Average Error: 26.1 → 25.6
Time: 3.5s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 1.04391992940771602 \cdot 10^{279}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]
\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} \le 1.04391992940771602 \cdot 10^{279}:\\
\;\;\;\;\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{b}{\sqrt{c \cdot c + d \cdot d}}\\

\end{array}
double f(double a, double b, double c, double d) {
        double r96387 = b;
        double r96388 = c;
        double r96389 = r96387 * r96388;
        double r96390 = a;
        double r96391 = d;
        double r96392 = r96390 * r96391;
        double r96393 = r96389 - r96392;
        double r96394 = r96388 * r96388;
        double r96395 = r96391 * r96391;
        double r96396 = r96394 + r96395;
        double r96397 = r96393 / r96396;
        return r96397;
}
↓
double f(double a, double b, double c, double d) {
        double r96398 = b;
        double r96399 = c;
        double r96400 = r96398 * r96399;
        double r96401 = a;
        double r96402 = d;
        double r96403 = r96401 * r96402;
        double r96404 = r96400 - r96403;
        double r96405 = r96399 * r96399;
        double r96406 = r96402 * r96402;
        double r96407 = r96405 + r96406;
        double r96408 = r96404 / r96407;
        double r96409 = 1.043919929407716e+279;
        bool r96410 = r96408 <= r96409;
        double r96411 = sqrt(r96407);
        double r96412 = r96404 / r96411;
        double r96413 = r96412 / r96411;
        double r96414 = r96398 / r96411;
        double r96415 = r96410 ? r96413 : r96414;
        return r96415;
}

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

Original26.1
Target0.5
Herbie25.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 1.043919929407716e+279

    1. Initial program 14.2

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

    3. Applied add-sqr-sqrt14.2

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

    4. Applied associate-/r*14.1

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

    if 1.043919929407716e+279 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

    1. Initial program 61.9

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

    3. Applied add-sqr-sqrt61.9

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

    4. Applied associate-/r*61.9

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

    5. Taylor expanded around inf 60.3

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

  4. Final simplification25.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 1.04391992940771602 \cdot 10^{279}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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))))