Average Error: 26.3 → 23.2
Time: 3.1s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{c}{\sqrt{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}}
double f(double a, double b, double c, double d) {
        double r100469 = b;
        double r100470 = c;
        double r100471 = r100469 * r100470;
        double r100472 = a;
        double r100473 = d;
        double r100474 = r100472 * r100473;
        double r100475 = r100471 - r100474;
        double r100476 = r100470 * r100470;
        double r100477 = r100473 * r100473;
        double r100478 = r100476 + r100477;
        double r100479 = r100475 / r100478;
        return r100479;
}
↓
double f(double a, double b, double c, double d) {
        double r100480 = b;
        double r100481 = c;
        double r100482 = r100481 * r100481;
        double r100483 = d;
        double r100484 = r100483 * r100483;
        double r100485 = r100482 + r100484;
        double r100486 = sqrt(r100485);
        double r100487 = r100480 / r100486;
        double r100488 = r100481 / r100486;
        double r100489 = r100487 * r100488;
        double r100490 = a;
        double r100491 = r100490 / r100486;
        double r100492 = r100483 / r100486;
        double r100493 = r100491 * r100492;
        double r100494 = r100489 - r100493;
        return r100494;
}

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

Original26.3
Target0.4
Herbie23.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.3

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

  3. Applied div-sub26.3

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

  5. Applied add-sqr-sqrt26.3

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

  6. Applied times-frac24.9

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

  8. Applied add-sqr-sqrt24.9

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

  9. Applied times-frac23.2

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

  10. Final simplification23.2

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

Reproduce

herbie shell --seed 2020039 
(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))))