Average Error: 25.5 → 25.5
Time: 14.6s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}{b \cdot c - a \cdot d}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}{b \cdot c - a \cdot d}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}
double f(double a, double b, double c, double d) {
        double r3586751 = b;
        double r3586752 = c;
        double r3586753 = r3586751 * r3586752;
        double r3586754 = a;
        double r3586755 = d;
        double r3586756 = r3586754 * r3586755;
        double r3586757 = r3586753 - r3586756;
        double r3586758 = r3586752 * r3586752;
        double r3586759 = r3586755 * r3586755;
        double r3586760 = r3586758 + r3586759;
        double r3586761 = r3586757 / r3586760;
        return r3586761;
}
↓
double f(double a, double b, double c, double d) {
        double r3586762 = 1.0;
        double r3586763 = d;
        double r3586764 = c;
        double r3586765 = r3586764 * r3586764;
        double r3586766 = fma(r3586763, r3586763, r3586765);
        double r3586767 = sqrt(r3586766);
        double r3586768 = b;
        double r3586769 = r3586768 * r3586764;
        double r3586770 = a;
        double r3586771 = r3586770 * r3586763;
        double r3586772 = r3586769 - r3586771;
        double r3586773 = r3586767 / r3586772;
        double r3586774 = r3586762 / r3586773;
        double r3586775 = r3586774 / r3586767;
        return r3586775;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.5
Target0.4
Herbie25.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.5

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

  2. Simplified25.5

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

  4. Applied add-sqr-sqrt25.5

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

  5. Applied associate-/r*25.4

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

  7. Applied *-un-lft-identity25.4

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

  8. Applied associate-/l*25.5

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

  9. Final simplification25.5

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

Reproduce

herbie shell --seed 2019141 +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))))