Average Error: 26.5 → 26.2
Time: 4.9s
Precision: binary64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.531747356501993 \cdot 10^{+130}:\\ \;\;\;\;\frac{-x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} - \frac{y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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.re \leq -2.531747356501993 \cdot 10^{+130}:\\
\;\;\;\;\frac{-x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y.re \cdot x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} - \frac{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 (<= y.re -2.531747356501993e+130)
   (/ (- x.im) (sqrt (+ (* y.re y.re) (* y.im y.im))))
   (/
    (-
     (/ (* y.re x.im) (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
     (/ (* 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_re <= -2.531747356501993e+130) {
		tmp = -x_46_im / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (((y_46_re * x_46_im) / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))) - ((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.re < -2.53174735650199306e130

    1. Initial program 43.5

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

    3. Applied add-sqr-sqrt_binary64_214643.5

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

    4. Applied associate-/r*_binary64_206843.5

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

    5. Simplified43.5

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

    6. Taylor expanded around -inf 42.0

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

    7. Simplified42.0

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

    if -2.53174735650199306e130 < y.re

    1. Initial program 23.6

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

    3. Applied add-sqr-sqrt_binary64_214623.6

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

    4. Applied associate-/r*_binary64_206823.5

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

    5. Simplified23.5

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

    7. Applied div-sub_binary64_212923.5

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

    8. Simplified23.5

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

    9. Simplified23.5

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

  4. Final simplification26.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.531747356501993 \cdot 10^{+130}:\\ \;\;\;\;\frac{-x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}} - \frac{y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(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))))