Average Error: 26.0 → 12.8
Time: 9.1s
Precision: binary64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.3717105610359733 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4138995451185585 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} + \frac{y.re}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.re}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}\\ \mathbf{elif}\;y.im \leq 3.7752606460619477 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 3.5175210954414894 \cdot 10^{+153}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} + \frac{y.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array}\]
\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.im \leq -1.3717105610359733 \cdot 10^{+154}:\\
\;\;\;\;\frac{x.im}{y.im}\\

\mathbf{elif}\;y.im \leq -1.4138995451185585 \cdot 10^{-126}:\\
\;\;\;\;\frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} + \frac{y.re}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.re}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}\\

\mathbf{elif}\;y.im \leq 3.7752606460619477 \cdot 10^{-66}:\\
\;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\

\mathbf{elif}\;y.im \leq 3.5175210954414894 \cdot 10^{+153}:\\
\;\;\;\;\frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} + \frac{y.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{y.im}\\

\end{array}
(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))))
↓
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.3717105610359733e+154)
   (/ x.im y.im)
   (if (<= y.im -1.4138995451185585e-126)
     (+
      (*
       (/ y.im (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
       (/ x.im (sqrt (+ (pow y.re 2.0) (pow y.im 2.0)))))
      (*
       (/
        y.re
        (*
         (cbrt (+ (pow y.re 2.0) (pow y.im 2.0)))
         (cbrt (+ (pow y.re 2.0) (pow y.im 2.0)))))
       (/ x.re (cbrt (+ (pow y.re 2.0) (pow y.im 2.0))))))
     (if (<= y.im 3.7752606460619477e-66)
       (+ (/ x.re y.re) (/ (* y.im x.im) (pow y.re 2.0)))
       (if (<= y.im 3.5175210954414894e+153)
         (+
          (*
           (/ y.im (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
           (/ x.im (sqrt (+ (pow y.re 2.0) (pow y.im 2.0)))))
          (*
           (/ y.re (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
           (/ x.re (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))))
         (/ x.im y.im))))))
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));
}
↓
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.3717105610359733e+154) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.4138995451185585e-126) {
		tmp = ((y_46_im / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))) * (x_46_im / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0)))) + ((y_46_re / (cbrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0)) * cbrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0)))) * (x_46_re / cbrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))));
	} else if (y_46_im <= 3.7752606460619477e-66) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / pow(y_46_re, 2.0));
	} else if (y_46_im <= 3.5175210954414894e+153) {
		tmp = ((y_46_im / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))) * (x_46_im / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0)))) + ((y_46_re / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))) * (x_46_re / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if y.im < -1.3717105610359733e154 or 3.51752109544148941e153 < y.im

    1. Initial program 44.5

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

    2. Taylor expanded around 0 13.7

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

    if -1.3717105610359733e154 < y.im < -1.4138995451185585e-126

    1. Initial program 17.9

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

    2. Taylor expanded around 0 17.9

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

    3. Simplified17.9

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

    5. Applied add-sqr-sqrt_binary64_112317.9

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

    6. Applied times-frac_binary64_110714.6

      \[\leadsto \color{blue}{\frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}} + \frac{y.re \cdot x.re}{{y.re}^{2} + {y.im}^{2}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt_binary64_113614.9

      \[\leadsto \frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} + \frac{y.re \cdot x.re}{\color{blue}{\left(\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}\right) \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}}}\]

    9. Applied times-frac_binary64_110713.1

      \[\leadsto \frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} + \color{blue}{\frac{y.re}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.re}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}}\]

    if -1.4138995451185585e-126 < y.im < 3.77526064606194773e-66

    1. Initial program 20.5

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

    2. Taylor expanded around inf 11.6

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

    if 3.77526064606194773e-66 < y.im < 3.51752109544148941e153

    1. Initial program 18.8

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

    2. Taylor expanded around 0 18.8

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

    3. Simplified18.8

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

    5. Applied add-sqr-sqrt_binary64_112318.8

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

    6. Applied times-frac_binary64_110714.6

      \[\leadsto \color{blue}{\frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}} + \frac{y.re \cdot x.re}{{y.re}^{2} + {y.im}^{2}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt_binary64_112314.6

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

    9. Applied times-frac_binary64_110712.9

      \[\leadsto \frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} + \color{blue}{\frac{y.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.3717105610359733 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4138995451185585 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} + \frac{y.re}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.re}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}\\ \mathbf{elif}\;y.im \leq 3.7752606460619477 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 3.5175210954414894 \cdot 10^{+153}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} + \frac{y.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2021060 
(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))))