Average Error: 25.4 → 25.3
Time: 10.2s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{b \cdot c - a \cdot d}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\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)}}
double f(double a, double b, double c, double d) {
        double r1599855 = b;
        double r1599856 = c;
        double r1599857 = r1599855 * r1599856;
        double r1599858 = a;
        double r1599859 = d;
        double r1599860 = r1599858 * r1599859;
        double r1599861 = r1599857 - r1599860;
        double r1599862 = r1599856 * r1599856;
        double r1599863 = r1599859 * r1599859;
        double r1599864 = r1599862 + r1599863;
        double r1599865 = r1599861 / r1599864;
        return r1599865;
}
↓
double f(double a, double b, double c, double d) {
        double r1599866 = b;
        double r1599867 = c;
        double r1599868 = r1599866 * r1599867;
        double r1599869 = a;
        double r1599870 = d;
        double r1599871 = r1599869 * r1599870;
        double r1599872 = r1599868 - r1599871;
        double r1599873 = r1599867 * r1599867;
        double r1599874 = fma(r1599870, r1599870, r1599873);
        double r1599875 = sqrt(r1599874);
        double r1599876 = r1599872 / r1599875;
        double r1599877 = r1599876 / r1599875;
        return r1599877;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.4
Target0.4
Herbie25.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.4

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

  2. Simplified25.4

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

  4. Applied add-sqr-sqrt25.4

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

  5. Applied associate-/r*25.3

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

  6. Final simplification25.3

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

Reproduce

herbie shell --seed 2019154 +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))))