_divideComplex, imaginary part

Percentage Accurate: 61.8% → 99.0%

Time: 12.6s

Alternatives: 15

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (/ 1.0 (hypot y.re y.im))
  (- (/ x.im (/ (hypot y.re y.im) y.re)) (/ x.re (/ (hypot 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 (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_im / (hypot(y_46_re, y_46_im) / y_46_re)) - (x_46_re / (hypot(y_46_re, y_46_im) / y_46_im)));
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (1.0 / Math.hypot(y_46_re, y_46_im)) * ((x_46_im / (Math.hypot(y_46_re, y_46_im) / y_46_re)) - (x_46_re / (Math.hypot(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 (1.0 / math.hypot(y_46_re, y_46_im)) * ((x_46_im / (math.hypot(y_46_re, y_46_im) / y_46_re)) - (x_46_re / (math.hypot(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(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(x_46_im / Float64(hypot(y_46_re, y_46_im) / y_46_re)) - Float64(x_46_re / Float64(hypot(y_46_re, y_46_im) / y_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_im / (hypot(y_46_re, y_46_im) / y_46_re)) - (x_46_re / (hypot(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[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.9%

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

   1. *-un-lft-identity 63.9%

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

   2. add-sqr-sqrt 63.9%

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

   3. times-frac 63.9%

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

   4. hypot-def 63.9%

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

   5. hypot-def 78.5%

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

3. Applied egg-rr 78.5%

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

   1. div-sub 78.5%

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

5. Applied egg-rr 78.5%

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

   1. associate-/l* 88.7%

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

   2. associate-/l* 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 97.7% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.9%

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

   1. *-un-lft-identity 63.9%

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

   2. add-sqr-sqrt 63.9%

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

   3. times-frac 63.9%

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

   4. hypot-def 63.9%

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

   5. hypot-def 78.5%

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

3. Applied egg-rr 78.5%

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

   1. div-sub 78.5%

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

5. Applied egg-rr 78.5%

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

   1. associate-/l* 88.7%

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

   2. associate-/l* 99.3%

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

7. Simplified 99.3%

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

   1. expm1-log1p-u 81.4%

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

   2. expm1-udef 35.3%

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

   3. associate-*l/ 35.3%

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

   4. *-un-lft-identity 35.3%

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

   5. div-inv 35.3%

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

   6. clear-num 35.3%

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

   7. div-inv 35.3%

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

   8. clear-num 35.3%

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

9. Applied egg-rr 35.3%

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

    1. expm1-def 81.6%

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

    2. expm1-log1p 99.5%

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

    3. associate-*r/ 88.8%

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

    4. associate-*l/ 98.7%

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

    5. *-commutative 98.7%

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

    6. associate-*r/ 88.2%

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

    7. associate-*l/ 98.0%

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

    8. *-commutative 98.0%

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

11. Simplified 98.0%

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

12. Final simplification 98.0%

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

Alternative 3: 87.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_1}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+236}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x.im - \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))) (t_1 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_1 (+ (* y.re y.re) (* y.im y.im))) 2e+236)
     (* t_0 (/ t_1 (hypot y.re y.im)))
     (* t_0 (- x.im (/ x.re (/ (hypot 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) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+236) {
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = t_0 * (x_46_im - (x_46_re / (hypot(y_46_re, y_46_im) / y_46_im)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+236) {
		tmp = t_0 * (t_1 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = t_0 * (x_46_im - (x_46_re / (Math.hypot(y_46_re, y_46_im) / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if (t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+236:
		tmp = t_0 * (t_1 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = t_0 * (x_46_im - (x_46_re / (math.hypot(y_46_re, y_46_im) / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 2e+236)
		tmp = Float64(t_0 * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(t_0 * Float64(x_46_im - Float64(x_46_re / Float64(hypot(y_46_re, y_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)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+236)
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	else
		tmp = t_0 * (x_46_im - (x_46_re / (hypot(y_46_re, y_46_im) / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+236], N[(t$95$0 * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$46$im - N[(x$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.00000000000000011e236

   1. Initial program 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. add-sqr-sqrt 78.9%

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

      3. times-frac 78.9%

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

      4. hypot-def 78.9%

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

      5. hypot-def 97.2%

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

   3. Applied egg-rr 97.2%

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

   if 2.00000000000000011e236 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 16.7%

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

      1. *-un-lft-identity 16.7%

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

      2. add-sqr-sqrt 16.7%

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

      3. times-frac 16.7%

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

      4. hypot-def 16.7%

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

      5. hypot-def 19.9%

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

   3. Applied egg-rr 19.9%

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

      1. div-sub 19.9%

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

   5. Applied egg-rr 19.9%

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

      1. associate-/l* 60.6%

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

      2. associate-/l* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y.re around inf 72.8%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+236}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \]

Alternative 4: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ x.re (/ y.re y.im))) y.re)))
   (if (<= y.im -7.5e+126)
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
     (if (<= y.im -1.15e+85)
       t_0
       (if (<= y.im -2.4e-19)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.im 1.45e-76)
           t_0
           (/
            (- (* y.re (/ x.im (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 t_0 = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	double tmp;
	if (y_46_im <= -7.5e+126) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -1.15e+85) {
		tmp = t_0;
	} else if (y_46_im <= -2.4e-19) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.45e-76) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) - x_46_re) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	double tmp;
	if (y_46_im <= -7.5e+126) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -1.15e+85) {
		tmp = t_0;
	} else if (y_46_im <= -2.4e-19) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.45e-76) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re * (x_46_im / Math.hypot(y_46_re, y_46_im))) - x_46_re) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	tmp = 0
	if y_46_im <= -7.5e+126:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= -1.15e+85:
		tmp = t_0
	elif y_46_im <= -2.4e-19:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 1.45e-76:
		tmp = t_0
	else:
		tmp = ((y_46_re * (x_46_im / math.hypot(y_46_re, y_46_im))) - x_46_re) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re)
	tmp = 0.0
	if (y_46_im <= -7.5e+126)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -1.15e+85)
		tmp = t_0;
	elseif (y_46_im <= -2.4e-19)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.45e-76)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / hypot(y_46_re, y_46_im))) - x_46_re) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	tmp = 0.0;
	if (y_46_im <= -7.5e+126)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -1.15e+85)
		tmp = t_0;
	elseif (y_46_im <= -2.4e-19)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 1.45e-76)
		tmp = t_0;
	else
		tmp = ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) - x_46_re) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$im, -7.5e+126], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.15e+85], t$95$0, If[LessEqual[y$46$im, -2.4e-19], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.45e-76], t$95$0, N[(N[(N[(y$46$re * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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.im < -7.5000000000000006e126

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. add-sqr-sqrt 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. hypot-def 68.6%

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

   3. Applied egg-rr 68.6%

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

      1. div-sub 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. associate-/l* 73.2%

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

      2. associate-/l* 97.4%

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

   7. Simplified 97.4%

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

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

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

      1. neg-mul-1 76.2%

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

      2. *-commutative 76.2%

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

      3. unpow2 76.2%

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

      4. associate-/l* 74.3%

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

      5. associate-*l/ 73.8%

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

      6. +-commutative 73.8%

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

      7. unsub-neg 73.8%

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

      8. associate-*l/ 74.3%

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

      9. associate-/l* 76.2%

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

      10. associate-/r* 85.7%

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

      11. *-commutative 85.7%

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

      12. associate-*r/ 87.6%

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

      13. div-sub 87.6%

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

   10. Simplified 87.6%

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

   if -7.5000000000000006e126 < y.im < -1.1499999999999999e85 or -2.40000000000000023e-19 < y.im < 1.4500000000000001e-76

   1. Initial program 68.4%

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

   2. Taylor expanded in y.re around inf 82.0%

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

      1. mul-1-neg 82.0%

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

      2. unsub-neg 82.0%

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

      3. unpow2 82.0%

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

      4. associate-/r* 85.3%

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

   4. Simplified 85.3%

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

      1. sub-div 88.9%

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

      2. associate-/l* 90.2%

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

   6. Applied egg-rr 90.2%

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

   if -1.1499999999999999e85 < y.im < -2.40000000000000023e-19

   1. Initial program 80.2%

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

   if 1.4500000000000001e-76 < y.im

   1. Initial program 65.2%

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

      1. *-un-lft-identity 65.2%

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

      2. add-sqr-sqrt 65.1%

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

      3. times-frac 65.2%

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

      4. hypot-def 65.2%

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

      5. hypot-def 77.5%

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

   3. Applied egg-rr 77.5%

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

      1. div-sub 77.6%

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

   5. Applied egg-rr 77.6%

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

      1. associate-/l* 83.7%

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

      2. associate-/l* 99.7%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 83.4%

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

      2. expm1-udef 36.6%

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

      3. associate-*l/ 36.6%

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

      4. *-un-lft-identity 36.6%

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

      5. div-inv 36.6%

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

      6. clear-num 36.6%

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

      7. div-inv 36.6%

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

      8. clear-num 36.6%

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

   9. Applied egg-rr 36.6%

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

       1. expm1-def 83.6%

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

       2. expm1-log1p 99.9%

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

       3. associate-*r/ 92.5%

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

       4. associate-*l/ 99.9%

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

       5. *-commutative 99.9%

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

       6. associate-*r/ 83.9%

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

       7. associate-*l/ 99.9%

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

       8. *-commutative 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in y.im around inf 91.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 5: 84.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(y.re \cdot x.im\right) \cdot \frac{1}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.2e+53)
   (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
   (if (<= y.re -3.6e-152)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 2e-53)
       (/ (- (* (* y.re x.im) (/ 1.0 y.im)) x.re) y.im)
       (*
        (/ 1.0 (hypot y.re y.im))
        (- x.im (/ x.re (/ (hypot 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) {
	double tmp;
	if (y_46_re <= -4.2e+53) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_re <= -3.6e-152) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 2e-53) {
		tmp = (((y_46_re * x_46_im) * (1.0 / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re / (hypot(y_46_re, y_46_im) / y_46_im)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.2e+53) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_re <= -3.6e-152) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 2e-53) {
		tmp = (((y_46_re * x_46_im) * (1.0 / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re / (Math.hypot(y_46_re, y_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 <= -4.2e+53:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	elif y_46_re <= -3.6e-152:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 2e-53:
		tmp = (((y_46_re * x_46_im) * (1.0 / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re / (math.hypot(y_46_re, y_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 <= -4.2e+53)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_re <= -3.6e-152)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 2e-53)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) * Float64(1.0 / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im - Float64(x_46_re / Float64(hypot(y_46_re, y_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 <= -4.2e+53)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	elseif (y_46_re <= -3.6e-152)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 2e-53)
		tmp = (((y_46_re * x_46_im) * (1.0 / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re / (hypot(y_46_re, y_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[LessEqual[y$46$re, -4.2e+53], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.6e-152], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2e-53], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im - N[(x$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.2000000000000004e53

   1. Initial program 52.9%

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

   2. Taylor expanded in y.re around inf 85.0%

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

      1. mul-1-neg 85.0%

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

      2. unsub-neg 85.0%

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

      3. unpow2 85.0%

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

      4. associate-/r* 86.6%

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

   4. Simplified 86.6%

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

      1. sub-div 86.6%

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

      2. associate-/l* 88.6%

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

   6. Applied egg-rr 88.6%

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

   if -4.2000000000000004e53 < y.re < -3.6e-152

   1. Initial program 81.8%

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

   if -3.6e-152 < y.re < 2.00000000000000006e-53

   1. Initial program 72.3%

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

   2. Taylor expanded in y.re around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

      4. unpow2 83.5%

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

      5. times-frac 86.9%

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

   4. Simplified 86.9%

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

      1. associate-*r/ 88.0%

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

      2. sub-div 91.5%

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

   6. Applied egg-rr 91.5%

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

      1. associate-*l/ 92.5%

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

      2. clear-num 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. associate-/r/ 92.4%

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

   10. Simplified 92.4%

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

   if 2.00000000000000006e-53 < y.re

   1. Initial program 50.8%

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

      1. *-un-lft-identity 50.8%

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

      2. add-sqr-sqrt 50.7%

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

      3. times-frac 50.7%

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

      4. hypot-def 50.7%

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

      5. hypot-def 69.4%

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

   3. Applied egg-rr 69.4%

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

      1. div-sub 69.4%

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

   5. Applied egg-rr 69.4%

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

      1. associate-/l* 89.4%

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

      2. associate-/l* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y.re around inf 88.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(y.re \cdot x.im\right) \cdot \frac{1}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \]

Alternative 6: 80.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{t_1}\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, x.re \cdot \left(-y.im\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ x.re (/ y.re y.im))) y.re))
        (t_1 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.im -7.5e+126)
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
     (if (<= y.im -1.15e+85)
       t_0
       (if (<= y.im -5.7e-19)
         (/ (- (* y.re x.im) (* y.im x.re)) t_1)
         (if (<= y.im 1.5e-84)
           t_0
           (if (<= y.im 1.02e+139)
             (/ (fma y.re x.im (* x.re (- y.im))) t_1)
             (- (/ y.re (* y.im (/ y.im x.im))) (/ x.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	double t_1 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -7.5e+126) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -1.15e+85) {
		tmp = t_0;
	} else if (y_46_im <= -5.7e-19) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / t_1;
	} else if (y_46_im <= 1.5e-84) {
		tmp = t_0;
	} else if (y_46_im <= 1.02e+139) {
		tmp = fma(y_46_re, x_46_im, (x_46_re * -y_46_im)) / t_1;
	} else {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re)
	t_1 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -7.5e+126)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -1.15e+85)
		tmp = t_0;
	elseif (y_46_im <= -5.7e-19)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / t_1);
	elseif (y_46_im <= 1.5e-84)
		tmp = t_0;
	elseif (y_46_im <= 1.02e+139)
		tmp = Float64(fma(y_46_re, x_46_im, Float64(x_46_re * Float64(-y_46_im))) / t_1);
	else
		tmp = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7.5e+126], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.15e+85], t$95$0, If[LessEqual[y$46$im, -5.7e-19], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y$46$im, 1.5e-84], t$95$0, If[LessEqual[y$46$im, 1.02e+139], N[(N[(y$46$re * x$46$im + N[(x$46$re * (-y$46$im)), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -7.5000000000000006e126

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. add-sqr-sqrt 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. hypot-def 68.6%

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

   3. Applied egg-rr 68.6%

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

      1. div-sub 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. associate-/l* 73.2%

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

      2. associate-/l* 97.4%

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

   7. Simplified 97.4%

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

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

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

      1. neg-mul-1 76.2%

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

      2. *-commutative 76.2%

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

      3. unpow2 76.2%

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

      4. associate-/l* 74.3%

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

      5. associate-*l/ 73.8%

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

      6. +-commutative 73.8%

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

      7. unsub-neg 73.8%

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

      8. associate-*l/ 74.3%

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

      9. associate-/l* 76.2%

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

      10. associate-/r* 85.7%

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

      11. *-commutative 85.7%

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

      12. associate-*r/ 87.6%

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

      13. div-sub 87.6%

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

   10. Simplified 87.6%

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

   if -7.5000000000000006e126 < y.im < -1.1499999999999999e85 or -5.69999999999999952e-19 < y.im < 1.5000000000000001e-84

   1. Initial program 68.4%

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

   2. Taylor expanded in y.re around inf 82.0%

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

      1. mul-1-neg 82.0%

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

      2. unsub-neg 82.0%

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

      3. unpow2 82.0%

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

      4. associate-/r* 85.3%

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

   4. Simplified 85.3%

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

      1. sub-div 88.9%

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

      2. associate-/l* 90.2%

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

   6. Applied egg-rr 90.2%

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

   if -1.1499999999999999e85 < y.im < -5.69999999999999952e-19

   1. Initial program 80.2%

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

   if 1.5000000000000001e-84 < y.im < 1.02e139

   1. Initial program 87.2%

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

   2. Taylor expanded in x.im around 0 87.2%

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

      1. mul-1-neg 87.2%

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

      2. distribute-rgt-neg-out 87.2%

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

      3. +-commutative 87.2%

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

      4. fma-udef 87.2%

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

      5. distribute-rgt-neg-out 87.2%

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

      6. *-commutative 87.2%

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

      7. distribute-rgt-neg-in 87.2%

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

   4. Simplified 87.2%

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

   if 1.02e139 < y.im

   1. Initial program 32.8%

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

   2. Taylor expanded in y.re around 0 81.9%

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. unpow2 81.9%

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

      5. times-frac 91.0%

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

   4. Simplified 91.0%

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

      1. *-commutative 91.0%

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

      2. clear-num 91.0%

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

      3. frac-times 91.2%

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

      4. *-un-lft-identity 91.2%

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

   6. Applied egg-rr 91.2%

      \[\leadsto \color{blue}{\frac{y.re}{\frac{y.im}{x.im} \cdot y.im}} - \frac{x.re}{y.im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq -5.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, x.re \cdot \left(-y.im\right)\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -7.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -8.3 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.95 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ x.re (/ y.re y.im))) y.re))
        (t_1
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -7.8e+126)
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
     (if (<= y.im -1.15e+85)
       t_0
       (if (<= y.im -8.3e-19)
         t_1
         (if (<= y.im 2.95e-80)
           t_0
           (if (<= y.im 1.02e+139)
             t_1
             (- (/ y.re (* y.im (/ y.im x.im))) (/ x.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	double t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -7.8e+126) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -1.15e+85) {
		tmp = t_0;
	} else if (y_46_im <= -8.3e-19) {
		tmp = t_1;
	} else if (y_46_im <= 2.95e-80) {
		tmp = t_0;
	} else if (y_46_im <= 1.02e+139) {
		tmp = t_1;
	} else {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    t_1 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-7.8d+126)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= (-1.15d+85)) then
        tmp = t_0
    else if (y_46im <= (-8.3d-19)) then
        tmp = t_1
    else if (y_46im <= 2.95d-80) then
        tmp = t_0
    else if (y_46im <= 1.02d+139) then
        tmp = t_1
    else
        tmp = (y_46re / (y_46im * (y_46im / x_46im))) - (x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	double t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -7.8e+126) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -1.15e+85) {
		tmp = t_0;
	} else if (y_46_im <= -8.3e-19) {
		tmp = t_1;
	} else if (y_46_im <= 2.95e-80) {
		tmp = t_0;
	} else if (y_46_im <= 1.02e+139) {
		tmp = t_1;
	} else {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -7.8e+126:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= -1.15e+85:
		tmp = t_0
	elif y_46_im <= -8.3e-19:
		tmp = t_1
	elif y_46_im <= 2.95e-80:
		tmp = t_0
	elif y_46_im <= 1.02e+139:
		tmp = t_1
	else:
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re)
	t_1 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -7.8e+126)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -1.15e+85)
		tmp = t_0;
	elseif (y_46_im <= -8.3e-19)
		tmp = t_1;
	elseif (y_46_im <= 2.95e-80)
		tmp = t_0;
	elseif (y_46_im <= 1.02e+139)
		tmp = t_1;
	else
		tmp = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -7.8e+126)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -1.15e+85)
		tmp = t_0;
	elseif (y_46_im <= -8.3e-19)
		tmp = t_1;
	elseif (y_46_im <= 2.95e-80)
		tmp = t_0;
	elseif (y_46_im <= 1.02e+139)
		tmp = t_1;
	else
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7.8e+126], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.15e+85], t$95$0, If[LessEqual[y$46$im, -8.3e-19], t$95$1, If[LessEqual[y$46$im, 2.95e-80], t$95$0, If[LessEqual[y$46$im, 1.02e+139], t$95$1, N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -7.79999999999999986e126

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. add-sqr-sqrt 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. hypot-def 68.6%

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

   3. Applied egg-rr 68.6%

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

      1. div-sub 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. associate-/l* 73.2%

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

      2. associate-/l* 97.4%

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

   7. Simplified 97.4%

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

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

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

      1. neg-mul-1 76.2%

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

      2. *-commutative 76.2%

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

      3. unpow2 76.2%

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

      4. associate-/l* 74.3%

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

      5. associate-*l/ 73.8%

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

      6. +-commutative 73.8%

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

      7. unsub-neg 73.8%

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

      8. associate-*l/ 74.3%

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

      9. associate-/l* 76.2%

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

      10. associate-/r* 85.7%

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

      11. *-commutative 85.7%

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

      12. associate-*r/ 87.6%

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

      13. div-sub 87.6%

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

   10. Simplified 87.6%

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

   if -7.79999999999999986e126 < y.im < -1.1499999999999999e85 or -8.3000000000000001e-19 < y.im < 2.95e-80

   1. Initial program 68.4%

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

   2. Taylor expanded in y.re around inf 82.0%

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

      1. mul-1-neg 82.0%

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

      2. unsub-neg 82.0%

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

      3. unpow2 82.0%

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

      4. associate-/r* 85.3%

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

   4. Simplified 85.3%

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

      1. sub-div 88.9%

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

      2. associate-/l* 90.2%

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

   6. Applied egg-rr 90.2%

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

   if -1.1499999999999999e85 < y.im < -8.3000000000000001e-19 or 2.95e-80 < y.im < 1.02e139

   1. Initial program 85.1%

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

   if 1.02e139 < y.im

   1. Initial program 32.8%

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

   2. Taylor expanded in y.re around 0 81.9%

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. unpow2 81.9%

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

      5. times-frac 91.0%

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

   4. Simplified 91.0%

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

      1. *-commutative 91.0%

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

      2. clear-num 91.0%

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

      3. frac-times 91.2%

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

      4. *-un-lft-identity 91.2%

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

   6. Applied egg-rr 91.2%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq -8.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.95 \cdot 10^{-80}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 8: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+127} \lor \neg \left(y.im \leq -5.3 \cdot 10^{+63} \lor \neg \left(y.im \leq -5.2 \cdot 10^{-18}\right) \land y.im \leq 1650000\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.2e+127)
         (not
          (or (<= y.im -5.3e+63)
              (and (not (<= y.im -5.2e-18)) (<= y.im 1650000.0)))))
   (/ (- x.re) y.im)
   (/ (- x.im (/ x.re (/ y.re y.im))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.2e+127) || !((y_46_im <= -5.3e+63) || (!(y_46_im <= -5.2e-18) && (y_46_im <= 1650000.0)))) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.2d+127)) .or. (.not. (y_46im <= (-5.3d+63)) .or. (.not. (y_46im <= (-5.2d-18))) .and. (y_46im <= 1650000.0d0))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.2e+127) || !((y_46_im <= -5.3e+63) || (!(y_46_im <= -5.2e-18) && (y_46_im <= 1650000.0)))) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.2e+127) or not ((y_46_im <= -5.3e+63) or (not (y_46_im <= -5.2e-18) and (y_46_im <= 1650000.0))):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.2e+127) || !((y_46_im <= -5.3e+63) || (!(y_46_im <= -5.2e-18) && (y_46_im <= 1650000.0))))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.2e+127) || ~(((y_46_im <= -5.3e+63) || (~((y_46_im <= -5.2e-18)) && (y_46_im <= 1650000.0)))))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.2e+127], N[Not[Or[LessEqual[y$46$im, -5.3e+63], And[N[Not[LessEqual[y$46$im, -5.2e-18]], $MachinePrecision], LessEqual[y$46$im, 1650000.0]]]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.2000000000000001e127 or -5.2999999999999999e63 < y.im < -5.2000000000000001e-18 or 1.65e6 < y.im

   1. Initial program 55.3%

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

   2. Taylor expanded in y.re around 0 76.0%

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

      1. associate-*r/ 76.0%

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

      2. neg-mul-1 76.0%

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

   4. Simplified 76.0%

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

   if -1.2000000000000001e127 < y.im < -5.2999999999999999e63 or -5.2000000000000001e-18 < y.im < 1.65e6

   1. Initial program 71.7%

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

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

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

      1. mul-1-neg 76.3%

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

      2. unsub-neg 76.3%

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

      3. unpow2 76.3%

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

      4. associate-/r* 79.1%

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

   4. Simplified 79.1%

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

      1. sub-div 82.2%

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

      2. associate-/l* 83.3%

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

   6. Applied egg-rr 83.3%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+127} \lor \neg \left(y.im \leq -5.3 \cdot 10^{+63} \lor \neg \left(y.im \leq -5.2 \cdot 10^{-18}\right) \land y.im \leq 1650000\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]

Alternative 9: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+126} \lor \neg \left(y.im \leq -9.2 \cdot 10^{+84} \lor \neg \left(y.im \leq -5.4 \cdot 10^{-18}\right) \land y.im \leq 1.18 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.5e+126)
         (not
          (or (<= y.im -9.2e+84)
              (and (not (<= y.im -5.4e-18)) (<= y.im 1.18e-54)))))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (/ (- x.im (/ x.re (/ y.re y.im))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.5e+126) || !((y_46_im <= -9.2e+84) || (!(y_46_im <= -5.4e-18) && (y_46_im <= 1.18e-54)))) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-7.5d+126)) .or. (.not. (y_46im <= (-9.2d+84)) .or. (.not. (y_46im <= (-5.4d-18))) .and. (y_46im <= 1.18d-54))) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.5e+126) || !((y_46_im <= -9.2e+84) || (!(y_46_im <= -5.4e-18) && (y_46_im <= 1.18e-54)))) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -7.5e+126) or not ((y_46_im <= -9.2e+84) or (not (y_46_im <= -5.4e-18) and (y_46_im <= 1.18e-54))):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -7.5e+126) || !((y_46_im <= -9.2e+84) || (!(y_46_im <= -5.4e-18) && (y_46_im <= 1.18e-54))))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -7.5e+126) || ~(((y_46_im <= -9.2e+84) || (~((y_46_im <= -5.4e-18)) && (y_46_im <= 1.18e-54)))))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -7.5e+126], N[Not[Or[LessEqual[y$46$im, -9.2e+84], And[N[Not[LessEqual[y$46$im, -5.4e-18]], $MachinePrecision], LessEqual[y$46$im, 1.18e-54]]]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.5000000000000006e126 or -9.1999999999999996e84 < y.im < -5.39999999999999977e-18 or 1.17999999999999996e-54 < y.im

   1. Initial program 59.5%

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

      1. *-un-lft-identity 59.5%

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

      2. add-sqr-sqrt 59.5%

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

      3. times-frac 59.5%

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

      4. hypot-def 59.5%

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

      5. hypot-def 76.0%

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

   3. Applied egg-rr 76.0%

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

      1. div-sub 76.0%

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

   5. Applied egg-rr 76.0%

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

      1. associate-/l* 82.3%

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

      2. associate-/l* 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in y.re around 0 74.8%

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

      1. neg-mul-1 74.8%

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

      2. *-commutative 74.8%

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

      3. unpow2 74.8%

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

      4. associate-/l* 75.0%

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

      5. associate-*l/ 74.8%

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

      6. +-commutative 74.8%

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

      7. unsub-neg 74.8%

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

      8. associate-*l/ 75.0%

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

      9. associate-/l* 74.8%

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

      10. associate-/r* 78.5%

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

      11. *-commutative 78.5%

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

      12. associate-*r/ 80.5%

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

      13. div-sub 80.5%

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

   10. Simplified 80.5%

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

   if -7.5000000000000006e126 < y.im < -9.1999999999999996e84 or -5.39999999999999977e-18 < y.im < 1.17999999999999996e-54

   1. Initial program 69.2%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. mul-1-neg 81.7%

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

      2. unsub-neg 81.7%

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

      3. unpow2 81.7%

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

      4. associate-/r* 84.9%

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

   4. Simplified 84.9%

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

      1. sub-div 88.4%

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

      2. associate-/l* 89.7%

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

   6. Applied egg-rr 89.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+126} \lor \neg \left(y.im \leq -9.2 \cdot 10^{+84} \lor \neg \left(y.im \leq -5.4 \cdot 10^{-18}\right) \land y.im \leq 1.18 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]

Alternative 10: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ t_1 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.12 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.36 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ x.re (/ y.re y.im))) y.re))
        (t_1 (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))
   (if (<= y.im -7.5e+126)
     t_1
     (if (<= y.im -1.12e+85)
       t_0
       (if (<= y.im -2e-18)
         (/ (- x.re) (+ y.im (/ (* y.re y.re) y.im)))
         (if (<= y.im 1.36e-52) 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_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	double t_1 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -7.5e+126) {
		tmp = t_1;
	} else if (y_46_im <= -1.12e+85) {
		tmp = t_0;
	} else if (y_46_im <= -2e-18) {
		tmp = -x_46_re / (y_46_im + ((y_46_re * y_46_re) / y_46_im));
	} else if (y_46_im <= 1.36e-52) {
		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_46im - (x_46re / (y_46re / y_46im))) / y_46re
    t_1 = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    if (y_46im <= (-7.5d+126)) then
        tmp = t_1
    else if (y_46im <= (-1.12d+85)) then
        tmp = t_0
    else if (y_46im <= (-2d-18)) then
        tmp = -x_46re / (y_46im + ((y_46re * y_46re) / y_46im))
    else if (y_46im <= 1.36d-52) 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_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	double t_1 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -7.5e+126) {
		tmp = t_1;
	} else if (y_46_im <= -1.12e+85) {
		tmp = t_0;
	} else if (y_46_im <= -2e-18) {
		tmp = -x_46_re / (y_46_im + ((y_46_re * y_46_re) / y_46_im));
	} else if (y_46_im <= 1.36e-52) {
		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_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	t_1 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -7.5e+126:
		tmp = t_1
	elif y_46_im <= -1.12e+85:
		tmp = t_0
	elif y_46_im <= -2e-18:
		tmp = -x_46_re / (y_46_im + ((y_46_re * y_46_re) / y_46_im))
	elif y_46_im <= 1.36e-52:
		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(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re)
	t_1 = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7.5e+126)
		tmp = t_1;
	elseif (y_46_im <= -1.12e+85)
		tmp = t_0;
	elseif (y_46_im <= -2e-18)
		tmp = Float64(Float64(-x_46_re) / Float64(y_46_im + Float64(Float64(y_46_re * y_46_re) / y_46_im)));
	elseif (y_46_im <= 1.36e-52)
		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_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	t_1 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -7.5e+126)
		tmp = t_1;
	elseif (y_46_im <= -1.12e+85)
		tmp = t_0;
	elseif (y_46_im <= -2e-18)
		tmp = -x_46_re / (y_46_im + ((y_46_re * y_46_re) / y_46_im));
	elseif (y_46_im <= 1.36e-52)
		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[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7.5e+126], t$95$1, If[LessEqual[y$46$im, -1.12e+85], t$95$0, If[LessEqual[y$46$im, -2e-18], N[((-x$46$re) / N[(y$46$im + N[(N[(y$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.36e-52], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.5000000000000006e126 or 1.36e-52 < y.im

   1. Initial program 56.1%

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

      1. *-un-lft-identity 56.1%

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

      2. add-sqr-sqrt 56.0%

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

      3. times-frac 56.1%

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

      4. hypot-def 56.1%

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

      5. hypot-def 73.7%

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

   3. Applied egg-rr 73.7%

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

      1. div-sub 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. associate-/l* 79.5%

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

      2. associate-/l* 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y.re around 0 74.7%

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

      1. neg-mul-1 74.7%

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

      2. *-commutative 74.7%

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

      3. unpow2 74.7%

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

      4. associate-/l* 74.8%

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

      5. associate-*l/ 74.7%

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

      6. +-commutative 74.7%

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

      7. unsub-neg 74.7%

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

      8. associate-*l/ 74.8%

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

      9. associate-/l* 74.7%

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

      10. associate-/r* 78.9%

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

      11. *-commutative 78.9%

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

      12. associate-*r/ 81.3%

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

      13. div-sub 81.3%

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

   10. Simplified 81.3%

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

   if -7.5000000000000006e126 < y.im < -1.11999999999999993e85 or -2.0000000000000001e-18 < y.im < 1.36e-52

   1. Initial program 69.2%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. mul-1-neg 81.7%

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

      2. unsub-neg 81.7%

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

      3. unpow2 81.7%

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

      4. associate-/r* 84.9%

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

   4. Simplified 84.9%

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

      1. sub-div 88.4%

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

      2. associate-/l* 89.7%

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

   6. Applied egg-rr 89.7%

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

   if -1.11999999999999993e85 < y.im < -2.0000000000000001e-18

   1. Initial program 80.2%

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

      1. *-un-lft-identity 80.2%

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

      2. add-sqr-sqrt 80.2%

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

      3. times-frac 80.2%

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

      4. hypot-def 80.2%

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

      5. hypot-def 89.8%

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

   3. Applied egg-rr 89.8%

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

      1. div-sub 89.8%

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

   5. Applied egg-rr 89.8%

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

      1. associate-/l* 99.5%

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

      2. associate-/l* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x.im around 0 75.8%

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

      1. mul-1-neg 75.8%

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

      2. associate-/l* 76.2%

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

      3. unpow2 76.2%

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

      4. unpow2 76.2%

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

   10. Simplified 76.2%

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

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

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

       1. +-commutative 76.2%

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

       2. unpow2 76.2%

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

   13. Simplified 76.2%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.12 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.36 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternative 11: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.3 \cdot 10^{-52} \lor \neg \left(y.re \leq 2.3 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y.re \cdot x.im\right) \cdot \frac{1}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.3e-52) (not (<= y.re 2.3e-49)))
   (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
   (/ (- (* (* y.re x.im) (/ 1.0 y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.3e-52) || !(y_46_re <= 2.3e-49)) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = (((y_46_re * x_46_im) * (1.0 / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-4.3d-52)) .or. (.not. (y_46re <= 2.3d-49))) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else
        tmp = (((y_46re * x_46im) * (1.0d0 / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.3e-52) || !(y_46_re <= 2.3e-49)) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = (((y_46_re * x_46_im) * (1.0 / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -4.3e-52) or not (y_46_re <= 2.3e-49):
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	else:
		tmp = (((y_46_re * x_46_im) * (1.0 / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -4.3e-52) || !(y_46_re <= 2.3e-49))
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) * Float64(1.0 / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -4.3e-52) || ~((y_46_re <= 2.3e-49)))
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	else
		tmp = (((y_46_re * x_46_im) * (1.0 / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -4.3e-52], N[Not[LessEqual[y$46$re, 2.3e-49]], $MachinePrecision]], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.3000000000000003e-52 or 2.2999999999999999e-49 < y.re

   1. Initial program 54.4%

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

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

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

      1. mul-1-neg 73.4%

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

      2. unsub-neg 73.4%

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

      3. unpow2 73.4%

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

      4. associate-/r* 74.7%

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

   4. Simplified 74.7%

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

      1. sub-div 74.7%

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

      2. associate-/l* 76.7%

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

   6. Applied egg-rr 76.7%

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

   if -4.3000000000000003e-52 < y.re < 2.2999999999999999e-49

   1. Initial program 75.9%

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

   2. Taylor expanded in y.re around 0 81.1%

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

      1. +-commutative 81.1%

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

      2. mul-1-neg 81.1%

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

      3. unsub-neg 81.1%

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

      4. unpow2 81.1%

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

      5. times-frac 84.7%

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

   4. Simplified 84.7%

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

      1. associate-*r/ 85.5%

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

      2. sub-div 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. associate-*l/ 89.0%

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

      2. clear-num 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. associate-/r/ 89.0%

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

   10. Simplified 89.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.3 \cdot 10^{-52} \lor \neg \left(y.re \leq 2.3 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y.re \cdot x.im\right) \cdot \frac{1}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternative 12: 63.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+121} \lor \neg \left(y.im \leq -2.3 \cdot 10^{+84} \lor \neg \left(y.im \leq -6.2 \cdot 10^{-18}\right) \land y.im \leq 6.2 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.1e+121)
         (not
          (or (<= y.im -2.3e+84)
              (and (not (<= y.im -6.2e-18)) (<= y.im 6.2e-33)))))
   (/ (- x.re) y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.1e+121) || !((y_46_im <= -2.3e+84) || (!(y_46_im <= -6.2e-18) && (y_46_im <= 6.2e-33)))) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-3.1d+121)) .or. (.not. (y_46im <= (-2.3d+84)) .or. (.not. (y_46im <= (-6.2d-18))) .and. (y_46im <= 6.2d-33))) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.1e+121) || !((y_46_im <= -2.3e+84) || (!(y_46_im <= -6.2e-18) && (y_46_im <= 6.2e-33)))) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -3.1e+121) or not ((y_46_im <= -2.3e+84) or (not (y_46_im <= -6.2e-18) and (y_46_im <= 6.2e-33))):
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -3.1e+121) || !((y_46_im <= -2.3e+84) || (!(y_46_im <= -6.2e-18) && (y_46_im <= 6.2e-33))))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -3.1e+121) || ~(((y_46_im <= -2.3e+84) || (~((y_46_im <= -6.2e-18)) && (y_46_im <= 6.2e-33)))))
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -3.1e+121], N[Not[Or[LessEqual[y$46$im, -2.3e+84], And[N[Not[LessEqual[y$46$im, -6.2e-18]], $MachinePrecision], LessEqual[y$46$im, 6.2e-33]]]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.10000000000000008e121 or -2.2999999999999999e84 < y.im < -6.20000000000000014e-18 or 6.19999999999999994e-33 < y.im

   1. Initial program 57.5%

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

   2. Taylor expanded in y.re around 0 71.3%

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

      1. associate-*r/ 71.3%

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

      2. neg-mul-1 71.3%

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

   4. Simplified 71.3%

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

   if -3.10000000000000008e121 < y.im < -2.2999999999999999e84 or -6.20000000000000014e-18 < y.im < 6.19999999999999994e-33

   1. Initial program 71.3%

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

   2. Taylor expanded in y.re around inf 74.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+121} \lor \neg \left(y.im \leq -2.3 \cdot 10^{+84} \lor \neg \left(y.im \leq -6.2 \cdot 10^{-18}\right) \land y.im \leq 6.2 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 13: 43.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+178}:\\ \;\;\;\;\frac{-x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.65e+178) (/ (- x.im) y.im) (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.65e+178) {
		tmp = -x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.65d+178)) then
        tmp = -x_46im / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.65e+178) {
		tmp = -x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.65e+178:
		tmp = -x_46_im / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.65e+178)
		tmp = Float64(Float64(-x_46_im) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.65e+178)
		tmp = -x_46_im / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.65e+178], N[((-x$46$im) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.6499999999999999e178

   1. Initial program 44.2%

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

      1. *-un-lft-identity 44.2%

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

      2. add-sqr-sqrt 44.2%

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

      3. times-frac 44.2%

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

      4. hypot-def 44.2%

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

      5. hypot-def 64.4%

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

   3. Applied egg-rr 64.4%

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

   4. Taylor expanded in y.re around -inf 26.7%

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

      1. neg-mul-1 26.7%

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

   6. Simplified 26.7%

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

   7. Taylor expanded in y.re around 0 27.0%

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

      1. associate-*r/ 27.0%

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

      2. mul-1-neg 27.0%

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

   9. Simplified 27.0%

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

   if -1.6499999999999999e178 < y.im

   1. Initial program 66.8%

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

   2. Taylor expanded in y.re around inf 49.6%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+178}:\\ \;\;\;\;\frac{-x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 14: 43.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.2e+157) (/ x.im y.im) (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.2e+157) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.2d+157)) then
        tmp = x_46im / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.2e+157) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.2e+157:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.2e+157)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.2e+157)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.2e+157], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.2e157

   1. Initial program 40.9%

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

      1. *-un-lft-identity 40.9%

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

      2. add-sqr-sqrt 40.9%

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

      3. times-frac 40.9%

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

      4. hypot-def 40.9%

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

      5. hypot-def 67.2%

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

   3. Applied egg-rr 67.2%

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

   4. Taylor expanded in y.re around -inf 24.8%

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

      1. neg-mul-1 24.8%

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

   6. Simplified 24.8%

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

   7. Taylor expanded in y.im around -inf 24.8%

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

   if -1.2e157 < y.im

   1. Initial program 67.6%

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

   2. Taylor expanded in y.re around inf 50.3%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 15: 10.0% 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 63.9%

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

   1. *-un-lft-identity 63.9%

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

   2. add-sqr-sqrt 63.9%

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

   3. times-frac 63.9%

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

   4. hypot-def 63.9%

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

   5. hypot-def 78.5%

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

3. Applied egg-rr 78.5%

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

4. Taylor expanded in y.re around -inf 34.9%

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

   1. neg-mul-1 34.9%

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

6. Simplified 34.9%

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

7. Taylor expanded in y.im around -inf 11.9%

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

8. Final simplification 11.9%

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

Reproduce

herbie shell --seed 2023217 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))