_divideComplex, real part

Percentage Accurate: 62.2% → 82.9%

Time: 13.1s

Alternatives: 11

Speedup: 2.1×

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: 62.2% 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: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+95}:\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -2.4 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-141}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

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 -1.2e+95)
     (* (+ x.re (/ x.im (/ y.re y.im))) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re -2.4e-43)
       t_0
       (if (<= y.re 8e-141)
         (* (/ -1.0 y.im) (- (- x.im) (/ y.re (/ y.im x.re))))
         (if (<= y.re 1.75e+108)
           t_0
           (/ (fma (/ x.im y.re) y.im x.re) (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 <= -1.2e+95) {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -2.4e-43) {
		tmp = t_0;
	} else if (y_46_re <= 8e-141) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	} else if (y_46_re <= 1.75e+108) {
		tmp = t_0;
	} else {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / 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 <= -1.2e+95)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -2.4e-43)
		tmp = t_0;
	elseif (y_46_re <= 8e-141)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(Float64(-x_46_im) - Float64(y_46_re / Float64(y_46_im / x_46_re))));
	elseif (y_46_re <= 1.75e+108)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / 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[(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, -1.2e+95], N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -2.4e-43], t$95$0, If[LessEqual[y$46$re, 8e-141], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[((-x$46$im) - N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.75e+108], t$95$0, N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.2e95

   1. Initial program 37.7%

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

      1. *-un-lft-identity 37.7%

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

      2. add-sqr-sqrt 37.7%

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

      3. times-frac 37.7%

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

      4. hypot-def 37.7%

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

      5. fma-def 37.7%

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

      6. hypot-def 57.8%

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

   3. Applied egg-rr 57.8%

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

   4. Taylor expanded in y.re around -inf 93.6%

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

      1. distribute-lft-out 93.6%

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

      2. associate-/l* 96.2%

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

   6. Simplified 96.2%

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

   if -1.2e95 < y.re < -2.4000000000000002e-43 or 8.0000000000000003e-141 < y.re < 1.7500000000000001e108

   1. Initial program 87.2%

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

   if -2.4000000000000002e-43 < y.re < 8.0000000000000003e-141

   1. Initial program 66.5%

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

      1. *-un-lft-identity 66.5%

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

      2. add-sqr-sqrt 66.5%

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

      3. times-frac 66.5%

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

      4. hypot-def 66.5%

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

      5. fma-def 66.5%

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

      6. hypot-def 77.7%

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

   3. Applied egg-rr 77.7%

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

   4. Taylor expanded in y.im around -inf 43.7%

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

      1. distribute-lft-out 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-/l* 42.7%

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

   6. Simplified 42.7%

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

   7. Taylor expanded in y.im around -inf 85.9%

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

   if 1.7500000000000001e108 < y.re

   1. Initial program 41.5%

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

      1. *-un-lft-identity 41.5%

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

      2. add-sqr-sqrt 41.5%

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

      3. times-frac 41.5%

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

      4. hypot-def 41.5%

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

      5. fma-def 41.5%

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

      6. hypot-def 60.6%

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

   3. Applied egg-rr 60.6%

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

   4. Taylor expanded in y.re around inf 83.8%

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

      1. associate-/l* 86.1%

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

   6. Simplified 86.1%

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

      1. expm1-log1p-u 75.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right)\right)\right)} \]

      2. expm1-udef 36.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right)\right)} - 1} \]

      3. associate-*l/ 36.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]

      4. *-un-lft-identity 36.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x.re + \frac{x.im}{\frac{y.re}{y.im}}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

      5. +-commutative 36.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{x.im}{\frac{y.re}{y.im}} + x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

      6. div-inv 36.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x.im \cdot \frac{1}{\frac{y.re}{y.im}}} + x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

      7. fma-def 36.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x.im, \frac{1}{\frac{y.re}{y.im}}, x.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

      8. clear-num 36.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \color{blue}{\frac{y.im}{y.re}}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   8. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 75.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} \]

      2. expm1-log1p 86.4%

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

      3. fma-udef 86.4%

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

      4. associate-*r/ 83.9%

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

      5. associate-*l/ 88.2%

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

      6. fma-udef 88.2%

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

   10. Simplified 88.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+95}:\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -2.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-141}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+108}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2: 85.8% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      INFINITY)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (* (/ y.re (hypot y.re y.im)) (/ x.re (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 tmp;
	if ((((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) INFINITY)) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_re / hypot(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 (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))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * Float64(x_46_re / 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_] := If[LessEqual[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], Infinity], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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))) < +inf.0

   1. Initial program 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. add-sqr-sqrt 79.0%

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

      3. times-frac 79.0%

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

      4. hypot-def 79.0%

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

      5. fma-def 79.0%

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

      6. hypot-def 93.6%

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

   3. Applied egg-rr 93.6%

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

   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. Taylor expanded in x.re around inf 1.3%

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

      1. *-commutative 1.3%

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

   4. Simplified 1.3%

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

      1. add-sqr-sqrt 1.3%

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

      2. hypot-udef 1.3%

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

      3. hypot-udef 1.3%

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

      4. times-frac 63.2%

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

   6. Applied egg-rr 63.2%

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

4. Final simplification 88.0%

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

Alternative 3: 82.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+98}:\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -1.12 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.36 \cdot 10^{-144}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \end{array} \end{array} \]

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 -1.15e+98)
     (* (+ x.re (/ x.im (/ y.re y.im))) (/ 1.0 y.re))
     (if (<= y.re -1.12e-43)
       t_0
       (if (<= y.re 1.36e-144)
         (* (/ -1.0 y.im) (- (- x.im) (/ y.re (/ y.im x.re))))
         (if (<= y.re 4.2e+146)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (+ x.re (* y.im (/ x.im y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 <= -1.15e+98) {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re);
	} else if (y_46_re <= -1.12e-43) {
		tmp = t_0;
	} else if (y_46_re <= 1.36e-144) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	} else if (y_46_re <= 4.2e+146) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re + (y_46_im * (x_46_im / y_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 <= -1.15e+98) {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re);
	} else if (y_46_re <= -1.12e-43) {
		tmp = t_0;
	} else if (y_46_re <= 1.36e-144) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	} else if (y_46_re <= 4.2e+146) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_re + (y_46_im * (x_46_im / y_46_re)));
	}
	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 <= -1.15e+98:
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re)
	elif y_46_re <= -1.12e-43:
		tmp = t_0
	elif y_46_re <= 1.36e-144:
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)))
	elif y_46_re <= 4.2e+146:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_re + (y_46_im * (x_46_im / y_46_re)))
	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 <= -1.15e+98)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) * Float64(1.0 / y_46_re));
	elseif (y_46_re <= -1.12e-43)
		tmp = t_0;
	elseif (y_46_re <= 1.36e-144)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(Float64(-x_46_im) - Float64(y_46_re / Float64(y_46_im / x_46_re))));
	elseif (y_46_re <= 4.2e+146)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	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 <= -1.15e+98)
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re);
	elseif (y_46_re <= -1.12e-43)
		tmp = t_0;
	elseif (y_46_re <= 1.36e-144)
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	elseif (y_46_re <= 4.2e+146)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re + (y_46_im * (x_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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, -1.15e+98], N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.12e-43], t$95$0, If[LessEqual[y$46$re, 1.36e-144], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[((-x$46$im) - N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.2e+146], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.15000000000000007e98

   1. Initial program 37.7%

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

      1. *-un-lft-identity 37.7%

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

      2. add-sqr-sqrt 37.7%

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

      3. times-frac 37.7%

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

      4. hypot-def 37.7%

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

      5. fma-def 37.7%

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

      6. hypot-def 57.8%

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

   3. Applied egg-rr 57.8%

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

   4. Taylor expanded in y.re around inf 32.7%

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

      1. associate-/l* 32.7%

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

   6. Simplified 32.7%

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

   7. Taylor expanded in y.re around inf 96.1%

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

   if -1.15000000000000007e98 < y.re < -1.12e-43 or 1.36e-144 < y.re < 4.2000000000000001e146

   1. Initial program 87.1%

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

   if -1.12e-43 < y.re < 1.36e-144

   1. Initial program 66.5%

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

      1. *-un-lft-identity 66.5%

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

      2. add-sqr-sqrt 66.5%

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

      3. times-frac 66.5%

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

      4. hypot-def 66.5%

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

      5. fma-def 66.5%

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

      6. hypot-def 77.7%

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

   3. Applied egg-rr 77.7%

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

   4. Taylor expanded in y.im around -inf 43.7%

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

      1. distribute-lft-out 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-/l* 42.7%

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

   6. Simplified 42.7%

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

   7. Taylor expanded in y.im around -inf 85.9%

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

   if 4.2000000000000001e146 < y.re

   1. Initial program 33.8%

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

      1. *-un-lft-identity 33.8%

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

      2. add-sqr-sqrt 33.8%

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

      3. times-frac 33.8%

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

      4. hypot-def 33.8%

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

      5. fma-def 33.8%

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

      6. hypot-def 53.8%

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

   3. Applied egg-rr 53.8%

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

   4. Taylor expanded in y.re around inf 83.4%

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

      1. associate-/l* 86.2%

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

      2. associate-/r/ 88.4%

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

   6. Simplified 88.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+98}:\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -1.12 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.36 \cdot 10^{-144}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+146}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \end{array} \]

Alternative 4: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.8 \cdot 10^{+94}:\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.95 \cdot 10^{-142}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.86 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \end{array} \end{array} \]

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 -1.8e+94)
     (* (+ x.re (/ x.im (/ y.re y.im))) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re -1.2e-43)
       t_0
       (if (<= y.re 2.95e-142)
         (* (/ -1.0 y.im) (- (- x.im) (/ y.re (/ y.im x.re))))
         (if (<= y.re 1.86e+146)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (+ x.re (* y.im (/ x.im y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 <= -1.8e+94) {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -1.2e-43) {
		tmp = t_0;
	} else if (y_46_re <= 2.95e-142) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	} else if (y_46_re <= 1.86e+146) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re + (y_46_im * (x_46_im / y_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 <= -1.8e+94) {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -1.2e-43) {
		tmp = t_0;
	} else if (y_46_re <= 2.95e-142) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	} else if (y_46_re <= 1.86e+146) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_re + (y_46_im * (x_46_im / y_46_re)));
	}
	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 <= -1.8e+94:
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= -1.2e-43:
		tmp = t_0
	elif y_46_re <= 2.95e-142:
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)))
	elif y_46_re <= 1.86e+146:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_re + (y_46_im * (x_46_im / y_46_re)))
	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 <= -1.8e+94)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -1.2e-43)
		tmp = t_0;
	elseif (y_46_re <= 2.95e-142)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(Float64(-x_46_im) - Float64(y_46_re / Float64(y_46_im / x_46_re))));
	elseif (y_46_re <= 1.86e+146)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	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 <= -1.8e+94)
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.2e-43)
		tmp = t_0;
	elseif (y_46_re <= 2.95e-142)
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	elseif (y_46_re <= 1.86e+146)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re + (y_46_im * (x_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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, -1.8e+94], N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.2e-43], t$95$0, If[LessEqual[y$46$re, 2.95e-142], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[((-x$46$im) - N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.86e+146], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.79999999999999996e94

   1. Initial program 37.7%

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

      1. *-un-lft-identity 37.7%

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

      2. add-sqr-sqrt 37.7%

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

      3. times-frac 37.7%

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

      4. hypot-def 37.7%

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

      5. fma-def 37.7%

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

      6. hypot-def 57.8%

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

   3. Applied egg-rr 57.8%

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

   4. Taylor expanded in y.re around -inf 93.6%

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

      1. distribute-lft-out 93.6%

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

      2. associate-/l* 96.2%

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

   6. Simplified 96.2%

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

   if -1.79999999999999996e94 < y.re < -1.2000000000000001e-43 or 2.94999999999999983e-142 < y.re < 1.86e146

   1. Initial program 87.1%

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

   if -1.2000000000000001e-43 < y.re < 2.94999999999999983e-142

   1. Initial program 66.5%

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

      1. *-un-lft-identity 66.5%

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

      2. add-sqr-sqrt 66.5%

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

      3. times-frac 66.5%

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

      4. hypot-def 66.5%

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

      5. fma-def 66.5%

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

      6. hypot-def 77.7%

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

   3. Applied egg-rr 77.7%

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

   4. Taylor expanded in y.im around -inf 43.7%

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

      1. distribute-lft-out 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-/l* 42.7%

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

   6. Simplified 42.7%

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

   7. Taylor expanded in y.im around -inf 85.9%

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

   if 1.86e146 < y.re

   1. Initial program 33.8%

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

      1. *-un-lft-identity 33.8%

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

      2. add-sqr-sqrt 33.8%

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

      3. times-frac 33.8%

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

      4. hypot-def 33.8%

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

      5. fma-def 33.8%

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

      6. hypot-def 53.8%

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

   3. Applied egg-rr 53.8%

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

   4. Taylor expanded in y.re around inf 83.4%

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

      1. associate-/l* 86.2%

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

      2. associate-/r/ 88.4%

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

   6. Simplified 88.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.8 \cdot 10^{+94}:\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.95 \cdot 10^{-142}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.86 \cdot 10^{+146}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \end{array} \]

Alternative 5: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.55 \cdot 10^{-142}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{+119}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 (/ x.im (/ y.re y.im))) (/ 1.0 y.re))))
   (if (<= y.re -3.9e+95)
     t_1
     (if (<= y.re -4.2e-41)
       t_0
       (if (<= y.re 2.55e-142)
         (* (/ -1.0 y.im) (- (- x.im) (/ y.re (/ y.im x.re))))
         (if (<= y.re 1.95e+119) 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 + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re);
	double tmp;
	if (y_46_re <= -3.9e+95) {
		tmp = t_1;
	} else if (y_46_re <= -4.2e-41) {
		tmp = t_0;
	} else if (y_46_re <= 2.55e-142) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	} else if (y_46_re <= 1.95e+119) {
		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 + (x_46im / (y_46re / y_46im))) * (1.0d0 / y_46re)
    if (y_46re <= (-3.9d+95)) then
        tmp = t_1
    else if (y_46re <= (-4.2d-41)) then
        tmp = t_0
    else if (y_46re <= 2.55d-142) then
        tmp = ((-1.0d0) / y_46im) * (-x_46im - (y_46re / (y_46im / x_46re)))
    else if (y_46re <= 1.95d+119) 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 + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re);
	double tmp;
	if (y_46_re <= -3.9e+95) {
		tmp = t_1;
	} else if (y_46_re <= -4.2e-41) {
		tmp = t_0;
	} else if (y_46_re <= 2.55e-142) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	} else if (y_46_re <= 1.95e+119) {
		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 + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re)
	tmp = 0
	if y_46_re <= -3.9e+95:
		tmp = t_1
	elif y_46_re <= -4.2e-41:
		tmp = t_0
	elif y_46_re <= 2.55e-142:
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)))
	elif y_46_re <= 1.95e+119:
		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 + Float64(x_46_im / Float64(y_46_re / y_46_im))) * Float64(1.0 / y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.9e+95)
		tmp = t_1;
	elseif (y_46_re <= -4.2e-41)
		tmp = t_0;
	elseif (y_46_re <= 2.55e-142)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(Float64(-x_46_im) - Float64(y_46_re / Float64(y_46_im / x_46_re))));
	elseif (y_46_re <= 1.95e+119)
		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 + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -3.9e+95)
		tmp = t_1;
	elseif (y_46_re <= -4.2e-41)
		tmp = t_0;
	elseif (y_46_re <= 2.55e-142)
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	elseif (y_46_re <= 1.95e+119)
		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 + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.9e+95], t$95$1, If[LessEqual[y$46$re, -4.2e-41], t$95$0, If[LessEqual[y$46$re, 2.55e-142], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[((-x$46$im) - N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.95e+119], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.8999999999999997e95 or 1.9499999999999999e119 < y.re

   1. Initial program 38.8%

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

      1. *-un-lft-identity 38.8%

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

      2. add-sqr-sqrt 38.8%

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

      3. times-frac 38.8%

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

      4. hypot-def 38.8%

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

      5. fma-def 38.8%

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

      6. hypot-def 57.9%

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

   3. Applied egg-rr 57.9%

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

   4. Taylor expanded in y.re around inf 61.0%

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

      1. associate-/l* 62.4%

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

   6. Simplified 62.4%

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

   7. Taylor expanded in y.re around inf 91.3%

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

   if -3.8999999999999997e95 < y.re < -4.20000000000000025e-41 or 2.55e-142 < y.re < 1.9499999999999999e119

   1. Initial program 86.6%

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

   if -4.20000000000000025e-41 < y.re < 2.55e-142

   1. Initial program 66.5%

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

      1. *-un-lft-identity 66.5%

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

      2. add-sqr-sqrt 66.5%

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

      3. times-frac 66.5%

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

      4. hypot-def 66.5%

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

      5. fma-def 66.5%

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

      6. hypot-def 77.7%

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

   3. Applied egg-rr 77.7%

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

   4. Taylor expanded in y.im around -inf 43.7%

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

      1. distribute-lft-out 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-/l* 42.7%

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

   6. Simplified 42.7%

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

   7. Taylor expanded in y.im around -inf 85.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+95}:\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.55 \cdot 10^{-142}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{1}{y.re}\\ \end{array} \]

Alternative 6: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-42} \lor \neg \left(y.re \leq 2 \cdot 10^{+51}\right):\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.4e-42) (not (<= y.re 2e+51)))
   (* (+ x.re (/ x.im (/ y.re y.im))) (/ 1.0 y.re))
   (* (/ -1.0 y.im) (- (- x.im) (/ y.re (/ y.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.4e-42) || !(y_46_re <= 2e+51)) {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re);
	} else {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.4d-42)) .or. (.not. (y_46re <= 2d+51))) then
        tmp = (x_46re + (x_46im / (y_46re / y_46im))) * (1.0d0 / y_46re)
    else
        tmp = ((-1.0d0) / y_46im) * (-x_46im - (y_46re / (y_46im / x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.4e-42) || !(y_46_re <= 2e+51)) {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re);
	} else {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.4e-42) or not (y_46_re <= 2e+51):
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re)
	else:
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2.4e-42) || !(y_46_re <= 2e+51))
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) * Float64(1.0 / y_46_re));
	else
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(Float64(-x_46_im) - Float64(y_46_re / Float64(y_46_im / x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.4e-42) || ~((y_46_re <= 2e+51)))
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / y_46_re);
	else
		tmp = (-1.0 / y_46_im) * (-x_46_im - (y_46_re / (y_46_im / x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.4e-42], N[Not[LessEqual[y$46$re, 2e+51]], $MachinePrecision]], N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[((-x$46$im) - N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.40000000000000003e-42 or 2e51 < y.re

   1. Initial program 54.5%

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

      1. *-un-lft-identity 54.5%

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

      2. add-sqr-sqrt 54.5%

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

      3. times-frac 54.5%

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

      4. hypot-def 54.5%

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

      5. fma-def 54.5%

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

      6. hypot-def 70.9%

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

   3. Applied egg-rr 70.9%

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

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

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

      1. associate-/l* 49.4%

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

   6. Simplified 49.4%

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

   7. Taylor expanded in y.re around inf 84.6%

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

   if -2.40000000000000003e-42 < y.re < 2e51

   1. Initial program 73.5%

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

      1. *-un-lft-identity 73.5%

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

      2. add-sqr-sqrt 73.5%

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

      3. times-frac 73.5%

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

      4. hypot-def 73.5%

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

      5. fma-def 73.5%

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

      6. hypot-def 82.3%

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

   3. Applied egg-rr 82.3%

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

   4. Taylor expanded in y.im around -inf 40.7%

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

      1. distribute-lft-out 40.7%

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

      2. *-commutative 40.7%

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

      3. associate-/l* 39.9%

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

   6. Simplified 39.9%

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

   7. Taylor expanded in y.im around -inf 79.6%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-42} \lor \neg \left(y.re \leq 2 \cdot 10^{+51}\right):\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \end{array} \]

Alternative 7: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+86} \lor \neg \left(y.im \leq 1.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{1}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.8e+86) (not (<= y.im 1.1e+18)))
   (/ x.im y.im)
   (* (+ x.re (/ x.im (/ y.re y.im))) (/ 1.0 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 <= -5.8e+86) || !(y_46_im <= 1.1e+18)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / 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 <= (-5.8d+86)) .or. (.not. (y_46im <= 1.1d+18))) then
        tmp = x_46im / y_46im
    else
        tmp = (x_46re + (x_46im / (y_46re / y_46im))) * (1.0d0 / 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 <= -5.8e+86) || !(y_46_im <= 1.1e+18)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / 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 <= -5.8e+86) or not (y_46_im <= 1.1e+18):
		tmp = x_46_im / y_46_im
	else:
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / 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 <= -5.8e+86) || !(y_46_im <= 1.1e+18))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) * Float64(1.0 / 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 <= -5.8e+86) || ~((y_46_im <= 1.1e+18)))
		tmp = x_46_im / y_46_im;
	else
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) * (1.0 / 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, -5.8e+86], N[Not[LessEqual[y$46$im, 1.1e+18]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.79999999999999981e86 or 1.1e18 < y.im

   1. Initial program 45.6%

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

   2. Taylor expanded in y.re around 0 66.0%

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

   if -5.79999999999999981e86 < y.im < 1.1e18

   1. Initial program 75.6%

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

      1. *-un-lft-identity 75.6%

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

      2. add-sqr-sqrt 75.6%

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

      3. times-frac 75.6%

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

      4. hypot-def 75.6%

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

      5. fma-def 75.6%

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

      6. hypot-def 83.3%

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

   3. Applied egg-rr 83.3%

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

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

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

      1. associate-/l* 45.7%

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

   6. Simplified 45.7%

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

   7. Taylor expanded in y.re around inf 78.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+86} \lor \neg \left(y.im \leq 1.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right) \cdot \frac{1}{y.re}\\ \end{array} \]

Alternative 8: 42.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+74} \lor \neg \left(y.re \leq 1.82 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.5e+74) (not (<= y.re 1.82e+171)))
   (/ x.im 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 <= -2.5e+74) || !(y_46_re <= 1.82e+171)) {
		tmp = x_46_im / 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 <= (-2.5d+74)) .or. (.not. (y_46re <= 1.82d+171))) then
        tmp = x_46im / 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 <= -2.5e+74) || !(y_46_re <= 1.82e+171)) {
		tmp = x_46_im / 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 <= -2.5e+74) or not (y_46_re <= 1.82e+171):
		tmp = x_46_im / 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 <= -2.5e+74) || !(y_46_re <= 1.82e+171))
		tmp = Float64(x_46_im / 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 <= -2.5e+74) || ~((y_46_re <= 1.82e+171)))
		tmp = x_46_im / 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, -2.5e+74], N[Not[LessEqual[y$46$re, 1.82e+171]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.49999999999999982e74 or 1.81999999999999996e171 < 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. Step-by-step derivation

      1. *-un-lft-identity 43.8%

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

      2. add-sqr-sqrt 43.8%

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

      3. times-frac 43.8%

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

      4. hypot-def 43.8%

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

      5. fma-def 43.8%

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

      6. hypot-def 61.5%

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

   3. Applied egg-rr 61.5%

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

   4. Taylor expanded in y.re around inf 55.3%

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

      1. associate-/l* 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in y.re around 0 23.5%

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

   if -2.49999999999999982e74 < y.re < 1.81999999999999996e171

   1. Initial program 73.2%

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

   2. Taylor expanded in y.re around 0 50.1%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+74} \lor \neg \left(y.re \leq 1.82 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9: 63.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -250 \lor \neg \left(y.re \leq 3.2 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -250.0) (not (<= y.re 3.2e+39)))
   (/ 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 <= -250.0) || !(y_46_re <= 3.2e+39)) {
		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 <= (-250.0d0)) .or. (.not. (y_46re <= 3.2d+39))) 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 <= -250.0) || !(y_46_re <= 3.2e+39)) {
		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 <= -250.0) or not (y_46_re <= 3.2e+39):
		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 <= -250.0) || !(y_46_re <= 3.2e+39))
		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 <= -250.0) || ~((y_46_re <= 3.2e+39)))
		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, -250.0], N[Not[LessEqual[y$46$re, 3.2e+39]], $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 < -250 or 3.19999999999999993e39 < y.re

   1. Initial program 52.5%

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

   2. Taylor expanded in y.re around inf 73.1%

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

   if -250 < y.re < 3.19999999999999993e39

   1. Initial program 74.5%

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

   2. Taylor expanded in y.re around 0 57.6%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -250 \lor \neg \left(y.re \leq 3.2 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10: 63.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -195:\\ \;\;\;\;\frac{1}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -195.0)
   (/ 1.0 (/ y.re x.re))
   (if (<= y.re 3.2e+37) (/ x.im y.im) (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -195.0) {
		tmp = 1.0 / (y_46_re / x_46_re);
	} else if (y_46_re <= 3.2e+37) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-195.0d0)) then
        tmp = 1.0d0 / (y_46re / x_46re)
    else if (y_46re <= 3.2d+37) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -195.0) {
		tmp = 1.0 / (y_46_re / x_46_re);
	} else if (y_46_re <= 3.2e+37) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -195.0:
		tmp = 1.0 / (y_46_re / x_46_re)
	elif y_46_re <= 3.2e+37:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -195.0)
		tmp = Float64(1.0 / Float64(y_46_re / x_46_re));
	elseif (y_46_re <= 3.2e+37)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -195.0)
		tmp = 1.0 / (y_46_re / x_46_re);
	elseif (y_46_re <= 3.2e+37)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -195.0], N[(1.0 / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.2e+37], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -195

   1. Initial program 57.7%

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

      1. *-un-lft-identity 57.7%

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

      2. add-sqr-sqrt 57.7%

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

      3. times-frac 57.7%

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

      4. hypot-def 57.7%

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

      5. fma-def 57.7%

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

      6. hypot-def 72.4%

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

   3. Applied egg-rr 72.4%

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

   4. Taylor expanded in y.re around -inf 81.4%

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

      1. mul-1-neg 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in y.re around inf 24.4%

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

      1. associate-*l/ 24.4%

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

      2. associate-/l* 25.7%

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

      3. add-sqr-sqrt 4.7%

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

      4. sqrt-unprod 43.0%

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

      5. sqr-neg 43.0%

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

      6. sqrt-unprod 51.7%

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

      7. add-sqr-sqrt 79.7%

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

   9. Applied egg-rr 79.7%

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

   if -195 < y.re < 3.20000000000000014e37

   1. Initial program 74.5%

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

   2. Taylor expanded in y.re around 0 57.6%

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

   if 3.20000000000000014e37 < y.re

   1. Initial program 47.4%

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

   2. Taylor expanded in y.re around inf 67.9%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -195:\\ \;\;\;\;\frac{1}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11: 42.0% 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 64.5%

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

2. Taylor expanded in y.re around 0 36.9%

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

3. Final simplification 36.9%

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

Reproduce

herbie shell --seed 2023305 
(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))))