Average Error: 25.9 → 1.0
Time: 4.5s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\mathsf{fma}\left(\frac{b}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, \frac{c}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, -\frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \left(\left(-\frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right)}{\mathsf{hypot}\left(c, d\right)}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\mathsf{fma}\left(\frac{b}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, \frac{c}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, -\frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \left(\left(-\frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right)}{\mathsf{hypot}\left(c, d\right)}
double f(double a, double b, double c, double d) {
        double r111340 = b;
        double r111341 = c;
        double r111342 = r111340 * r111341;
        double r111343 = a;
        double r111344 = d;
        double r111345 = r111343 * r111344;
        double r111346 = r111342 - r111345;
        double r111347 = r111341 * r111341;
        double r111348 = r111344 * r111344;
        double r111349 = r111347 + r111348;
        double r111350 = r111346 / r111349;
        return r111350;
}
↓
double f(double a, double b, double c, double d) {
        double r111351 = b;
        double r111352 = c;
        double r111353 = d;
        double r111354 = hypot(r111352, r111353);
        double r111355 = sqrt(r111354);
        double r111356 = r111351 / r111355;
        double r111357 = r111352 / r111355;
        double r111358 = r111353 / r111355;
        double r111359 = a;
        double r111360 = r111359 / r111355;
        double r111361 = r111358 * r111360;
        double r111362 = -r111361;
        double r111363 = fma(r111356, r111357, r111362);
        double r111364 = -r111360;
        double r111365 = r111364 + r111360;
        double r111366 = r111358 * r111365;
        double r111367 = r111363 + r111366;
        double r111368 = r111367 / r111354;
        return r111368;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.9
Target0.5
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.9

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

  3. Applied add-sqr-sqrt25.9

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

  4. Applied *-un-lft-identity25.9

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

  5. Applied times-frac25.9

    \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

  6. Simplified25.9

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

  7. Simplified17.0

    \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity17.0

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

  10. Applied associate-*l*17.0

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

  11. Simplified16.9

    \[\leadsto 1 \cdot \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}}\]
  12. Using strategy rm

  13. Applied div-sub16.9

    \[\leadsto 1 \cdot \frac{\color{blue}{\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}}}{\mathsf{hypot}\left(c, d\right)}\]
  14. Using strategy rm

  15. Applied add-sqr-sqrt17.0

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

  16. Applied times-frac9.4

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

  17. Applied add-sqr-sqrt9.5

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

  18. Applied times-frac1.0

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

  19. Applied prod-diff1.0

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

  20. Simplified1.0

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

  21. Final simplification1.0

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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))))