Average Error: 25.3 → 22.1
Time: 24.5s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b \cdot \frac{c}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}} - \frac{a}{\frac{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}{d}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b \cdot \frac{c}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}} - \frac{a}{\frac{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}{d}}}{\sqrt{\mathsf{fma}\left(d, d, c \cdot c\right)}}
double f(double a, double b, double c, double d) {
        double r4814297 = b;
        double r4814298 = c;
        double r4814299 = r4814297 * r4814298;
        double r4814300 = a;
        double r4814301 = d;
        double r4814302 = r4814300 * r4814301;
        double r4814303 = r4814299 - r4814302;
        double r4814304 = r4814298 * r4814298;
        double r4814305 = r4814301 * r4814301;
        double r4814306 = r4814304 + r4814305;
        double r4814307 = r4814303 / r4814306;
        return r4814307;
}
↓
double f(double a, double b, double c, double d) {
        double r4814308 = b;
        double r4814309 = c;
        double r4814310 = d;
        double r4814311 = r4814309 * r4814309;
        double r4814312 = fma(r4814310, r4814310, r4814311);
        double r4814313 = sqrt(r4814312);
        double r4814314 = r4814309 / r4814313;
        double r4814315 = r4814308 * r4814314;
        double r4814316 = a;
        double r4814317 = r4814313 / r4814310;
        double r4814318 = r4814316 / r4814317;
        double r4814319 = r4814315 - r4814318;
        double r4814320 = r4814319 / r4814313;
        return r4814320;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.3
Target0.5
Herbie22.1
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.3

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

  2. Simplified25.3

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

  4. Applied add-sqr-sqrt25.3

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

  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 div-sub25.3

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

  9. Applied *-un-lft-identity25.3

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

  10. Applied sqrt-prod25.3

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

  11. Applied times-frac23.8

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

  12. Simplified23.8

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

  14. Applied associate-/l*22.1

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

  15. Final simplification22.1

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

Reproduce

herbie shell --seed 2019144 +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))))