_divideComplex, real part

Percentage Accurate: 63.2% → 85.2%

Time: 8.3s

Alternatives: 9

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_0 (- INFINITY))
     (fma (/ y.re y.im) (/ x.re y.im) (/ x.im y.im))
     (if (<= t_0 5e+267)
       (/
        (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im))
        (hypot y.re y.im))
       (/ (* x.im (/ y.im (hypot y.re y.im))) (hypot y.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((y_46_re / y_46_im), (x_46_re / y_46_im), (x_46_im / y_46_im));
	} else if (t_0 <= 5e+267) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im * (y_46_im / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(y_46_re / y_46_im), Float64(x_46_re / y_46_im), Float64(x_46_im / y_46_im));
	elseif (t_0 <= 5e+267)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_im * Float64(y_46_im / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+267], N[(N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 61.9%

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

   2. Taylor expanded in y.re around 0 60.5%

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

      1. *-commutative 60.5%

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

      2. unpow2 60.5%

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

      3. times-frac 80.3%

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

      4. fma-def 85.3%

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

   4. Simplified 85.3%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 4.9999999999999999e267

   1. Initial program 76.7%

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

      1. *-un-lft-identity 76.7%

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

      2. add-sqr-sqrt 76.7%

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

      3. times-frac 76.7%

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

      4. hypot-def 76.7%

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

      5. fma-def 76.7%

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

      6. hypot-def 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.1%

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

      2. *-un-lft-identity 99.1%

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

   5. Applied egg-rr 99.1%

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

   if 4.9999999999999999e267 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 7.3%

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

   2. Taylor expanded in x.re around 0 6.7%

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

      1. add-sqr-sqrt 6.7%

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

      2. hypot-udef 6.7%

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

      3. hypot-udef 6.7%

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

      4. times-frac 63.1%

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

   4. Applied egg-rr 63.1%

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

      1. associate-*r/ 63.2%

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

   6. Applied egg-rr 63.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \left(\frac{x.re}{y.im} \cdot \frac{1}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -7.8e-21)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im 1.15e+98)
     (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
     (+ (/ x.im y.im) (* y.re (* (/ x.re y.im) (/ 1.0 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 <= -7.8e-21) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.15e+98) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) * (1.0 / 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 <= (-7.8d-21)) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else if (y_46im <= 1.15d+98) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else
        tmp = (x_46im / y_46im) + (y_46re * ((x_46re / y_46im) * (1.0d0 / 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 <= -7.8e-21) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.15e+98) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) * (1.0 / 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 <= -7.8e-21:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= 1.15e+98:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	else:
		tmp = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) * (1.0 / 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 <= -7.8e-21)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= 1.15e+98)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(Float64(x_46_re / y_46_im) * Float64(1.0 / 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 <= -7.8e-21)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 1.15e+98)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	else
		tmp = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) * (1.0 / 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, -7.8e-21], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.15e+98], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.8000000000000001e-21

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. *-commutative 83.0%

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

      3. unpow2 83.0%

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

      4. times-frac 89.6%

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

   4. Simplified 89.6%

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

      1. associate-*l/ 89.7%

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

   6. Applied egg-rr 89.7%

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

   if -7.8000000000000001e-21 < y.im < 1.15000000000000007e98

   1. Initial program 72.0%

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

      1. *-un-lft-identity 72.0%

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

      2. add-sqr-sqrt 72.0%

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

      3. times-frac 72.0%

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

      4. hypot-def 72.0%

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

      5. fma-def 72.0%

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

      6. hypot-def 82.7%

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

   3. Applied egg-rr 82.7%

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

   4. Taylor expanded in y.re around inf 78.3%

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

      1. unpow2 78.3%

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

      2. times-frac 79.7%

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

   6. Simplified 79.7%

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

   if 1.15000000000000007e98 < y.im

   1. Initial program 36.6%

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

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

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

      1. +-commutative 72.6%

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

      2. *-commutative 72.6%

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

      3. unpow2 72.6%

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

   4. Simplified 72.6%

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

      1. times-frac 80.1%

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

      2. div-inv 80.0%

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

      3. associate-*l* 81.9%

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

   6. Applied egg-rr 81.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \left(\frac{x.re}{y.im} \cdot \frac{1}{y.im}\right)\\ \end{array} \]

Alternative 3: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.1 \cdot 10^{-18} \lor \neg \left(y.im \leq 3 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -6.1e-18) (not (<= y.im 3e-46)))
   (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.1e-18) || !(y_46_im <= 3e-46)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-6.1d-18)) .or. (.not. (y_46im <= 3d-46))) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.1e-18) || !(y_46_im <= 3e-46)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -6.1e-18) or not (y_46_im <= 3e-46):
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -6.1e-18) || !(y_46_im <= 3e-46))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -6.1e-18) || ~((y_46_im <= 3e-46)))
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -6.1e-18], N[Not[LessEqual[y$46$im, 3e-46]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.0999999999999999e-18 or 2.99999999999999987e-46 < y.im

   1. Initial program 49.6%

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

   2. Taylor expanded in y.re around 0 72.5%

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

      1. +-commutative 72.5%

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

      2. *-commutative 72.5%

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

      3. unpow2 72.5%

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

      4. times-frac 78.1%

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

   4. Simplified 78.1%

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

   if -6.0999999999999999e-18 < y.im < 2.99999999999999987e-46

   1. Initial program 75.4%

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

   2. Taylor expanded in y.re around inf 72.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.1 \cdot 10^{-18} \lor \neg \left(y.im \leq 3 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 4: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9.4e-14)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im 5.4e-42)
     (/ x.re y.re)
     (+ (/ x.im y.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_im <= -9.4e-14) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 5.4e-42) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_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_46im <= (-9.4d-14)) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else if (y_46im <= 5.4d-42) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_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_im <= -9.4e-14) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 5.4e-42) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_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_im <= -9.4e-14:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= 5.4e-42:
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_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_im <= -9.4e-14)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= 5.4e-42)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / 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)
	tmp = 0.0;
	if (y_46_im <= -9.4e-14)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 5.4e-42)
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_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[LessEqual[y$46$im, -9.4e-14], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.4e-42], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9.4000000000000003e-14

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. *-commutative 83.0%

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

      3. unpow2 83.0%

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

      4. times-frac 89.6%

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

   4. Simplified 89.6%

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

      1. associate-*l/ 89.7%

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

   6. Applied egg-rr 89.7%

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

   if -9.4000000000000003e-14 < y.im < 5.39999999999999998e-42

   1. Initial program 75.4%

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

   2. Taylor expanded in y.re around inf 72.7%

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

   if 5.39999999999999998e-42 < y.im

   1. Initial program 46.2%

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

   2. Taylor expanded in y.re around 0 63.5%

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

      1. +-commutative 63.5%

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

      2. *-commutative 63.5%

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

      3. unpow2 63.5%

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

      4. times-frac 68.1%

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

   4. Simplified 68.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.3e-12)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im 3.8e-45)
     (/ x.re y.re)
     (+ (/ x.im y.im) (/ (/ x.re y.im) (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.3e-12) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 3.8e-45) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-3.3d-12)) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else if (y_46im <= 3.8d-45) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((x_46re / y_46im) / (y_46im / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.3e-12) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 3.8e-45) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -3.3e-12:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= 3.8e-45:
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.3e-12)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= 3.8e-45)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) / Float64(y_46_im / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -3.3e-12)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 3.8e-45)
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -3.3e-12], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.8e-45], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.3000000000000001e-12

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. *-commutative 83.0%

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

      3. unpow2 83.0%

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

      4. times-frac 89.6%

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

   4. Simplified 89.6%

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

      1. associate-*l/ 89.7%

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

   6. Applied egg-rr 89.7%

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

   if -3.3000000000000001e-12 < y.im < 3.79999999999999997e-45

   1. Initial program 75.4%

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

   2. Taylor expanded in y.re around inf 72.7%

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

   if 3.79999999999999997e-45 < y.im

   1. Initial program 46.2%

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

   2. Taylor expanded in y.re around 0 63.5%

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

      1. +-commutative 63.5%

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

      2. *-commutative 63.5%

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

      3. unpow2 63.5%

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

      4. times-frac 68.1%

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

   4. Simplified 68.1%

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

      1. *-commutative 68.1%

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

      2. clear-num 68.0%

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

      3. un-div-inv 68.1%

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

   6. Applied egg-rr 68.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \end{array} \]

Alternative 6: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.3e-18)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im 2.4e-40)
     (/ x.re y.re)
     (+ (/ x.im y.im) (/ (/ y.re y.im) (/ y.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.3e-18) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 2.4e-40) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) / (y_46_im / x_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.3d-18)) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else if (y_46im <= 2.4d-40) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) / (y_46im / x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.3e-18) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 2.4e-40) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) / (y_46_im / x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.3e-18:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= 2.4e-40:
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) / (y_46_im / x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.3e-18)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= 2.4e-40)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.3e-18)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 2.4e-40)
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) / (y_46_im / x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.3e-18], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.4e-40], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.3e-18

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. *-commutative 83.0%

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

      3. unpow2 83.0%

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

      4. times-frac 89.6%

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

   4. Simplified 89.6%

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

      1. associate-*l/ 89.7%

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

   6. Applied egg-rr 89.7%

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

   if -1.3e-18 < y.im < 2.39999999999999991e-40

   1. Initial program 75.4%

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

   2. Taylor expanded in y.re around inf 72.7%

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

   if 2.39999999999999991e-40 < y.im

   1. Initial program 46.2%

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

   2. Taylor expanded in y.re around 0 63.5%

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

      1. +-commutative 63.5%

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

      2. *-commutative 63.5%

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

      3. unpow2 63.5%

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

   4. Simplified 63.5%

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

   5. Taylor expanded in y.re around 0 63.5%

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

      1. unpow2 63.5%

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

      2. times-frac 68.1%

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

      3. *-commutative 68.1%

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

      4. associate-*r/ 68.0%

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

      5. associate-/l* 69.2%

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

   7. Simplified 69.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \end{array} \]

Alternative 7: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.3 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.3e-16)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im 1.2e+98)
     (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
     (+ (/ x.im y.im) (/ (/ y.re y.im) (/ y.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.3e-16) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.2e+98) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) / (y_46_im / x_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.3d-16)) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else if (y_46im <= 1.2d+98) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) / (y_46im / x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.3e-16) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.2e+98) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) / (y_46_im / x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.3e-16:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= 1.2e+98:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) / (y_46_im / x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.3e-16)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= 1.2e+98)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.3e-16)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 1.2e+98)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) / (y_46_im / x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.3e-16], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.2e+98], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.2999999999999999e-16

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. *-commutative 83.0%

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

      3. unpow2 83.0%

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

      4. times-frac 89.6%

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

   4. Simplified 89.6%

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

      1. associate-*l/ 89.7%

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

   6. Applied egg-rr 89.7%

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

   if -1.2999999999999999e-16 < y.im < 1.1999999999999999e98

   1. Initial program 72.0%

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

      1. *-un-lft-identity 72.0%

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

      2. add-sqr-sqrt 72.0%

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

      3. times-frac 72.0%

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

      4. hypot-def 72.0%

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

      5. fma-def 72.0%

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

      6. hypot-def 82.7%

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

   3. Applied egg-rr 82.7%

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

   4. Taylor expanded in y.re around inf 78.3%

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

      1. unpow2 78.3%

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

      2. times-frac 79.7%

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

   6. Simplified 79.7%

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

   if 1.1999999999999999e98 < y.im

   1. Initial program 36.6%

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

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

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

      1. +-commutative 72.6%

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

      2. *-commutative 72.6%

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

      3. unpow2 72.6%

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

   4. Simplified 72.6%

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

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

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

      1. unpow2 72.6%

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

      2. times-frac 80.1%

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

      3. *-commutative 80.1%

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

      4. associate-*r/ 79.9%

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

      5. associate-/l* 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.3 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \end{array} \]

Alternative 8: 63.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -8.5e-10)
   (/ x.im y.im)
   (if (<= y.im 1.15e+98) (/ x.re y.re) (/ x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -8.5e-10) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.15e+98) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-8.5d-10)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 1.15d+98) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -8.5e-10) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.15e+98) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -8.5e-10:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 1.15e+98:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -8.5e-10)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1.15e+98)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -8.5e-10)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 1.15e+98)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -8.5e-10], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.15e+98], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.4999999999999996e-10 or 1.15000000000000007e98 < y.im

   1. Initial program 46.5%

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

   2. Taylor expanded in y.re around 0 72.0%

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

   if -8.4999999999999996e-10 < y.im < 1.15000000000000007e98

   1. Initial program 72.0%

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

   2. Taylor expanded in y.re around inf 65.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9: 44.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.6%

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

2. Taylor expanded in y.re around 0 46.6%

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

3. Final simplification 46.6%

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

Reproduce

herbie shell --seed 2023193 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))