_divideComplex, imaginary part

Percentage Accurate: 61.2% → 82.1%

Time: 11.0s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% 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: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;y.im \leq -8.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{-y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -5.1 \cdot 10^{-93}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+131}:\\ \;\;\;\;\frac{t\_0}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= y.im -8.4e+79)
     (/ (* (/ (- y.im) (hypot y.re y.im)) x.re) (hypot y.re y.im))
     (if (<= y.im -5.1e-93)
       (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 2.3e-153)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 8e+131)
           (/ t_0 (pow (hypot y.re y.im) 2.0))
           (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_im <= -8.4e+79) {
		tmp = ((-y_46_im / hypot(y_46_re, y_46_im)) * x_46_re) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -5.1e-93) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 2.3e-153) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 8e+131) {
		tmp = t_0 / pow(hypot(y_46_re, y_46_im), 2.0);
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_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 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_im <= -8.4e+79) {
		tmp = ((-y_46_im / Math.hypot(y_46_re, y_46_im)) * x_46_re) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -5.1e-93) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 2.3e-153) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 8e+131) {
		tmp = t_0 / Math.pow(Math.hypot(y_46_re, y_46_im), 2.0);
	} else {
		tmp = ((x_46_im * (y_46_re / 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):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if y_46_im <= -8.4e+79:
		tmp = ((-y_46_im / math.hypot(y_46_re, y_46_im)) * x_46_re) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -5.1e-93:
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 2.3e-153:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 8e+131:
		tmp = t_0 / math.pow(math.hypot(y_46_re, y_46_im), 2.0)
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (y_46_im <= -8.4e+79)
		tmp = Float64(Float64(Float64(Float64(-y_46_im) / hypot(y_46_re, y_46_im)) * x_46_re) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -5.1e-93)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 2.3e-153)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 8e+131)
		tmp = Float64(t_0 / (hypot(y_46_re, y_46_im) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / 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)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if (y_46_im <= -8.4e+79)
		tmp = ((-y_46_im / hypot(y_46_re, y_46_im)) * x_46_re) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -5.1e-93)
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 2.3e-153)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 8e+131)
		tmp = t_0 / (hypot(y_46_re, y_46_im) ^ 2.0);
	else
		tmp = ((x_46_im * (y_46_re / 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_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8.4e+79], N[(N[(N[((-y$46$im) / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -5.1e-93], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.3e-153], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8e+131], N[(t$95$0 / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -8.40000000000000032e79

   1. Initial program 41.3%

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

   3. Taylor expanded in x.im around 0 41.5%

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

      1. *-commutative 41.5%

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

      2. associate-/l* 47.4%

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

      3. associate-*r* 47.4%

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

      4. *-commutative 47.4%

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

      5. associate-*r/ 47.4%

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

      6. mul-1-neg 47.4%

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

      7. rem-square-sqrt 47.4%

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

      8. +-commutative 47.4%

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

      9. unpow2 47.4%

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

      10. unpow2 47.4%

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

      11. hypot-undefine 47.4%

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

      12. +-commutative 47.4%

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

      13. unpow2 47.4%

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

      14. unpow2 47.4%

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

      15. hypot-undefine 47.4%

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

      16. unpow2 47.4%

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

   5. Simplified 47.4%

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

      1. neg-mul-1 47.4%

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

      2. unpow2 47.4%

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

      3. times-frac 90.1%

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

      4. hypot-undefine 47.4%

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

      5. +-commutative 47.4%

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

      6. hypot-undefine 90.1%

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

      7. hypot-undefine 47.4%

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

      8. +-commutative 47.4%

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

      9. hypot-undefine 90.1%

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

   7. Applied egg-rr 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*l/ 90.1%

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

      3. associate-*l/ 92.8%

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

      4. associate-*r/ 92.8%

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

      5. associate-*l/ 92.6%

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

      6. frac-2neg 92.6%

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

      7. metadata-eval 92.6%

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

      8. associate-*l/ 92.8%

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

      9. *-un-lft-identity 92.8%

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

   9. Applied egg-rr 92.8%

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

   if -8.40000000000000032e79 < y.im < -5.10000000000000023e-93

   1. Initial program 77.5%

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

   if -5.10000000000000023e-93 < y.im < 2.29999999999999997e-153

   1. Initial program 68.8%

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

   3. Taylor expanded in y.re around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. unsub-neg 93.0%

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

      3. unsub-neg 93.0%

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

      4. remove-double-neg 93.0%

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

      5. mul-1-neg 93.0%

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

      6. neg-mul-1 93.0%

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

      7. mul-1-neg 93.0%

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

      8. distribute-lft-in 93.0%

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

      9. distribute-lft-in 93.0%

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

      10. neg-mul-1 93.0%

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

      11. mul-1-neg 93.0%

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

      12. remove-double-neg 93.0%

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

      13. associate-*r/ 93.0%

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

      14. associate-*r* 93.0%

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

      15. neg-mul-1 93.0%

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

   5. Simplified 93.0%

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

   if 2.29999999999999997e-153 < y.im < 7.9999999999999993e131

   1. Initial program 80.3%

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

      1. add-sqr-sqrt 80.3%

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

      2. pow2 80.3%

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

      3. hypot-define 80.3%

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

   4. Applied egg-rr 80.3%

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

   if 7.9999999999999993e131 < y.im

   1. Initial program 21.4%

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

   3. Taylor expanded in y.re around 0 73.5%

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

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. unpow2 73.5%

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

      5. associate-/r* 78.9%

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

      6. div-sub 78.9%

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

      7. *-commutative 78.9%

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

      8. associate-/l* 84.3%

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

   5. Simplified 84.3%

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

   6. Taylor expanded in y.re around 0 78.9%

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

      1. associate-*r/ 84.3%

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

   8. Simplified 84.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{-y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -5.1 \cdot 10^{-93}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+131}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -9.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{-y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.35 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -9.2e+79)
     (/ (* (/ (- y.im) (hypot y.re y.im)) x.re) (hypot y.re y.im))
     (if (<= y.im -1.42e-95)
       t_0
       (if (<= y.im 2e-153)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 2.35e+132)
           t_0
           (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -9.2e+79) {
		tmp = ((-y_46_im / hypot(y_46_re, y_46_im)) * x_46_re) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -1.42e-95) {
		tmp = t_0;
	} else if (y_46_im <= 2e-153) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.35e+132) {
		tmp = t_0;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_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 = ((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 <= -9.2e+79) {
		tmp = ((-y_46_im / Math.hypot(y_46_re, y_46_im)) * x_46_re) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -1.42e-95) {
		tmp = t_0;
	} else if (y_46_im <= 2e-153) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.35e+132) {
		tmp = t_0;
	} else {
		tmp = ((x_46_im * (y_46_re / 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):
	t_0 = ((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 <= -9.2e+79:
		tmp = ((-y_46_im / math.hypot(y_46_re, y_46_im)) * x_46_re) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -1.42e-95:
		tmp = t_0
	elif y_46_im <= 2e-153:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 2.35e+132:
		tmp = t_0
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(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 <= -9.2e+79)
		tmp = Float64(Float64(Float64(Float64(-y_46_im) / hypot(y_46_re, y_46_im)) * x_46_re) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -1.42e-95)
		tmp = t_0;
	elseif (y_46_im <= 2e-153)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 2.35e+132)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / 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)
	t_0 = ((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 <= -9.2e+79)
		tmp = ((-y_46_im / hypot(y_46_re, y_46_im)) * x_46_re) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -1.42e-95)
		tmp = t_0;
	elseif (y_46_im <= 2e-153)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 2.35e+132)
		tmp = t_0;
	else
		tmp = ((x_46_im * (y_46_re / 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_] := Block[{t$95$0 = 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, -9.2e+79], N[(N[(N[((-y$46$im) / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.42e-95], t$95$0, If[LessEqual[y$46$im, 2e-153], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.35e+132], t$95$0, N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.2000000000000002e79

   1. Initial program 41.3%

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

   3. Taylor expanded in x.im around 0 41.5%

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

      1. *-commutative 41.5%

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

      2. associate-/l* 47.4%

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

      3. associate-*r* 47.4%

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

      4. *-commutative 47.4%

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

      5. associate-*r/ 47.4%

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

      6. mul-1-neg 47.4%

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

      7. rem-square-sqrt 47.4%

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

      8. +-commutative 47.4%

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

      9. unpow2 47.4%

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

      10. unpow2 47.4%

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

      11. hypot-undefine 47.4%

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

      12. +-commutative 47.4%

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

      13. unpow2 47.4%

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

      14. unpow2 47.4%

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

      15. hypot-undefine 47.4%

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

      16. unpow2 47.4%

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

   5. Simplified 47.4%

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

      1. neg-mul-1 47.4%

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

      2. unpow2 47.4%

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

      3. times-frac 90.1%

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

      4. hypot-undefine 47.4%

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

      5. +-commutative 47.4%

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

      6. hypot-undefine 90.1%

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

      7. hypot-undefine 47.4%

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

      8. +-commutative 47.4%

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

      9. hypot-undefine 90.1%

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

   7. Applied egg-rr 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*l/ 90.1%

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

      3. associate-*l/ 92.8%

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

      4. associate-*r/ 92.8%

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

      5. associate-*l/ 92.6%

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

      6. frac-2neg 92.6%

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

      7. metadata-eval 92.6%

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

      8. associate-*l/ 92.8%

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

      9. *-un-lft-identity 92.8%

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

   9. Applied egg-rr 92.8%

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

   if -9.2000000000000002e79 < y.im < -1.42000000000000007e-95 or 2.00000000000000008e-153 < y.im < 2.35e132

   1. Initial program 79.1%

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

   if -1.42000000000000007e-95 < y.im < 2.00000000000000008e-153

   1. Initial program 68.8%

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

   3. Taylor expanded in y.re around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. unsub-neg 93.0%

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

      3. unsub-neg 93.0%

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

      4. remove-double-neg 93.0%

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

      5. mul-1-neg 93.0%

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

      6. neg-mul-1 93.0%

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

      7. mul-1-neg 93.0%

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

      8. distribute-lft-in 93.0%

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

      9. distribute-lft-in 93.0%

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

      10. neg-mul-1 93.0%

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

      11. mul-1-neg 93.0%

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

      12. remove-double-neg 93.0%

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

      13. associate-*r/ 93.0%

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

      14. associate-*r* 93.0%

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

      15. neg-mul-1 93.0%

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

   5. Simplified 93.0%

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

   if 2.35e132 < y.im

   1. Initial program 21.4%

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

   3. Taylor expanded in y.re around 0 73.5%

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

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. unpow2 73.5%

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

      5. associate-/r* 78.9%

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

      6. div-sub 78.9%

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

      7. *-commutative 78.9%

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

      8. associate-/l* 84.3%

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

   5. Simplified 84.3%

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

   6. Taylor expanded in y.re around 0 78.9%

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

      1. associate-*r/ 84.3%

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

   8. Simplified 84.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{-y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.35 \cdot 10^{+132}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -112000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-47}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* x.re (/ y.im y.re))) y.re)))
   (if (<= y.re -112000000000.0)
     t_0
     (if (<= y.re 1.95e-47)
       (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
       (if (<= y.re 1.4e+91)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         t_0)))))

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_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -112000000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 1.95e-47) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.4e+91) {
		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 {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    if (y_46re <= (-112000000000.0d0)) then
        tmp = t_0
    else if (y_46re <= 1.95d-47) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46re <= 1.4d+91) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -112000000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 1.95e-47) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.4e+91) {
		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 {
		tmp = t_0;
	}
	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_im / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -112000000000.0:
		tmp = t_0
	elif y_46_re <= 1.95e-47:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 1.4e+91:
		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:
		tmp = t_0
	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_im / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -112000000000.0)
		tmp = t_0;
	elseif (y_46_re <= 1.95e-47)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.4e+91)
		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)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -112000000000.0)
		tmp = t_0;
	elseif (y_46_re <= 1.95e-47)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 1.4e+91)
		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
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -112000000000.0], t$95$0, If[LessEqual[y$46$re, 1.95e-47], 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$46$re, 1.4e+91], 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], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.12e11 or 1.3999999999999999e91 < y.re

   1. Initial program 42.3%

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

   3. Taylor expanded in y.re around inf 78.2%

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

      1. mul-1-neg 78.2%

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

      2. unsub-neg 78.2%

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

      3. associate-/l* 81.1%

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

   5. Simplified 81.1%

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

   if -1.12e11 < y.re < 1.94999999999999989e-47

   1. Initial program 71.0%

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

   3. Taylor expanded in y.re around 0 78.4%

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

      1. +-commutative 78.4%

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

      2. mul-1-neg 78.4%

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

      3. unsub-neg 78.4%

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

      4. unpow2 78.4%

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

      5. associate-/r* 82.8%

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

      6. div-sub 85.3%

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

      7. *-commutative 85.3%

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

      8. associate-/l* 85.3%

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

   5. Simplified 85.3%

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

   if 1.94999999999999989e-47 < y.re < 1.3999999999999999e91

   1. Initial program 96.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -112000000000:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-47}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{-50} \lor \neg \left(y.im \leq 9.8 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4e-50) (not (<= y.im 9.8e+92)))
   (/ x.re (- y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4e-50) || !(y_46_im <= 9.8e+92)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4d-50)) .or. (.not. (y_46im <= 9.8d+92))) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4e-50) || !(y_46_im <= 9.8e+92)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4e-50) or not (y_46_im <= 9.8e+92):
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4e-50) || !(y_46_im <= 9.8e+92))
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4e-50) || ~((y_46_im <= 9.8e+92)))
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4e-50], N[Not[LessEqual[y$46$im, 9.8e+92]], $MachinePrecision]], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.00000000000000003e-50 or 9.8000000000000003e92 < y.im

   1. Initial program 47.6%

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

   3. Taylor expanded in y.re around 0 73.1%

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

      1. associate-*r/ 73.1%

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

      2. neg-mul-1 73.1%

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

   5. Simplified 73.1%

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

   if -4.00000000000000003e-50 < y.im < 9.8000000000000003e92

   1. Initial program 72.6%

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

   3. Taylor expanded in y.re around inf 80.8%

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

      1. mul-1-neg 80.8%

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

      2. unsub-neg 80.8%

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

      3. associate-/l* 80.8%

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

   5. Simplified 80.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{-50} \lor \neg \left(y.im \leq 9.8 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4300000000 \lor \neg \left(y.re \leq 2.1 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4300000000.0) (not (<= y.re 2.1e-27)))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4300000000.0) || !(y_46_re <= 2.1e-27)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / 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 <= (-4300000000.0d0)) .or. (.not. (y_46re <= 2.1d-27))) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = ((x_46im * (y_46re / 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 <= -4300000000.0) || !(y_46_re <= 2.1e-27)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / 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 <= -4300000000.0) or not (y_46_re <= 2.1e-27):
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -4300000000.0) || !(y_46_re <= 2.1e-27))
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / 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 <= -4300000000.0) || ~((y_46_re <= 2.1e-27)))
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = ((x_46_im * (y_46_re / 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, -4300000000.0], N[Not[LessEqual[y$46$re, 2.1e-27]], $MachinePrecision]], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.3e9 or 2.10000000000000015e-27 < y.re

   1. Initial program 52.2%

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

   3. Taylor expanded in y.re around inf 77.0%

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

      1. mul-1-neg 77.0%

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

      2. unsub-neg 77.0%

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

      3. associate-/l* 79.1%

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

   5. Simplified 79.1%

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

   if -4.3e9 < y.re < 2.10000000000000015e-27

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around 0 77.9%

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

      1. +-commutative 77.9%

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

      2. mul-1-neg 77.9%

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

      3. unsub-neg 77.9%

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

      4. unpow2 77.9%

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

      5. associate-/r* 82.3%

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

      6. div-sub 84.8%

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

      7. *-commutative 84.8%

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

      8. associate-/l* 84.8%

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

   5. Simplified 84.8%

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

   6. Taylor expanded in y.re around 0 84.8%

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

      1. associate-*r/ 84.1%

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

   8. Simplified 84.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4300000000 \lor \neg \left(y.re \leq 2.1 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{-91}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 82000000:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1e-91)
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
   (if (<= y.im 82000000.0)
     (/ (- x.im (* x.re (/ y.im y.re))) y.re)
     (/ (- (* y.re (/ x.im 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_im <= -1e-91) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 82000000.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / 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_46im <= (-1d-91)) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 82000000.0d0) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = ((y_46re * (x_46im / 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_im <= -1e-91) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 82000000.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / 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_im <= -1e-91:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 82000000.0:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = ((y_46_re * (x_46_im / 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_im <= -1e-91)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 82000000.0)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / 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_im <= -1e-91)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 82000000.0)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = ((y_46_re * (x_46_im / 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[LessEqual[y$46$im, -1e-91], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 82000000.0], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.00000000000000002e-91

   1. Initial program 59.2%

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

   3. Taylor expanded in y.re around 0 75.8%

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

      1. +-commutative 75.8%

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

      2. mul-1-neg 75.8%

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

      3. unsub-neg 75.8%

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

      4. unpow2 75.8%

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

      5. associate-/r* 77.0%

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

      6. div-sub 77.0%

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

      7. *-commutative 77.0%

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

      8. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in y.re around 0 77.0%

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

      1. associate-*r/ 77.2%

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

   8. Simplified 77.2%

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

   if -1.00000000000000002e-91 < y.im < 8.2e7

   1. Initial program 73.1%

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

   3. Taylor expanded in y.re around inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. associate-/l* 87.5%

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

   5. Simplified 87.5%

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

   if 8.2e7 < y.im

   1. Initial program 40.5%

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

   3. Taylor expanded in y.re around 0 69.4%

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

      1. +-commutative 69.4%

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

      2. mul-1-neg 69.4%

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

      3. unsub-neg 69.4%

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

      4. unpow2 69.4%

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

      5. associate-/r* 72.8%

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

      6. div-sub 72.8%

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

      7. *-commutative 72.8%

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

      8. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{-91}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 82000000:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.25 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 420000000:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.25e-92)
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
   (if (<= y.im 420000000.0)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (/ (- (* y.re (/ x.im 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_im <= -2.25e-92) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 420000000.0) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / 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_46im <= (-2.25d-92)) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 420000000.0d0) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = ((y_46re * (x_46im / 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_im <= -2.25e-92) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 420000000.0) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / 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_im <= -2.25e-92:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 420000000.0:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	else:
		tmp = ((y_46_re * (x_46_im / 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_im <= -2.25e-92)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 420000000.0)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / 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_im <= -2.25e-92)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 420000000.0)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = ((y_46_re * (x_46_im / 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[LessEqual[y$46$im, -2.25e-92], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 420000000.0], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.25e-92

   1. Initial program 59.2%

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

   3. Taylor expanded in y.re around 0 75.8%

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

      1. +-commutative 75.8%

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

      2. mul-1-neg 75.8%

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

      3. unsub-neg 75.8%

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

      4. unpow2 75.8%

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

      5. associate-/r* 77.0%

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

      6. div-sub 77.0%

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

      7. *-commutative 77.0%

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

      8. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in y.re around 0 77.0%

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

      1. associate-*r/ 77.2%

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

   8. Simplified 77.2%

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

   if -2.25e-92 < y.im < 4.2e8

   1. Initial program 73.1%

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

   3. Taylor expanded in y.re around inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. unsub-neg 88.3%

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

      4. remove-double-neg 88.3%

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

      5. mul-1-neg 88.3%

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

      6. neg-mul-1 88.3%

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

      7. mul-1-neg 88.3%

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

      8. distribute-lft-in 88.3%

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

      9. distribute-lft-in 88.3%

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

      10. neg-mul-1 88.3%

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

      11. mul-1-neg 88.3%

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

      12. remove-double-neg 88.3%

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

      13. associate-*r/ 88.3%

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

      14. associate-*r* 88.3%

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

      15. neg-mul-1 88.3%

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

   5. Simplified 88.3%

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

   if 4.2e8 < y.im

   1. Initial program 40.5%

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

   3. Taylor expanded in y.re around 0 69.4%

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

      1. +-commutative 69.4%

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

      2. mul-1-neg 69.4%

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

      3. unsub-neg 69.4%

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

      4. unpow2 69.4%

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

      5. associate-/r* 72.8%

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

      6. div-sub 72.8%

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

      7. *-commutative 72.8%

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

      8. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.25 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 420000000:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.9 \cdot 10^{-50} \lor \neg \left(y.im \leq 7500000000\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.9e-50) (not (<= y.im 7500000000.0)))
   (/ 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 <= -3.9e-50) || !(y_46_im <= 7500000000.0)) {
		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 <= (-3.9d-50)) .or. (.not. (y_46im <= 7500000000.0d0))) 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 <= -3.9e-50) || !(y_46_im <= 7500000000.0)) {
		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 <= -3.9e-50) or not (y_46_im <= 7500000000.0):
		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 <= -3.9e-50) || !(y_46_im <= 7500000000.0))
		tmp = Float64(x_46_re / Float64(-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 <= -3.9e-50) || ~((y_46_im <= 7500000000.0)))
		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, -3.9e-50], N[Not[LessEqual[y$46$im, 7500000000.0]], $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 < -3.90000000000000021e-50 or 7.5e9 < y.im

   1. Initial program 50.3%

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

   3. Taylor expanded in y.re around 0 69.3%

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

      1. associate-*r/ 69.3%

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

      2. neg-mul-1 69.3%

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

   5. Simplified 69.3%

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

   if -3.90000000000000021e-50 < y.im < 7.5e9

   1. Initial program 72.7%

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

   3. Taylor expanded in y.re around inf 71.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.9 \cdot 10^{-50} \lor \neg \left(y.im \leq 7500000000\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in y.re around inf 46.1%

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

4. Final simplification 46.1%

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

Reproduce

herbie shell --seed 2024100 
(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))))