_divideComplex, imaginary part

Percentage Accurate: 62.2% → 86.6%

Time: 11.2s

Alternatives: 9

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.2% 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: 86.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := x.im \cdot y.re - x.re \cdot y.im\\ t_2 := \frac{t_1}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0 \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im)))
        (t_1 (- (* x.im y.re) (* x.re y.im)))
        (t_2 (/ t_1 (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_2 (- INFINITY))
     (-
      (* t_0 (/ y.re (/ (hypot y.re y.im) x.im)))
      (* y.im (/ x.re (pow (hypot y.re y.im) 2.0))))
     (if (<= t_2 INFINITY)
       (* t_0 (/ t_1 (hypot y.re y.im)))
       (- (* (/ y.re y.im) (/ 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 t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double t_2 = t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * (x_46_re / pow(hypot(y_46_re, y_46_im), 2.0)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / 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 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double t_2 = t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 * (y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * (x_46_re / Math.pow(Math.hypot(y_46_re, y_46_im), 2.0)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_1 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) * (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):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	t_2 = t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (t_0 * (y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * (x_46_re / math.pow(math.hypot(y_46_re, y_46_im), 2.0)))
	elif t_2 <= math.inf:
		tmp = t_0 * (t_1 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = ((y_46_re / y_46_im) * (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)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	t_2 = Float64(t_1 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im))) - Float64(y_46_im * Float64(x_46_re / (hypot(y_46_re, y_46_im) ^ 2.0))));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_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 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	t_2 = t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (t_0 * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * (x_46_re / (hypot(y_46_re, y_46_im) ^ 2.0)));
	elseif (t_2 <= Inf)
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	else
		tmp = ((y_46_re / y_46_im) * (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_] := 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[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t$95$0 * N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 19.5%

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

      1. div-sub 10.4%

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

      2. *-un-lft-identity 10.4%

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

      3. add-sqr-sqrt 10.4%

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

      4. times-frac 10.4%

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

      5. fma-neg 10.4%

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

      6. hypot-def 10.4%

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

      7. hypot-def 19.3%

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

      8. associate-/l* 41.3%

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

      9. add-sqr-sqrt 41.3%

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

      10. pow2 41.3%

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

      11. hypot-def 41.3%

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

   3. Applied egg-rr 41.3%

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

      1. fma-neg 41.3%

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

      2. *-commutative 41.3%

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

      3. associate-/l* 77.7%

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

      4. associate-/r/ 77.4%

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

      5. *-commutative 77.4%

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

   5. Simplified 77.4%

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

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

   1. Initial program 82.1%

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

      1. *-un-lft-identity 82.1%

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

      2. add-sqr-sqrt 82.1%

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

      3. times-frac 82.0%

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

      4. hypot-def 82.0%

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

      5. hypot-def 97.1%

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

   3. Applied egg-rr 97.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in y.re around 0 56.5%

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

      1. +-commutative 56.5%

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

      2. mul-1-neg 56.5%

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

      3. unsub-neg 56.5%

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

      4. unpow2 56.5%

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

      5. times-frac 63.5%

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

   4. Simplified 63.5%

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

4. Final simplification 90.2%

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

Alternative 2: 86.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ t_1 := \frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im)))
        (t_1 (/ t_0 (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_1 (- INFINITY))
     (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re)))
     (if (<= t_1 -1e-245)
       t_1
       (if (<= t_1 INFINITY)
         (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
         (- (* (/ y.re y.im) (/ 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 t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	} else if (t_1 <= -1e-245) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / 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 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	} else if (t_1 <= -1e-245) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) * (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):
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re))
	elif t_1 <= -1e-245:
		tmp = t_1
	elif t_1 <= math.inf:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = ((y_46_re / y_46_im) * (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)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	t_1 = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)));
	elseif (t_1 <= -1e-245)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_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 * y_46_re) - (x_46_re * y_46_im);
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	elseif (t_1 <= -1e-245)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = ((y_46_re / y_46_im) * (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_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-245], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 19.5%

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

   2. Taylor expanded in y.re around inf 56.0%

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

      1. mul-1-neg 56.0%

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

      2. unsub-neg 56.0%

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

      3. unpow2 56.0%

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

      4. times-frac 71.0%

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

   4. Simplified 71.0%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < -9.9999999999999993e-246

   1. Initial program 99.6%

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

   if -9.9999999999999993e-246 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. add-sqr-sqrt 77.6%

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

      3. times-frac 77.7%

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

      4. hypot-def 77.7%

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

      5. hypot-def 96.6%

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

   3. Applied egg-rr 96.6%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in y.re around 0 56.5%

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

      1. +-commutative 56.5%

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

      2. mul-1-neg 56.5%

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

      3. unsub-neg 56.5%

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

      4. unpow2 56.5%

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

      5. times-frac 63.5%

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

   4. Simplified 63.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -\infty:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -1 \cdot 10^{-245}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 3: 81.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_2 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-114}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+24}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im)))
        (t_1 (/ (- x.im (* y.im (/ x.re y.re))) y.re))
        (t_2 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.re -2.4e+73)
     t_1
     (if (<= y.re -1.45e-114)
       (/ t_0 (fma y.re y.re (* y.im y.im)))
       (if (<= y.re 6.8e-29)
         t_2
         (if (<= y.re 4.3e+24)
           (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
           (if (<= y.re 2.25e+52) t_2 t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -2.4e+73) {
		tmp = t_1;
	} else if (y_46_re <= -1.45e-114) {
		tmp = t_0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_re <= 6.8e-29) {
		tmp = t_2;
	} else if (y_46_re <= 4.3e+24) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 2.25e+52) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	t_1 = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	t_2 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.4e+73)
		tmp = t_1;
	elseif (y_46_re <= -1.45e-114)
		tmp = Float64(t_0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 6.8e-29)
		tmp = t_2;
	elseif (y_46_re <= 4.3e+24)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 2.25e+52)
		tmp = t_2;
	else
		tmp = t_1;
	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 * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$2 = 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$46$re, -2.4e+73], t$95$1, If[LessEqual[y$46$re, -1.45e-114], N[(t$95$0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.8e-29], t$95$2, If[LessEqual[y$46$re, 4.3e+24], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.25e+52], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.40000000000000002e73 or 2.25e52 < y.re

   1. Initial program 53.2%

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

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

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

      1. mul-1-neg 78.4%

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

      2. unsub-neg 78.4%

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

      3. unpow2 78.4%

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

      4. times-frac 86.6%

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

   4. Simplified 86.6%

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

      1. associate-*r/ 86.7%

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

      2. sub-div 86.7%

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

   6. Applied egg-rr 86.7%

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

   if -2.40000000000000002e73 < y.re < -1.44999999999999998e-114

   1. Initial program 83.6%

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

      1. fma-def 83.6%

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

   3. Simplified 83.6%

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

   if -1.44999999999999998e-114 < y.re < 6.79999999999999945e-29 or 4.29999999999999987e24 < y.re < 2.25e52

   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. Taylor expanded in y.re around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. mul-1-neg 82.4%

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

      3. unsub-neg 82.4%

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

      4. unpow2 82.4%

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

      5. times-frac 88.3%

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

   4. Simplified 88.3%

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

      1. associate-*r/ 90.1%

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

      2. sub-div 90.2%

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

   6. Applied egg-rr 90.2%

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

   if 6.79999999999999945e-29 < y.re < 4.29999999999999987e24

   1. Initial program 99.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 4: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_2 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.5 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8.4 \cdot 10^{-35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (- x.im (* y.im (/ x.re y.re))) y.re))
        (t_2 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.re -2.4e+80)
     t_1
     (if (<= y.re -1.5e-115)
       t_0
       (if (<= y.re 8.4e-35)
         t_2
         (if (<= y.re 4.3e+24) t_0 (if (<= y.re 2.25e+52) t_2 t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -2.4e+80) {
		tmp = t_1;
	} else if (y_46_re <= -1.5e-115) {
		tmp = t_0;
	} else if (y_46_re <= 8.4e-35) {
		tmp = t_2;
	} else if (y_46_re <= 4.3e+24) {
		tmp = t_0;
	} else if (y_46_re <= 2.25e+52) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    t_2 = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    if (y_46re <= (-2.4d+80)) then
        tmp = t_1
    else if (y_46re <= (-1.5d-115)) then
        tmp = t_0
    else if (y_46re <= 8.4d-35) then
        tmp = t_2
    else if (y_46re <= 4.3d+24) then
        tmp = t_0
    else if (y_46re <= 2.25d+52) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -2.4e+80) {
		tmp = t_1;
	} else if (y_46_re <= -1.5e-115) {
		tmp = t_0;
	} else if (y_46_re <= 8.4e-35) {
		tmp = t_2;
	} else if (y_46_re <= 4.3e+24) {
		tmp = t_0;
	} else if (y_46_re <= 2.25e+52) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_re <= -2.4e+80:
		tmp = t_1
	elif y_46_re <= -1.5e-115:
		tmp = t_0
	elif y_46_re <= 8.4e-35:
		tmp = t_2
	elif y_46_re <= 4.3e+24:
		tmp = t_0
	elif y_46_re <= 2.25e+52:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(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)))
	t_1 = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	t_2 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.4e+80)
		tmp = t_1;
	elseif (y_46_re <= -1.5e-115)
		tmp = t_0;
	elseif (y_46_re <= 8.4e-35)
		tmp = t_2;
	elseif (y_46_re <= 4.3e+24)
		tmp = t_0;
	elseif (y_46_re <= 2.25e+52)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_re <= -2.4e+80)
		tmp = t_1;
	elseif (y_46_re <= -1.5e-115)
		tmp = t_0;
	elseif (y_46_re <= 8.4e-35)
		tmp = t_2;
	elseif (y_46_re <= 4.3e+24)
		tmp = t_0;
	elseif (y_46_re <= 2.25e+52)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(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]}, Block[{t$95$1 = N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$2 = 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$46$re, -2.4e+80], t$95$1, If[LessEqual[y$46$re, -1.5e-115], t$95$0, If[LessEqual[y$46$re, 8.4e-35], t$95$2, If[LessEqual[y$46$re, 4.3e+24], t$95$0, If[LessEqual[y$46$re, 2.25e+52], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.39999999999999979e80 or 2.25e52 < y.re

   1. Initial program 53.2%

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

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

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

      1. mul-1-neg 78.4%

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

      2. unsub-neg 78.4%

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

      3. unpow2 78.4%

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

      4. times-frac 86.6%

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

   4. Simplified 86.6%

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

      1. associate-*r/ 86.7%

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

      2. sub-div 86.7%

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

   6. Applied egg-rr 86.7%

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

   if -2.39999999999999979e80 < y.re < -1.5000000000000001e-115 or 8.3999999999999999e-35 < y.re < 4.29999999999999987e24

   1. Initial program 87.0%

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

   if -1.5000000000000001e-115 < y.re < 8.3999999999999999e-35 or 4.29999999999999987e24 < y.re < 2.25e52

   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. Taylor expanded in y.re around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. mul-1-neg 82.4%

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

      3. unsub-neg 82.4%

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

      4. unpow2 82.4%

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

      5. times-frac 88.3%

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

   4. Simplified 88.3%

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

      1. associate-*r/ 90.1%

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

      2. sub-div 90.2%

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

   6. Applied egg-rr 90.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 5: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{-5} \lor \neg \left(y.re \leq 1.4 \cdot 10^{-23}\right) \land \left(y.re \leq 2.25 \cdot 10^{+24} \lor \neg \left(y.re \leq 2.2 \cdot 10^{+52}\right)\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -9.2e-5)
         (and (not (<= y.re 1.4e-23))
              (or (<= y.re 2.25e+24) (not (<= y.re 2.2e+52)))))
   (/ (- x.im (* y.im (/ x.re 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 <= -9.2e-5) || (!(y_46_re <= 1.4e-23) && ((y_46_re <= 2.25e+24) || !(y_46_re <= 2.2e+52)))) {
		tmp = (x_46_im - (y_46_im * (x_46_re / 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 <= (-9.2d-5)) .or. (.not. (y_46re <= 1.4d-23)) .and. (y_46re <= 2.25d+24) .or. (.not. (y_46re <= 2.2d+52))) then
        tmp = (x_46im - (y_46im * (x_46re / 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 <= -9.2e-5) || (!(y_46_re <= 1.4e-23) && ((y_46_re <= 2.25e+24) || !(y_46_re <= 2.2e+52)))) {
		tmp = (x_46_im - (y_46_im * (x_46_re / 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 <= -9.2e-5) or (not (y_46_re <= 1.4e-23) and ((y_46_re <= 2.25e+24) or not (y_46_re <= 2.2e+52))):
		tmp = (x_46_im - (y_46_im * (x_46_re / 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 <= -9.2e-5) || (!(y_46_re <= 1.4e-23) && ((y_46_re <= 2.25e+24) || !(y_46_re <= 2.2e+52))))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / 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 <= -9.2e-5) || (~((y_46_re <= 1.4e-23)) && ((y_46_re <= 2.25e+24) || ~((y_46_re <= 2.2e+52)))))
		tmp = (x_46_im - (y_46_im * (x_46_re / 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, -9.2e-5], And[N[Not[LessEqual[y$46$re, 1.4e-23]], $MachinePrecision], Or[LessEqual[y$46$re, 2.25e+24], N[Not[LessEqual[y$46$re, 2.2e+52]], $MachinePrecision]]]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / 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 < -9.20000000000000001e-5 or 1.3999999999999999e-23 < y.re < 2.2500000000000001e24 or 2.2e52 < y.re

   1. Initial program 60.3%

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

   2. Taylor expanded in y.re around inf 75.1%

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

      1. mul-1-neg 75.1%

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

      2. unsub-neg 75.1%

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

      3. unpow2 75.1%

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

      4. times-frac 81.5%

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

   4. Simplified 81.5%

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

      1. associate-*r/ 81.6%

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

      2. sub-div 81.5%

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

   6. Applied egg-rr 81.5%

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

   if -9.20000000000000001e-5 < y.re < 1.3999999999999999e-23 or 2.2500000000000001e24 < y.re < 2.2e52

   1. Initial program 68.6%

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

   2. Taylor expanded in y.re around 0 79.9%

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

      1. +-commutative 79.9%

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

      2. mul-1-neg 79.9%

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

      3. unsub-neg 79.9%

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

      4. unpow2 79.9%

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

      5. times-frac 85.4%

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

   4. Simplified 85.4%

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

      1. associate-*r/ 86.9%

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

      2. sub-div 86.9%

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

   6. Applied egg-rr 86.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{-5} \lor \neg \left(y.re \leq 1.4 \cdot 10^{-23}\right) \land \left(y.re \leq 2.25 \cdot 10^{+24} \lor \neg \left(y.re \leq 2.2 \cdot 10^{+52}\right)\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternative 6: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -0.96:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im (/ y.re y.im)) x.re) y.im))
        (t_1 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -0.96)
     t_1
     (if (<= y.re 1.4e-23)
       t_0
       (if (<= y.re 2.05e+24)
         (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re)))
         (if (<= y.re 2.2e+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 * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -0.96) {
		tmp = t_1;
	} else if (y_46_re <= 1.4e-23) {
		tmp = t_0;
	} else if (y_46_re <= 2.05e+24) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	} else if (y_46_re <= 2.2e+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 * (y_46re / y_46im)) - x_46re) / y_46im
    t_1 = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    if (y_46re <= (-0.96d0)) then
        tmp = t_1
    else if (y_46re <= 1.4d-23) then
        tmp = t_0
    else if (y_46re <= 2.05d+24) then
        tmp = (x_46im / y_46re) - ((x_46re / y_46re) * (y_46im / y_46re))
    else if (y_46re <= 2.2d+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 * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -0.96) {
		tmp = t_1;
	} else if (y_46_re <= 1.4e-23) {
		tmp = t_0;
	} else if (y_46_re <= 2.05e+24) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	} else if (y_46_re <= 2.2e+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 * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -0.96:
		tmp = t_1
	elif y_46_re <= 1.4e-23:
		tmp = t_0
	elif y_46_re <= 2.05e+24:
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re))
	elif y_46_re <= 2.2e+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(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	t_1 = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -0.96)
		tmp = t_1;
	elseif (y_46_re <= 1.4e-23)
		tmp = t_0;
	elseif (y_46_re <= 2.05e+24)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)));
	elseif (y_46_re <= 2.2e+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 * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -0.96)
		tmp = t_1;
	elseif (y_46_re <= 1.4e-23)
		tmp = t_0;
	elseif (y_46_re <= 2.05e+24)
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	elseif (y_46_re <= 2.2e+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[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -0.96], t$95$1, If[LessEqual[y$46$re, 1.4e-23], t$95$0, If[LessEqual[y$46$re, 2.05e+24], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.2e+52], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.95999999999999996 or 2.2e52 < y.re

   1. Initial program 57.0%

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

   2. Taylor expanded in y.re around inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. unsub-neg 74.6%

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

      3. unpow2 74.6%

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

      4. times-frac 81.6%

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

   4. Simplified 81.6%

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

      1. associate-*r/ 81.6%

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

      2. sub-div 81.6%

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

   6. Applied egg-rr 81.6%

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

   if -0.95999999999999996 < y.re < 1.3999999999999999e-23 or 2.05e24 < y.re < 2.2e52

   1. Initial program 68.6%

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

   2. Taylor expanded in y.re around 0 79.9%

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

      1. +-commutative 79.9%

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

      2. mul-1-neg 79.9%

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

      3. unsub-neg 79.9%

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

      4. unpow2 79.9%

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

      5. times-frac 85.4%

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

   4. Simplified 85.4%

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

      1. associate-*r/ 86.9%

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

      2. sub-div 86.9%

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

   6. Applied egg-rr 86.9%

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

   if 1.3999999999999999e-23 < y.re < 2.05e24

   1. Initial program 99.8%

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

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

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

      1. mul-1-neg 80.8%

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

      2. unsub-neg 80.8%

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

      3. unpow2 80.8%

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

      4. times-frac 80.8%

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

   4. Simplified 80.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.96:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 7: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.45 \cdot 10^{-53} \lor \neg \left(y.re \leq 1.25 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.45e-53) (not (<= y.re 1.25e-24)))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (- (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.45e-53) || !(y_46_re <= 1.25e-24)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = -(x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.45d-53)) .or. (.not. (y_46re <= 1.25d-24))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = -(x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.45e-53) || !(y_46_re <= 1.25e-24)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = -(x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.45e-53) or not (y_46_re <= 1.25e-24):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = -(x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.45e-53) || !(y_46_re <= 1.25e-24))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = Float64(-Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.45e-53) || ~((y_46_re <= 1.25e-24)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = -(x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.45e-53], N[Not[LessEqual[y$46$re, 1.25e-24]], $MachinePrecision]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], (-N[(x$46$re / y$46$im), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.45000000000000019e-53 or 1.24999999999999995e-24 < y.re

   1. Initial program 60.4%

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

   2. Taylor expanded in y.re around inf 71.0%

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

      1. mul-1-neg 71.0%

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

      2. unsub-neg 71.0%

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

      3. unpow2 71.0%

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

      4. times-frac 76.7%

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

   4. Simplified 76.7%

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

      1. associate-*r/ 76.7%

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

      2. sub-div 76.7%

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

   6. Applied egg-rr 76.7%

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

   if -3.45000000000000019e-53 < y.re < 1.24999999999999995e-24

   1. Initial program 69.6%

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

   2. Taylor expanded in y.re around 0 73.7%

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

      1. associate-*r/ 73.7%

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

      2. neg-mul-1 73.7%

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

   4. Simplified 73.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.45 \cdot 10^{-53} \lor \neg \left(y.re \leq 1.25 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]

Alternative 8: 63.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2000000:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.2e+33)
   (/ x.im y.re)
   (if (<= y.re 2000000.0) (- (/ 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_re <= -3.2e+33) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2000000.0) {
		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_46re <= (-3.2d+33)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 2000000.0d0) 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_re <= -3.2e+33) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2000000.0) {
		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_re <= -3.2e+33:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 2000000.0:
		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_re <= -3.2e+33)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 2000000.0)
		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_re <= -3.2e+33)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 2000000.0)
		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[LessEqual[y$46$re, -3.2e+33], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2000000.0], (-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.re < -3.20000000000000017e33 or 2e6 < y.re

   1. Initial program 55.9%

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

   2. Taylor expanded in y.re around inf 70.0%

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

   if -3.20000000000000017e33 < y.re < 2e6

   1. Initial program 72.5%

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

   2. Taylor expanded in y.re around 0 68.9%

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

      1. associate-*r/ 68.9%

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

      2. neg-mul-1 68.9%

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

   4. Simplified 68.9%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2000000:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

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 64.5%

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

2. Taylor expanded in y.re around inf 42.4%

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

3. Final simplification 42.4%

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

Reproduce

herbie shell --seed 2023274 
(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))))