Average Error: 26.4 → 25.1
Time: 17.2s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} = -\infty:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 1.40075589692360536 \cdot 10^{294}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} = -\infty:\\
\;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 1.40075589692360536 \cdot 10^{294}:\\
\;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r101246 = b;
        double r101247 = c;
        double r101248 = r101246 * r101247;
        double r101249 = a;
        double r101250 = d;
        double r101251 = r101249 * r101250;
        double r101252 = r101248 - r101251;
        double r101253 = r101247 * r101247;
        double r101254 = r101250 * r101250;
        double r101255 = r101253 + r101254;
        double r101256 = r101252 / r101255;
        return r101256;
}
↓
double f(double a, double b, double c, double d) {
        double r101257 = b;
        double r101258 = c;
        double r101259 = r101257 * r101258;
        double r101260 = a;
        double r101261 = d;
        double r101262 = r101260 * r101261;
        double r101263 = r101259 - r101262;
        double r101264 = r101258 * r101258;
        double r101265 = r101261 * r101261;
        double r101266 = r101264 + r101265;
        double r101267 = r101263 / r101266;
        double r101268 = -inf.0;
        bool r101269 = r101267 <= r101268;
        double r101270 = sqrt(r101266);
        double r101271 = r101257 / r101270;
        double r101272 = 1.4007558969236054e+294;
        bool r101273 = r101267 <= r101272;
        double r101274 = r101263 / r101270;
        double r101275 = r101274 / r101270;
        double r101276 = -r101260;
        double r101277 = r101276 / r101270;
        double r101278 = r101273 ? r101275 : r101277;
        double r101279 = r101269 ? r101271 : r101278;
        return r101279;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original26.4
Target0.5
Herbie25.1
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < -inf.0

    1. Initial program 64.0

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt64.0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*64.0

      \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    5. Taylor expanded around inf 54.2

      \[\leadsto \frac{\color{blue}{b}}{\sqrt{c \cdot c + d \cdot d}}\]

    if -inf.0 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 1.4007558969236054e+294

    1. Initial program 11.5

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt11.5

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*11.4

      \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    if 1.4007558969236054e+294 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

    1. Initial program 63.4

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt63.4

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

    4. Applied associate-/r*63.4

      \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

    5. Taylor expanded around 0 60.2

      \[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c \cdot c + d \cdot d}}\]

    6. Simplified60.2

      \[\leadsto \frac{\color{blue}{-a}}{\sqrt{c \cdot c + d \cdot d}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} = -\infty:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 1.40075589692360536 \cdot 10^{294}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (a b c d)
  :name "Complex division, imag part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))