Average Error: 25.4 → 25.4
Time: 15.6s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{1}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}} \cdot \mathsf{fma}\left(b, c, \left(-a\right) \cdot d\right)}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{1}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}} \cdot \mathsf{fma}\left(b, c, \left(-a\right) \cdot d\right)}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}
double f(double a, double b, double c, double d) {
        double r2919497 = b;
        double r2919498 = c;
        double r2919499 = r2919497 * r2919498;
        double r2919500 = a;
        double r2919501 = d;
        double r2919502 = r2919500 * r2919501;
        double r2919503 = r2919499 - r2919502;
        double r2919504 = r2919498 * r2919498;
        double r2919505 = r2919501 * r2919501;
        double r2919506 = r2919504 + r2919505;
        double r2919507 = r2919503 / r2919506;
        return r2919507;
}
↓
double f(double a, double b, double c, double d) {
        double r2919508 = 1.0;
        double r2919509 = d;
        double r2919510 = c;
        double r2919511 = r2919510 * r2919510;
        double r2919512 = fma(r2919509, r2919509, r2919511);
        double r2919513 = sqrt(r2919512);
        double r2919514 = r2919508 / r2919513;
        double r2919515 = b;
        double r2919516 = a;
        double r2919517 = -r2919516;
        double r2919518 = r2919517 * r2919509;
        double r2919519 = fma(r2919515, r2919510, r2919518);
        double r2919520 = r2919514 * r2919519;
        double r2919521 = r2919520 / r2919513;
        return r2919521;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

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

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. Using strategy rm

  7. Applied fma-neg25.3

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

  9. Applied div-inv25.4

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

  10. Final simplification25.4

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

Reproduce

herbie shell --seed 2019152 +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))))