Average Error: 26.1 → 26.1
Time: 10.2s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
double f(double a, double b, double c, double d) {
        double r74969 = b;
        double r74970 = c;
        double r74971 = r74969 * r74970;
        double r74972 = a;
        double r74973 = d;
        double r74974 = r74972 * r74973;
        double r74975 = r74971 - r74974;
        double r74976 = r74970 * r74970;
        double r74977 = r74973 * r74973;
        double r74978 = r74976 + r74977;
        double r74979 = r74975 / r74978;
        return r74979;
}
↓
double f(double a, double b, double c, double d) {
        double r74980 = b;
        double r74981 = c;
        double r74982 = r74980 * r74981;
        double r74983 = a;
        double r74984 = d;
        double r74985 = r74983 * r74984;
        double r74986 = r74982 - r74985;
        double r74987 = r74981 * r74981;
        double r74988 = r74984 * r74984;
        double r74989 = r74987 + r74988;
        double r74990 = r74986 / r74989;
        return r74990;
}

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
Target0.4
Herbie26.1
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.1

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

  3. Applied add-sqr-sqrt26.1

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

  4. Applied associate-/r*26.0

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

  5. Final simplification26.1

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

Reproduce

herbie shell --seed 2019291 
(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))))