_divideComplex, real part

Percentage Accurate: 61.8% → 81.1%

Time: 11.1s

Alternatives: 7

Speedup: 1.1×

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: 61.8% 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: 81.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -8.2 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.re (* (/ x.im y.re) y.im)) y.re)))
   (if (<= y.re -1.15e+85)
     t_0
     (if (<= y.re -8.2e-112)
       (* (fma x.re y.re (* x.im y.im)) (/ 1.0 (pow (hypot y.re y.im) 2.0)))
       (if (<= y.re 2.2e-24) (/ (+ x.im (/ (* y.re x.re) y.im)) y.im) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re + ((x_46_im / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.15e+85) {
		tmp = t_0;
	} else if (y_46_re <= -8.2e-112) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) * (1.0 / pow(hypot(y_46_re, y_46_im), 2.0));
	} else if (y_46_re <= 2.2e-24) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re + Float64(Float64(x_46_im / y_46_re) * y_46_im)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.15e+85)
		tmp = t_0;
	elseif (y_46_re <= -8.2e-112)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) * Float64(1.0 / (hypot(y_46_re, y_46_im) ^ 2.0)));
	elseif (y_46_re <= 2.2e-24)
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / y_46_im);
	else
		tmp = t_0;
	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$46$re + N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.15e+85], t$95$0, If[LessEqual[y$46$re, -8.2e-112], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.2e-24], N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.1499999999999999e85 or 2.20000000000000002e-24 < y.re

   1. Initial program 47.9%

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

      1. fma-define 47.9%

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

      2. fma-define 47.9%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in y.re around inf 77.7%

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

      1. *-commutative 77.7%

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

   7. Simplified 77.7%

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

      1. associate-/l* 82.6%

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

      2. *-commutative 82.6%

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

   9. Applied egg-rr 82.6%

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

   if -1.1499999999999999e85 < y.re < -8.19999999999999991e-112

   1. Initial program 81.7%

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

      1. fma-define 81.7%

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

      2. fma-define 81.7%

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

   3. Simplified 81.7%

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

      1. fma-define 81.7%

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

      2. fma-define 81.7%

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

      3. div-inv 81.8%

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

      4. fma-define 81.8%

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

      5. fma-define 81.8%

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

      6. add-sqr-sqrt 81.8%

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

      7. pow2 81.8%

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

      8. fma-define 81.8%

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

      9. hypot-define 81.8%

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

   6. Applied egg-rr 81.8%

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

   if -8.19999999999999991e-112 < y.re < 2.20000000000000002e-24

   1. Initial program 69.6%

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

      1. fma-define 69.6%

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

      2. fma-define 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in y.im around inf 89.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -8.2 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -7.8 \cdot 10^{+82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-112}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.re (* (/ x.im y.re) y.im)) y.re)))
   (if (<= y.re -7.8e+82)
     t_0
     (if (<= y.re -1e-112)
       (/ (+ (* x.im y.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 6.2e-24) (/ (+ x.im (/ (* y.re x.re) y.im)) y.im) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re + ((x_46_im / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -7.8e+82) {
		tmp = t_0;
	} else if (y_46_re <= -1e-112) {
		tmp = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 6.2e-24) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re + ((x_46im / y_46re) * y_46im)) / y_46re
    if (y_46re <= (-7.8d+82)) then
        tmp = t_0
    else if (y_46re <= (-1d-112)) then
        tmp = ((x_46im * y_46im) + (y_46re * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 6.2d-24) then
        tmp = (x_46im + ((y_46re * x_46re) / y_46im)) / y_46im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re + ((x_46_im / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -7.8e+82) {
		tmp = t_0;
	} else if (y_46_re <= -1e-112) {
		tmp = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 6.2e-24) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re + ((x_46_im / y_46_re) * y_46_im)) / y_46_re
	tmp = 0
	if y_46_re <= -7.8e+82:
		tmp = t_0
	elif y_46_re <= -1e-112:
		tmp = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 6.2e-24:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re + Float64(Float64(x_46_im / y_46_re) * y_46_im)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -7.8e+82)
		tmp = t_0;
	elseif (y_46_re <= -1e-112)
		tmp = Float64(Float64(Float64(x_46_im * y_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 6.2e-24)
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re + ((x_46_im / y_46_re) * y_46_im)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -7.8e+82)
		tmp = t_0;
	elseif (y_46_re <= -1e-112)
		tmp = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 6.2e-24)
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re + N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -7.8e+82], t$95$0, If[LessEqual[y$46$re, -1e-112], N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-24], N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.79999999999999951e82 or 6.2000000000000001e-24 < y.re

   1. Initial program 47.9%

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

      1. fma-define 47.9%

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

      2. fma-define 47.9%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in y.re around inf 77.7%

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

      1. *-commutative 77.7%

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

   7. Simplified 77.7%

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

      1. associate-/l* 82.6%

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

      2. *-commutative 82.6%

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

   9. Applied egg-rr 82.6%

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

   if -7.79999999999999951e82 < y.re < -9.9999999999999995e-113

   1. Initial program 81.7%

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

   if -9.9999999999999995e-113 < y.re < 6.2000000000000001e-24

   1. Initial program 69.6%

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

      1. fma-define 69.6%

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

      2. fma-define 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in y.im around inf 89.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-112}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-68} \lor \neg \left(y.re \leq 3.25 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.6e-68) (not (<= y.re 3.25e-24)))
   (/ (+ x.re (* (/ x.im y.re) y.im)) y.re)
   (/ (+ x.im (/ (* y.re x.re) y.im)) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.6e-68) || !(y_46_re <= 3.25e-24)) {
		tmp = (x_46_re + ((x_46_im / y_46_re) * y_46_im)) / y_46_re;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-4.6d-68)) .or. (.not. (y_46re <= 3.25d-24))) then
        tmp = (x_46re + ((x_46im / y_46re) * y_46im)) / y_46re
    else
        tmp = (x_46im + ((y_46re * x_46re) / y_46im)) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.6e-68) || !(y_46_re <= 3.25e-24)) {
		tmp = (x_46_re + ((x_46_im / y_46_re) * y_46_im)) / y_46_re;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -4.6e-68) or not (y_46_re <= 3.25e-24):
		tmp = (x_46_re + ((x_46_im / y_46_re) * y_46_im)) / y_46_re
	else:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -4.6e-68) || !(y_46_re <= 3.25e-24))
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im / y_46_re) * y_46_im)) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -4.6e-68) || ~((y_46_re <= 3.25e-24)))
		tmp = (x_46_re + ((x_46_im / y_46_re) * y_46_im)) / y_46_re;
	else
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -4.6e-68], N[Not[LessEqual[y$46$re, 3.25e-24]], $MachinePrecision]], N[(N[(x$46$re + N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.59999999999999994e-68 or 3.25e-24 < y.re

   1. Initial program 56.0%

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

      1. fma-define 56.0%

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

      2. fma-define 56.0%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in y.re around inf 75.2%

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

      1. *-commutative 75.2%

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

   7. Simplified 75.2%

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

      1. associate-/l* 79.0%

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

      2. *-commutative 79.0%

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

   9. Applied egg-rr 79.0%

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

   if -4.59999999999999994e-68 < y.re < 3.25e-24

   1. Initial program 70.4%

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

      1. fma-define 70.4%

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

      2. fma-define 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in y.im around inf 86.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-68} \lor \neg \left(y.re \leq 3.25 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.02 \cdot 10^{-60} \lor \neg \left(y.re \leq 8 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.02e-60) (not (<= y.re 8e-24)))
   (/ x.re y.re)
   (/ (+ x.im (/ (* y.re x.re) y.im)) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.02e-60) || !(y_46_re <= 8e-24)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.02d-60)) .or. (.not. (y_46re <= 8d-24))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im + ((y_46re * x_46re) / y_46im)) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.02e-60) || !(y_46_re <= 8e-24)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.02e-60) or not (y_46_re <= 8e-24):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.02e-60) || !(y_46_re <= 8e-24))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.02e-60) || ~((y_46_re <= 8e-24)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.02e-60], N[Not[LessEqual[y$46$re, 8e-24]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.01999999999999994e-60 or 7.99999999999999939e-24 < y.re

   1. Initial program 55.4%

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

      1. fma-define 55.4%

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

      2. fma-define 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in y.re around inf 67.1%

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

   if -1.01999999999999994e-60 < y.re < 7.99999999999999939e-24

   1. Initial program 70.9%

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

      1. fma-define 70.9%

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

      2. fma-define 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in y.im around inf 86.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.02 \cdot 10^{-60} \lor \neg \left(y.re \leq 8 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{-62} \lor \neg \left(y.re \leq 10^{-23}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.9e-62) (not (<= y.re 1e-23)))
   (/ x.re y.re)
   (/ (+ x.im (* x.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) {
	double tmp;
	if ((y_46_re <= -2.9e-62) || !(y_46_re <= 1e-23)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.9d-62)) .or. (.not. (y_46re <= 1d-23))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.9e-62) || !(y_46_re <= 1e-23)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.9e-62) or not (y_46_re <= 1e-23):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2.9e-62) || !(y_46_re <= 1e-23))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.9e-62) || ~((y_46_re <= 1e-23)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.9e-62], N[Not[LessEqual[y$46$re, 1e-23]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.89999999999999986e-62 or 9.9999999999999996e-24 < y.re

   1. Initial program 55.4%

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

      1. fma-define 55.4%

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

      2. fma-define 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in y.re around inf 67.1%

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

   if -2.89999999999999986e-62 < y.re < 9.9999999999999996e-24

   1. Initial program 70.9%

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

      1. fma-define 70.9%

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

      2. fma-define 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in y.im around inf 86.0%

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

      1. associate-/l* 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{-62} \lor \neg \left(y.re \leq 10^{-23}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{-67} \lor \neg \left(y.re \leq 10^{-55}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.4e-67) (not (<= y.re 1e-55)))
   (/ x.re y.re)
   (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.4e-67) || !(y_46_re <= 1e-55)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-4.4d-67)) .or. (.not. (y_46re <= 1d-55))) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.4e-67) || !(y_46_re <= 1e-55)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -4.4e-67) or not (y_46_re <= 1e-55):
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -4.4e-67) || !(y_46_re <= 1e-55))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -4.4e-67) || ~((y_46_re <= 1e-55)))
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -4.4e-67], N[Not[LessEqual[y$46$re, 1e-55]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.4000000000000002e-67 or 9.99999999999999995e-56 < y.re

   1. Initial program 56.8%

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

      1. fma-define 56.8%

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

      2. fma-define 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in y.re around inf 65.9%

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

   if -4.4000000000000002e-67 < y.re < 9.99999999999999995e-56

   1. Initial program 70.3%

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

      1. fma-define 70.3%

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

      2. fma-define 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in y.re around 0 69.5%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{-67} \lor \neg \left(y.re \leq 10^{-55}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

   1. fma-define 62.7%

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

   2. fma-define 62.7%

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

3. Simplified 62.7%

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

5. Taylor expanded in y.re around 0 42.3%

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

Reproduce

herbie shell --seed 2024144 
(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))))