Average Error: 26.3 → 18.0
Time: 4.3s
Precision: binary64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.462055629753792 \cdot 10^{+124} \lor \neg \left(y.im \leq 3.54338862761989 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.im \leq -7.462055629753792 \cdot 10^{+124} \lor \neg \left(y.im \leq 3.54338862761989 \cdot 10^{+69}\right):\\
\;\;\;\;\frac{-x.re}{y.im}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\end{array}
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
↓
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.462055629753792e+124)
         (not (<= y.im 3.54338862761989e+69)))
   (/ (- x.re) y.im)
   (/
    (/
     (- (* x.im y.re) (* y.im x.re))
     (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
    (sqrt (+ (* y.re y.re) (* y.im y.im))))))
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}
↓
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.462055629753792e+124) || !(y_46_im <= 3.54338862761989e+69)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))) / sqrt((y_46_re * y_46_re) + (y_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 2 regimes

  2. if y.im < -7.4620556297537919e124 or 3.5433886276198903e69 < y.im

    1. Initial program 40.1

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt_binary64_112340.1

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

    4. Applied associate-/r*_binary64_104540.1

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

    5. Simplified40.1

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

    6. Taylor expanded around 0 41.0

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

    7. Simplified41.0

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

    8. Taylor expanded around 0 17.2

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

    if -7.4620556297537919e124 < y.im < 3.5433886276198903e69

    1. Initial program 18.6

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt_binary64_112318.6

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

    4. Applied associate-/r*_binary64_104518.5

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

    5. Simplified18.5

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.462055629753792 \cdot 10^{+124} \lor \neg \left(y.im \leq 3.54338862761989 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020356 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))