Average Error: 26.2 → 24.7
Time: 11.4s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\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}}
double f(double a, double b, double c, double d) {
        double r81490 = b;
        double r81491 = c;
        double r81492 = r81490 * r81491;
        double r81493 = a;
        double r81494 = d;
        double r81495 = r81493 * r81494;
        double r81496 = r81492 - r81495;
        double r81497 = r81491 * r81491;
        double r81498 = r81494 * r81494;
        double r81499 = r81497 + r81498;
        double r81500 = r81496 / r81499;
        return r81500;
}
↓
double f(double a, double b, double c, double d) {
        double r81501 = b;
        double r81502 = c;
        double r81503 = r81501 * r81502;
        double r81504 = r81502 * r81502;
        double r81505 = d;
        double r81506 = r81505 * r81505;
        double r81507 = r81504 + r81506;
        double r81508 = r81503 / r81507;
        double r81509 = a;
        double r81510 = sqrt(r81507);
        double r81511 = r81509 / r81510;
        double r81512 = r81505 / r81510;
        double r81513 = r81511 * r81512;
        double r81514 = r81508 - r81513;
        return r81514;
}

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
Target0.5
Herbie24.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.2

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

  3. Applied div-sub26.2

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

  5. Applied add-sqr-sqrt26.2

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

  6. Applied times-frac24.7

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

  7. Final simplification24.7

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

Reproduce

herbie shell --seed 2019325 
(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))))