Average Error: 26.2 → 23.7
Time: 3.1s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -4.002330886974353 \cdot 10^{-17} \lor \neg \left(c \le 3.4366687886863465 \cdot 10^{63}\right):\\ \;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\begin{array}{l}
\mathbf{if}\;c \le -4.002330886974353 \cdot 10^{-17} \lor \neg \left(c \le 3.4366687886863465 \cdot 10^{63}\right):\\
\;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\

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

\end{array}
double f(double a, double b, double c, double d) {
        double r119374 = b;
        double r119375 = c;
        double r119376 = r119374 * r119375;
        double r119377 = a;
        double r119378 = d;
        double r119379 = r119377 * r119378;
        double r119380 = r119376 - r119379;
        double r119381 = r119375 * r119375;
        double r119382 = r119378 * r119378;
        double r119383 = r119381 + r119382;
        double r119384 = r119380 / r119383;
        return r119384;
}
↓
double f(double a, double b, double c, double d) {
        double r119385 = c;
        double r119386 = -4.002330886974353e-17;
        bool r119387 = r119385 <= r119386;
        double r119388 = 3.4366687886863465e+63;
        bool r119389 = r119385 <= r119388;
        double r119390 = !r119389;
        bool r119391 = r119387 || r119390;
        double r119392 = b;
        double r119393 = 2.0;
        double r119394 = pow(r119385, r119393);
        double r119395 = d;
        double r119396 = pow(r119395, r119393);
        double r119397 = r119394 + r119396;
        double r119398 = r119397 / r119385;
        double r119399 = r119392 / r119398;
        double r119400 = a;
        double r119401 = r119400 * r119395;
        double r119402 = r119385 * r119385;
        double r119403 = r119395 * r119395;
        double r119404 = r119402 + r119403;
        double r119405 = r119401 / r119404;
        double r119406 = r119399 - r119405;
        double r119407 = r119392 * r119385;
        double r119408 = r119407 / r119404;
        double r119409 = sqrt(r119404);
        double r119410 = r119400 / r119409;
        double r119411 = r119395 / r119409;
        double r119412 = r119410 * r119411;
        double r119413 = r119408 - r119412;
        double r119414 = r119391 ? r119406 : r119413;
        return r119414;
}

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

Original26.2
Target0.4
Herbie23.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if c < -4.002330886974353e-17 or 3.4366687886863465e+63 < c

    1. Initial program 33.9

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

    3. Applied div-sub33.9

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

    5. Applied associate-/l*30.5

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

    6. Simplified30.5

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

    if -4.002330886974353e-17 < c < 3.4366687886863465e+63

    1. Initial program 19.0

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

    3. Applied div-sub19.0

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

    5. Applied add-sqr-sqrt19.0

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

    6. Applied times-frac17.2

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

  4. Final simplification23.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -4.002330886974353 \cdot 10^{-17} \lor \neg \left(c \le 3.4366687886863465 \cdot 10^{63}\right):\\ \;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))))