Average Error: 25.9 → 1.0
Time: 4.4s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\mathsf{fma}\left(\frac{b}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, \frac{c}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, -\frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \left(\left(-\frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right)}{\mathsf{hypot}\left(c, d\right) \cdot 1}\]
\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) \cdot 1}
double f(double a, double b, double c, double d) {
        double r126238 = b;
        double r126239 = c;
        double r126240 = r126238 * r126239;
        double r126241 = a;
        double r126242 = d;
        double r126243 = r126241 * r126242;
        double r126244 = r126240 - r126243;
        double r126245 = r126239 * r126239;
        double r126246 = r126242 * r126242;
        double r126247 = r126245 + r126246;
        double r126248 = r126244 / r126247;
        return r126248;
}
↓
double f(double a, double b, double c, double d) {
        double r126249 = b;
        double r126250 = c;
        double r126251 = d;
        double r126252 = hypot(r126250, r126251);
        double r126253 = sqrt(r126252);
        double r126254 = r126249 / r126253;
        double r126255 = r126250 / r126253;
        double r126256 = r126251 / r126253;
        double r126257 = a;
        double r126258 = r126257 / r126253;
        double r126259 = r126256 * r126258;
        double r126260 = -r126259;
        double r126261 = fma(r126254, r126255, r126260);
        double r126262 = -r126258;
        double r126263 = r126262 + r126258;
        double r126264 = r126256 * r126263;
        double r126265 = r126261 + r126264;
        double r126266 = 1.0;
        double r126267 = r126252 * r126266;
        double r126268 = r126265 / r126267;
        return r126268;
}

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 associate-*l/16.9

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

  10. Simplified16.9

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

  12. Applied div-sub16.9

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

  14. Applied add-sqr-sqrt17.0

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

  15. Applied times-frac9.4

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

  16. Applied add-sqr-sqrt9.5

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

  17. Applied times-frac1.0

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

  18. Applied prod-diff1.0

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

  19. Simplified1.0

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

  20. Final simplification1.0

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

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))))