Average Error: 26.2 → 14.8
Time: 14.2s
Precision: binary64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.2563987087522574 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -8.232708701844971 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 1.8192329406445362 \cdot 10^{-149}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 2.0139063475555305 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \end{array}\]
\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 -3.2563987087522574 \cdot 10^{+81}:\\
\;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\

\mathbf{elif}\;y.im \leq -8.232708701844971 \cdot 10^{-48}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{elif}\;y.im \leq 1.8192329406445362 \cdot 10^{-149}:\\
\;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}\\

\mathbf{elif}\;y.im \leq 2.0139063475555305 \cdot 10^{+69}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{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.im -3.2563987087522574e+81)
   (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ x.re y.im))
   (if (<= y.im -8.232708701844971e-48)
     (/
      (/
       (- (* x.im y.re) (* y.im x.re))
       (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
      (sqrt (+ (* y.re y.re) (* y.im y.im))))
     (if (<= y.im 1.8192329406445362e-149)
       (- (/ x.im y.re) (/ (* y.im x.re) (pow y.re 2.0)))
       (if (<= y.im 2.0139063475555305e+69)
         (/
          (/
           (- (* x.im y.re) (* y.im x.re))
           (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
          (sqrt (+ (* y.re y.re) (* y.im y.im))))
         (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ x.re 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 <= -3.2563987087522574e+81) {
		tmp = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_im <= -8.232708701844971e-48) {
		tmp = (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.8192329406445362e-149) {
		tmp = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / pow(y_46_re, 2.0));
	} else if (y_46_im <= 2.0139063475555305e+69) {
		tmp = (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / 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 3 regimes

  2. if y.im < -3.256398708752257e81 or 2.01390634755553049e69 < y.im

    1. Initial program 37.9

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

    3. Applied add-sqr-sqrt_binary64_214637.9

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

    4. Applied *-un-lft-identity_binary64_212437.9

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

    5. Applied times-frac_binary64_213037.9

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

    6. Simplified37.9

      \[\leadsto \color{blue}{\frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

    7. Simplified37.9

      \[\leadsto \frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \color{blue}{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}\]

    8. Taylor expanded around 0 17.5

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

    9. Simplified16.9

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

    if -3.256398708752257e81 < y.im < -8.23270870184497108e-48 or 1.8192329406445362e-149 < y.im < 2.01390634755553049e69

    1. Initial program 14.7

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

    3. Applied add-sqr-sqrt_binary64_214614.7

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

    4. Applied associate-/r*_binary64_206814.6

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

    5. Simplified14.6

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

    if -8.23270870184497108e-48 < y.im < 1.8192329406445362e-149

    1. Initial program 21.8

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

    2. Taylor expanded around inf 12.3

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.2563987087522574 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -8.232708701844971 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 1.8192329406445362 \cdot 10^{-149}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 2.0139063475555305 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))