re <= -8.2e-138)
		tmp = t_0;
	elseif (y_46_re <= 3.1e-158)
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	elseif (y_46_re <= 1.2e+81)
		tmp = t_0;
	else
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / hypot(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_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.4e+78], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -8.2e-138], t$95$0, If[LessEqual[y$46$re, 3.1e-158], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.2e+81], t$95$0, N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.39999999999999989e78

   1. Initial program 36.4%

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

   3. Taylor expanded in y.re around inf 83.0%

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

      1. associate-/l* 83.4%

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

      2. associate-/r/ 85.9%

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

   5. Simplified 85.9%

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

      1. *-un-lft-identity 85.9%

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

      2. pow2 85.9%

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

      3. times-frac 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. associate-*l/ 88.0%

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

      2. *-lft-identity 88.0%

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

   9. Simplified 88.0%

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

   if -6.39999999999999989e78 < y.re < -8.19999999999999998e-138 or 3.10000000000000018e-158 < y.re < 1.19999999999999995e81

   1. Initial program 86.3%

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

   if -8.19999999999999998e-138 < y.re < 3.10000000000000018e-158

   1. Initial program 65.3%

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

      1. *-un-lft-identity 65.3%

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

      2. associate-*r/ 65.3%

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

      3. fma-def 65.3%

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

      4. add-sqr-sqrt 65.3%

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

      5. times-frac 65.3%

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

      6. fma-def 65.3%

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

      7. hypot-def 65.3%

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

      8. fma-def 65.3%

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

      9. fma-def 65.3%

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

      10. hypot-def 84.4%

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

   4. Applied egg-rr 84.4%

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

   5. Taylor expanded in y.re around 0 50.3%

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

   6. Taylor expanded in y.re around 0 93.5%

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

   if 1.19999999999999995e81 < y.re

   1. Initial program 40.1%

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

      1. *-un-lft-identity 40.1%

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

      2. associate-*r/ 40.1%

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

      3. fma-def 40.1%

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

      4. add-sqr-sqrt 40.1%

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

      5. times-frac 40.0%

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

      6. fma-def 40.0%

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

      7. hypot-def 40.0%

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

      8. fma-def 40.0%

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

      9. fma-def 40.0%

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

      10. hypot-def 59.4%

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

   4. Applied egg-rr 59.4%

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

      1. associate-*l/ 59.5%

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

      2. *-un-lft-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

   7. Taylor expanded in y.re around inf 76.0%

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

      1. associate-/l* 88.0%

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

   9. Simplified 88.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.4 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -8.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{-x.im}{\frac{y.re}{y.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -7.2e+78)
     (/ (- (/ (- x.im) (/ y.re y.im)) x.re) (hypot y.re y.im))
     (if (<= y.re -9e-138)
       t_0
       (if (<= y.re 3.1e-156)
         (* (/ 1.0 y.im) (+ x.im (/ (* x.re y.re) y.im)))
         (if (<= y.re 1.6e+80)
           t_0
           (/ (+ x.re (/ x.im (/ y.re y.im))) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -7.2e+78) {
		tmp = ((-x_46_im / (y_46_re / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -9e-138) {
		tmp = t_0;
	} else if (y_46_re <= 3.1e-156) {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.6e+80) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	}
	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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -7.2e+78) {
		tmp = ((-x_46_im / (y_46_re / y_46_im)) - x_46_re) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -9e-138) {
		tmp = t_0;
	} else if (y_46_re <= 3.1e-156) {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.6e+80) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -7.2e+78:
		tmp = ((-x_46_im / (y_46_re / y_46_im)) - x_46_re) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -9e-138:
		tmp = t_0
	elif y_46_re <= 3.1e-156:
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im))
	elif y_46_re <= 1.6e+80:
		tmp = t_0
	else:
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / math.hypot(y_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(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -7.2e+78)
		tmp = Float64(Float64(Float64(Float64(-x_46_im) / Float64(y_46_re / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -9e-138)
		tmp = t_0;
	elseif (y_46_re <= 3.1e-156)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)));
	elseif (y_46_re <= 1.6e+80)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) / hypot(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)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -7.2e+78)
		tmp = ((-x_46_im / (y_46_re / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -9e-138)
		tmp = t_0;
	elseif (y_46_re <= 3.1e-156)
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	elseif (y_46_re <= 1.6e+80)
		tmp = t_0;
	else
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / hypot(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_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2e+78], N[(N[(N[((-x$46$im) / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -9e-138], t$95$0, If[LessEqual[y$46$re, 3.1e-156], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.6e+80], t$95$0, N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.20000000000000039e78

   1. Initial program 36.4%

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

      1. *-un-lft-identity 36.4%

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

      2. associate-*r/ 36.4%

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

      3. fma-def 36.4%

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

      4. add-sqr-sqrt 36.4%

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

      5. times-frac 36.4%

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

      6. fma-def 36.4%

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

      7. hypot-def 36.4%

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

      8. fma-def 36.4%

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

      9. fma-def 36.4%

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

      10. hypot-def 46.4%

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

   4. Applied egg-rr 46.4%

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

      1. associate-*l/ 46.4%

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

      2. *-un-lft-identity 46.4%

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

   6. Applied egg-rr 46.4%

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

   7. Taylor expanded in y.re around -inf 85.3%

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

      1. neg-mul-1 85.3%

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

      2. +-commutative 85.3%

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

      3. unsub-neg 85.3%

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

      4. mul-1-neg 85.3%

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

      5. associate-/l* 90.5%

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

      6. distribute-neg-frac 90.5%

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

   9. Simplified 90.5%

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

   if -7.20000000000000039e78 < y.re < -9.00000000000000016e-138 or 3.0999999999999998e-156 < y.re < 1.59999999999999995e80

   1. Initial program 86.3%

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

   if -9.00000000000000016e-138 < y.re < 3.0999999999999998e-156

   1. Initial program 65.3%

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

      1. *-un-lft-identity 65.3%

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

      2. associate-*r/ 65.3%

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

      3. fma-def 65.3%

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

      4. add-sqr-sqrt 65.3%

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

      5. times-frac 65.3%

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

      6. fma-def 65.3%

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

      7. hypot-def 65.3%

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

      8. fma-def 65.3%

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

      9. fma-def 65.3%

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

      10. hypot-def 84.4%

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

   4. Applied egg-rr 84.4%

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

   5. Taylor expanded in y.re around 0 50.3%

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

   6. Taylor expanded in y.re around 0 93.5%

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

   if 1.59999999999999995e80 < y.re

   1. Initial program 40.1%

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

      1. *-un-lft-identity 40.1%

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

      2. associate-*r/ 40.1%

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

      3. fma-def 40.1%

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

      4. add-sqr-sqrt 40.1%

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

      5. times-frac 40.0%

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

      6. fma-def 40.0%

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

      7. hypot-def 40.0%

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

      8. fma-def 40.0%

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

      9. fma-def 40.0%

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

      10. hypot-def 59.4%

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

   4. Applied egg-rr 59.4%

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

      1. associate-*l/ 59.5%

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

      2. *-un-lft-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

   7. Taylor expanded in y.re around inf 76.0%

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

      1. associate-/l* 88.0%

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

   9. Simplified 88.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{-x.im}{\frac{y.re}{y.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ x.re y.re) (* y.im (/ (/ x.im y.re) y.re)))))
   (if (<= y.re -2.1e+79)
     t_1
     (if (<= y.re -8.6e-138)
       t_0
       (if (<= y.re 5.5e-158)
         (* (/ 1.0 y.im) (+ x.im (/ (* x.re y.re) y.im)))
         (if (<= y.re 2.8e+81) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	double tmp;
	if (y_46_re <= -2.1e+79) {
		tmp = t_1;
	} else if (y_46_re <= -8.6e-138) {
		tmp = t_0;
	} else if (y_46_re <= 5.5e-158) {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 2.8e+81) {
		tmp = t_0;
	} 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) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46re / y_46re) + (y_46im * ((x_46im / y_46re) / y_46re))
    if (y_46re <= (-2.1d+79)) then
        tmp = t_1
    else if (y_46re <= (-8.6d-138)) then
        tmp = t_0
    else if (y_46re <= 5.5d-158) then
        tmp = (1.0d0 / y_46im) * (x_46im + ((x_46re * y_46re) / y_46im))
    else if (y_46re <= 2.8d+81) then
        tmp = t_0
    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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	double tmp;
	if (y_46_re <= -2.1e+79) {
		tmp = t_1;
	} else if (y_46_re <= -8.6e-138) {
		tmp = t_0;
	} else if (y_46_re <= 5.5e-158) {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 2.8e+81) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re))
	tmp = 0
	if y_46_re <= -2.1e+79:
		tmp = t_1
	elif y_46_re <= -8.6e-138:
		tmp = t_0
	elif y_46_re <= 5.5e-158:
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im))
	elif y_46_re <= 2.8e+81:
		tmp = t_0
	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(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_im / y_46_re) / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -2.1e+79)
		tmp = t_1;
	elseif (y_46_re <= -8.6e-138)
		tmp = t_0;
	elseif (y_46_re <= 5.5e-158)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)));
	elseif (y_46_re <= 2.8e+81)
		tmp = t_0;
	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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -2.1e+79)
		tmp = t_1;
	elseif (y_46_re <= -8.6e-138)
		tmp = t_0;
	elseif (y_46_re <= 5.5e-158)
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	elseif (y_46_re <= 2.8e+81)
		tmp = t_0;
	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[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.1e+79], t$95$1, If[LessEqual[y$46$re, -8.6e-138], t$95$0, If[LessEqual[y$46$re, 5.5e-158], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.8e+81], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.10000000000000008e79 or 2.79999999999999995e81 < y.re

   1. Initial program 38.3%

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

   3. Taylor expanded in y.re around inf 78.4%

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

      1. associate-/l* 80.1%

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

      2. associate-/r/ 81.4%

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

   5. Simplified 81.4%

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

      1. *-un-lft-identity 81.4%

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

      2. pow2 81.4%

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

      3. times-frac 87.2%

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

   7. Applied egg-rr 87.2%

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

      1. associate-*l/ 87.2%

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

      2. *-lft-identity 87.2%

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

   9. Simplified 87.2%

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

   if -2.10000000000000008e79 < y.re < -8.6000000000000001e-138 or 5.50000000000000025e-158 < y.re < 2.79999999999999995e81

   1. Initial program 86.3%

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

   if -8.6000000000000001e-138 < y.re < 5.50000000000000025e-158

   1. Initial program 65.3%

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

      1. *-un-lft-identity 65.3%

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

      2. associate-*r/ 65.3%

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

      3. fma-def 65.3%

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

      4. add-sqr-sqrt 65.3%

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

      5. times-frac 65.3%

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

      6. fma-def 65.3%

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

      7. hypot-def 65.3%

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

      8. fma-def 65.3%

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

      9. fma-def 65.3%

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

      10. hypot-def 84.4%

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

   4. Applied egg-rr 84.4%

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

   5. Taylor expanded in y.re around 0 50.3%

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

   6. Taylor expanded in y.re around 0 93.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{-36}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{1}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -7.6e-16)
   (* (/ 1.0 y.im) (+ x.im (* y.re (/ x.re y.im))))
   (if (<= y.im 1.02e-36)
     (+ (/ x.re y.re) (/ (/ (* x.im y.im) y.re) y.re))
     (* (/ 1.0 y.im) (+ x.im (* x.re (/ 1.0 (/ 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 <= -7.6e-16) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	} else if (y_46_im <= 1.02e-36) {
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (1.0 / (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 <= (-7.6d-16)) then
        tmp = (1.0d0 / y_46im) * (x_46im + (y_46re * (x_46re / y_46im)))
    else if (y_46im <= 1.02d-36) then
        tmp = (x_46re / y_46re) + (((x_46im * y_46im) / y_46re) / y_46re)
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re * (1.0d0 / (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 <= -7.6e-16) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	} else if (y_46_im <= 1.02e-36) {
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (1.0 / (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 <= -7.6e-16:
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)))
	elif y_46_im <= 1.02e-36:
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re)
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (1.0 / (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 <= -7.6e-16)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))));
	elseif (y_46_im <= 1.02e-36)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(Float64(x_46_im * y_46_im) / y_46_re) / y_46_re));
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re * Float64(1.0 / Float64(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 <= -7.6e-16)
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	elseif (y_46_im <= 1.02e-36)
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (1.0 / (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[LessEqual[y$46$im, -7.6e-16], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.02e-36], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re * N[(1.0 / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.60000000000000024e-16

   1. Initial program 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. associate-*r/ 53.6%

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

      3. fma-def 53.6%

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

      4. add-sqr-sqrt 53.6%

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

      5. times-frac 53.5%

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

      6. fma-def 53.5%

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

      7. hypot-def 53.5%

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

      8. fma-def 53.5%

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

      9. fma-def 53.5%

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

      10. hypot-def 73.6%

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

   4. Applied egg-rr 73.6%

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

   5. Taylor expanded in y.re around 0 21.4%

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

   6. Taylor expanded in y.re around 0 76.2%

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

      1. associate-/l* 77.7%

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

      2. associate-/r/ 79.1%

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

   8. Applied egg-rr 79.1%

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

   if -7.60000000000000024e-16 < y.im < 1.02e-36

   1. Initial program 71.0%

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

   3. Taylor expanded in y.re around inf 79.0%

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

      1. associate-/l* 78.2%

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

      2. associate-/r/ 71.7%

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

   5. Simplified 71.7%

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

      1. pow2 71.7%

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

      2. associate-*l/ 79.0%

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

      3. associate-/r* 83.5%

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

   7. Applied egg-rr 83.5%

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

   if 1.02e-36 < y.im

   1. Initial program 65.6%

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

      1. *-un-lft-identity 65.6%

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

      2. associate-*r/ 65.6%

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

      3. fma-def 65.6%

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

      4. add-sqr-sqrt 65.6%

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

      5. times-frac 65.7%

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

      6. fma-def 65.7%

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

      7. hypot-def 65.7%

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

      8. fma-def 65.7%

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

      9. fma-def 65.7%

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

      10. hypot-def 75.3%

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

   4. Applied egg-rr 75.3%

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

   5. Taylor expanded in y.re around 0 74.2%

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

   6. Taylor expanded in y.re around 0 72.6%

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

      1. associate-/l* 76.8%

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

      2. div-inv 76.9%

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

   8. Applied egg-rr 76.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{-36}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{1}{\frac{y.im}{y.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+51} \lor \neg \left(y.re \leq 1.8 \cdot 10^{+81}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.8e+51) (not (<= y.re 1.8e+81)))
   (/ x.re y.re)
   (* (/ 1.0 y.im) (+ x.im (* y.re (/ x.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.8e+51) || !(y_46_re <= 1.8e+81)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.8d+51)) .or. (.not. (y_46re <= 1.8d+81))) then
        tmp = x_46re / y_46re
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (y_46re * (x_46re / y_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.8e+51) || !(y_46_re <= 1.8e+81)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.8e+51) or not (y_46_re <= 1.8e+81):
		tmp = x_46_re / y_46_re
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.8e+51) || !(y_46_re <= 1.8e+81))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(y_46_re * Float64(x_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_re <= -3.8e+51) || ~((y_46_re <= 1.8e+81)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.8e+51], N[Not[LessEqual[y$46$re, 1.8e+81]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.7999999999999997e51 or 1.80000000000000003e81 < y.re

   1. Initial program 43.8%

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

   3. Taylor expanded in y.re around inf 78.7%

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

   if -3.7999999999999997e51 < y.re < 1.80000000000000003e81

   1. Initial program 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. associate-*r/ 76.2%

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

      3. fma-def 76.2%

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

      4. add-sqr-sqrt 76.2%

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

      5. times-frac 76.1%

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

      6. fma-def 76.1%

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

      7. hypot-def 76.1%

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

      8. fma-def 76.1%

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

      9. fma-def 76.1%

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

      10. hypot-def 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in y.re around 0 44.5%

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

   6. Taylor expanded in y.re around 0 76.5%

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

      1. associate-/l* 76.5%

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

      2. associate-/r/ 76.5%

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

   8. Applied egg-rr 76.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+51} \lor \neg \left(y.re \leq 1.8 \cdot 10^{+81}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+47} \lor \neg \left(y.re \leq 6.4 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -8e+47) (not (<= y.re 6.4e+79)))
   (/ x.re y.re)
   (* (/ 1.0 y.im) (+ x.im (/ (* x.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_re <= -8e+47) || !(y_46_re <= 6.4e+79)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_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_46re <= (-8d+47)) .or. (.not. (y_46re <= 6.4d+79))) then
        tmp = x_46re / y_46re
    else
        tmp = (1.0d0 / y_46im) * (x_46im + ((x_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_re <= -8e+47) || !(y_46_re <= 6.4e+79)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_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_re <= -8e+47) or not (y_46_re <= 6.4e+79):
		tmp = x_46_re / y_46_re
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_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_re <= -8e+47) || !(y_46_re <= 6.4e+79))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(Float64(x_46_re * 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_re <= -8e+47) || ~((y_46_re <= 6.4e+79)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_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$46$re, -8e+47], N[Not[LessEqual[y$46$re, 6.4e+79]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.0000000000000004e47 or 6.40000000000000005e79 < y.re

   1. Initial program 43.8%

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

   3. Taylor expanded in y.re around inf 78.7%

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

   if -8.0000000000000004e47 < y.re < 6.40000000000000005e79

   1. Initial program 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. associate-*r/ 76.2%

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

      3. fma-def 76.2%

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

      4. add-sqr-sqrt 76.2%

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

      5. times-frac 76.1%

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

      6. fma-def 76.1%

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

      7. hypot-def 76.1%

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

      8. fma-def 76.1%

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

      9. fma-def 76.1%

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

      10. hypot-def 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in y.re around 0 44.5%

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

   6. Taylor expanded in y.re around 0 76.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+47} \lor \neg \left(y.re \leq 6.4 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{+47} \lor \neg \left(y.re \leq 6.5 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6e+47) (not (<= y.re 6.5e+79)))
   (+ (/ x.re y.re) (* y.im (/ (/ x.im y.re) y.re)))
   (* (/ 1.0 y.im) (+ x.im (/ (* x.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_re <= -6e+47) || !(y_46_re <= 6.5e+79)) {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_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_46re <= (-6d+47)) .or. (.not. (y_46re <= 6.5d+79))) then
        tmp = (x_46re / y_46re) + (y_46im * ((x_46im / y_46re) / y_46re))
    else
        tmp = (1.0d0 / y_46im) * (x_46im + ((x_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_re <= -6e+47) || !(y_46_re <= 6.5e+79)) {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_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_re <= -6e+47) or not (y_46_re <= 6.5e+79):
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re))
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_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_re <= -6e+47) || !(y_46_re <= 6.5e+79))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_im / y_46_re) / y_46_re)));
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(Float64(x_46_re * 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_re <= -6e+47) || ~((y_46_re <= 6.5e+79)))
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	else
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_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$46$re, -6e+47], N[Not[LessEqual[y$46$re, 6.5e+79]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.0000000000000003e47 or 6.49999999999999954e79 < y.re

   1. Initial program 43.8%

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

   3. Taylor expanded in y.re around inf 79.3%

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

      1. associate-/l* 80.9%

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

      2. associate-/r/ 82.0%

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

   5. Simplified 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. pow2 82.0%

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

      3. times-frac 87.3%

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

   7. Applied egg-rr 87.3%

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

      1. associate-*l/ 87.4%

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

      2. *-lft-identity 87.4%

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

   9. Simplified 87.4%

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

   if -6.0000000000000003e47 < y.re < 6.49999999999999954e79

   1. Initial program 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. associate-*r/ 76.2%

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

      3. fma-def 76.2%

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

      4. add-sqr-sqrt 76.2%

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

      5. times-frac 76.1%

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

      6. fma-def 76.1%

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

      7. hypot-def 76.1%

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

      8. fma-def 76.1%

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

      9. fma-def 76.1%

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

      10. hypot-def 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in y.re around 0 44.5%

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

   6. Taylor expanded in y.re around 0 76.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{+47} \lor \neg \left(y.re \leq 6.5 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.85 \cdot 10^{-16} \lor \neg \left(y.im \leq 2 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.85e-16) (not (<= y.im 2e-34)))
   (* (/ 1.0 y.im) (+ x.im (* y.re (/ x.re y.im))))
   (+ (/ x.re y.re) (/ (/ (* x.im y.im) y.re) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.85e-16) || !(y_46_im <= 2e-34)) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	} else {
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.85d-16)) .or. (.not. (y_46im <= 2d-34))) then
        tmp = (1.0d0 / y_46im) * (x_46im + (y_46re * (x_46re / y_46im)))
    else
        tmp = (x_46re / y_46re) + (((x_46im * y_46im) / y_46re) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.85e-16) || !(y_46_im <= 2e-34)) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	} else {
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.85e-16) or not (y_46_im <= 2e-34):
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)))
	else:
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.85e-16) || !(y_46_im <= 2e-34))
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(Float64(x_46_im * y_46_im) / y_46_re) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.85e-16) || ~((y_46_im <= 2e-34)))
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	else
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.85e-16], N[Not[LessEqual[y$46$im, 2e-34]], $MachinePrecision]], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.85e-16 or 1.99999999999999986e-34 < y.im

   1. Initial program 59.8%

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

      1. *-un-lft-identity 59.8%

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

      2. associate-*r/ 59.8%

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

      3. fma-def 59.8%

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

      4. add-sqr-sqrt 59.8%

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

      5. times-frac 59.8%

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

      6. fma-def 59.8%

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

      7. hypot-def 59.8%

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

      8. fma-def 59.8%

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

      9. fma-def 59.8%

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

      10. hypot-def 74.5%

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

   4. Applied egg-rr 74.5%

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

   5. Taylor expanded in y.re around 0 48.6%

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

   6. Taylor expanded in y.re around 0 74.4%

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

      1. associate-/l* 77.3%

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

      2. associate-/r/ 77.9%

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

   8. Applied egg-rr 77.9%

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

   if -1.85e-16 < y.im < 1.99999999999999986e-34

   1. Initial program 71.0%

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

   3. Taylor expanded in y.re around inf 79.0%

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

      1. associate-/l* 78.2%

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

      2. associate-/r/ 71.7%

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

   5. Simplified 71.7%

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

      1. pow2 71.7%

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

      2. associate-*l/ 79.0%

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

      3. associate-/r* 83.5%

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

   7. Applied egg-rr 83.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.85 \cdot 10^{-16} \lor \neg \left(y.im \leq 2 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+49} \lor \neg \left(y.re \leq 6.4 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.05e+49) (not (<= y.re 6.4e+79)))
   (/ x.re y.re)
   (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.05e+49) || !(y_46_re <= 6.4e+79)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.05d+49)) .or. (.not. (y_46re <= 6.4d+79))) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.05e+49) || !(y_46_re <= 6.4e+79)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.05e+49) or not (y_46_re <= 6.4e+79):
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.05e+49) || !(y_46_re <= 6.4e+79))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.05e+49) || ~((y_46_re <= 6.4e+79)))
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.05e+49], N[Not[LessEqual[y$46$re, 6.4e+79]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.05000000000000005e49 or 6.40000000000000005e79 < y.re

   1. Initial program 43.8%

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

   3. Taylor expanded in y.re around inf 78.7%

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

   if -1.05000000000000005e49 < y.re < 6.40000000000000005e79

   1. Initial program 76.2%

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

   3. Taylor expanded in y.re around 0 62.7%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+49} \lor \neg \left(y.re \leq 6.4 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.5% 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 64.9%

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

3. Taylor expanded in y.re around 0 46.2%

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

4. Final simplification 46.2%

   \[\leadsto \frac{x.im}{y.im} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))