Average Error: 25.7 → 25.7
Time: 11.6s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}{b \cdot c - a \cdot d}}}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}{b \cdot c - a \cdot d}}}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}
double f(double a, double b, double c, double d) {
        double r3139969 = b;
        double r3139970 = c;
        double r3139971 = r3139969 * r3139970;
        double r3139972 = a;
        double r3139973 = d;
        double r3139974 = r3139972 * r3139973;
        double r3139975 = r3139971 - r3139974;
        double r3139976 = r3139970 * r3139970;
        double r3139977 = r3139973 * r3139973;
        double r3139978 = r3139976 + r3139977;
        double r3139979 = r3139975 / r3139978;
        return r3139979;
}
↓
double f(double a, double b, double c, double d) {
        double r3139980 = 1.0;
        double r3139981 = d;
        double r3139982 = c;
        double r3139983 = r3139982 * r3139982;
        double r3139984 = fma(r3139981, r3139981, r3139983);
        double r3139985 = sqrt(r3139984);
        double r3139986 = b;
        double r3139987 = r3139986 * r3139982;
        double r3139988 = a;
        double r3139989 = r3139988 * r3139981;
        double r3139990 = r3139987 - r3139989;
        double r3139991 = r3139985 / r3139990;
        double r3139992 = r3139980 / r3139991;
        double r3139993 = r3139992 / r3139985;
        return r3139993;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.7
Target0.5
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.7

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

  2. Simplified25.7

    \[\leadsto \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt25.7

    \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)} \cdot \sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}}\]

  5. Applied associate-/r*25.6

    \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}}\]
  6. Using strategy rm

  7. Applied clear-num25.7

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}{b \cdot c - a \cdot d}}}}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}\]

  8. Final simplification25.7

    \[\leadsto \frac{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}{b \cdot c - a \cdot d}}}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, imag part"

  :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))))