Average Error: 25.4 → 15.0
Time: 11.6s
Precision: binary64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.9125655249550264 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.116085593577531 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 5.31062868416609 \cdot 10^{-58}:\\ \;\;\;\;\frac{y.re \cdot x.im}{{y.im}^{2}} + \frac{x.re}{y.im} \cdot \left(\frac{{y.re}^{2}}{{y.im}^{2}} + -1\right)\\ \mathbf{elif}\;y.re \leq 2.432657971452133 \cdot 10^{+89}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\frac{{y.re}^{2}}{y.im}}\\ \end{array}\]
\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 -1.9125655249550264 \cdot 10^{+124}:\\
\;\;\;\;\frac{x.im}{y.re}\\

\mathbf{elif}\;y.re \leq -4.116085593577531 \cdot 10^{-93}:\\
\;\;\;\;\frac{\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}}\\

\mathbf{elif}\;y.re \leq 5.31062868416609 \cdot 10^{-58}:\\
\;\;\;\;\frac{y.re \cdot x.im}{{y.im}^{2}} + \frac{x.re}{y.im} \cdot \left(\frac{{y.re}^{2}}{{y.im}^{2}} + -1\right)\\

\mathbf{elif}\;y.re \leq 2.432657971452133 \cdot 10^{+89}:\\
\;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\frac{{y.re}^{2}}{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 -1.9125655249550264e+124)
   (/ x.im y.re)
   (if (<= y.re -4.116085593577531e-93)
     (/
      (/
       (- (* y.re x.im) (* y.im x.re))
       (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
      (sqrt (+ (* y.re y.re) (* y.im y.im))))
     (if (<= y.re 5.31062868416609e-58)
       (+
        (/ (* y.re x.im) (pow y.im 2.0))
        (* (/ x.re y.im) (+ (/ (pow y.re 2.0) (pow y.im 2.0)) -1.0)))
       (if (<= y.re 2.432657971452133e+89)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         (- (/ x.im y.re) (/ x.re (/ (pow y.re 2.0) 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 <= -1.9125655249550264e+124) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -4.116085593577531e-93) {
		tmp = (((y_46_re * x_46_im) - (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));
	} else if (y_46_re <= 5.31062868416609e-58) {
		tmp = ((y_46_re * x_46_im) / pow(y_46_im, 2.0)) + ((x_46_re / y_46_im) * ((pow(y_46_re, 2.0) / pow(y_46_im, 2.0)) + -1.0));
	} else if (y_46_re <= 2.432657971452133e+89) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (pow(y_46_re, 2.0) / 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 5 regimes

  2. if y.re < -1.9125655249550264e124

    1. Initial program 40.5

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

    2. Taylor expanded around inf 15.9

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

    if -1.9125655249550264e124 < y.re < -4.11608559357753139e-93

    1. Initial program 16.3

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

    3. Applied add-sqr-sqrt_binary64_214616.3

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

    4. Applied associate-/r*_binary64_206816.2

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

    5. Simplified16.2

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

    if -4.11608559357753139e-93 < y.re < 5.3106286841660903e-58

    1. Initial program 20.7

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

    2. Taylor expanded around 0 16.4

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

    3. Simplified13.1

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

    if 5.3106286841660903e-58 < y.re < 2.4326579714521331e89

    1. Initial program 15.1

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

    if 2.4326579714521331e89 < y.re

    1. Initial program 37.8

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

    3. Applied add-sqr-sqrt_binary64_214637.8

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

    4. Applied associate-/r*_binary64_206837.8

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

    5. Simplified37.8

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

    6. Taylor expanded around inf 17.4

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

    7. Simplified16.7

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{x.re}{\frac{{y.re}^{2}}{y.im}}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.9125655249550264 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.116085593577531 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 5.31062868416609 \cdot 10^{-58}:\\ \;\;\;\;\frac{y.re \cdot x.im}{{y.im}^{2}} + \frac{x.re}{y.im} \cdot \left(\frac{{y.re}^{2}}{{y.im}^{2}} + -1\right)\\ \mathbf{elif}\;y.re \leq 2.432657971452133 \cdot 10^{+89}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\frac{{y.re}^{2}}{y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021060 
(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))))