Average Error: 26.5 → 16.2
Time: 3.6s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -4.4263537617510411 \cdot 10^{138} \lor \neg \left(c \le 8.23624815355425478 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;c \le -4.4263537617510411 \cdot 10^{138} \lor \neg \left(c \le 8.23624815355425478 \cdot 10^{-31}\right):\\
\;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\

\mathbf{else}:\\
\;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r93441 = b;
        double r93442 = c;
        double r93443 = r93441 * r93442;
        double r93444 = a;
        double r93445 = d;
        double r93446 = r93444 * r93445;
        double r93447 = r93443 - r93446;
        double r93448 = r93442 * r93442;
        double r93449 = r93445 * r93445;
        double r93450 = r93448 + r93449;
        double r93451 = r93447 / r93450;
        return r93451;
}
↓
double f(double a, double b, double c, double d) {
        double r93452 = c;
        double r93453 = -4.426353761751041e+138;
        bool r93454 = r93452 <= r93453;
        double r93455 = 8.236248153554255e-31;
        bool r93456 = r93452 <= r93455;
        double r93457 = !r93456;
        bool r93458 = r93454 || r93457;
        double r93459 = b;
        double r93460 = d;
        double r93461 = 2.0;
        double r93462 = pow(r93460, r93461);
        double r93463 = r93462 / r93452;
        double r93464 = r93463 + r93452;
        double r93465 = r93459 / r93464;
        double r93466 = a;
        double r93467 = r93466 * r93460;
        double r93468 = r93452 * r93452;
        double r93469 = r93460 * r93460;
        double r93470 = r93468 + r93469;
        double r93471 = r93467 / r93470;
        double r93472 = r93465 - r93471;
        double r93473 = pow(r93452, r93461);
        double r93474 = r93473 + r93462;
        double r93475 = r93474 / r93452;
        double r93476 = r93459 / r93475;
        double r93477 = r93470 / r93460;
        double r93478 = r93466 / r93477;
        double r93479 = r93476 - r93478;
        double r93480 = r93458 ? r93472 : r93479;
        return r93480;
}

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

Original26.5
Target0.4
Herbie16.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if c < -4.426353761751041e+138 or 8.236248153554255e-31 < c

    1. Initial program 34.9

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

    3. Applied div-sub34.9

      \[\leadsto \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}}\]
    4. Using strategy rm

    5. Applied associate-/l*32.2

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

    6. Simplified32.2

      \[\leadsto \frac{b}{\color{blue}{\frac{{c}^{2} + {d}^{2}}{c}}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\]

    7. Taylor expanded around 0 15.5

      \[\leadsto \frac{b}{\color{blue}{\frac{{d}^{2}}{c} + c}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\]

    if -4.426353761751041e+138 < c < 8.236248153554255e-31

    1. Initial program 19.9

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

    3. Applied div-sub19.9

      \[\leadsto \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}}\]
    4. Using strategy rm

    5. Applied associate-/l*19.3

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

    6. Simplified19.3

      \[\leadsto \frac{b}{\color{blue}{\frac{{c}^{2} + {d}^{2}}{c}}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\]
    7. Using strategy rm

    8. Applied associate-/l*16.7

      \[\leadsto \frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \color{blue}{\frac{a}{\frac{c \cdot c + d \cdot d}{d}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -4.4263537617510411 \cdot 10^{138} \lor \neg \left(c \le 8.23624815355425478 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :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))))