Complex division, imag part

Average Error: 25.6 → 12.9

Time: 11.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.0127401874663891 \cdot 10^{+151}:\\ \;\;\;\;-\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 1.6977188157374524 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r1567260 = b;
        double r1567261 = c;
        double r1567262 = r1567260 * r1567261;
        double r1567263 = a;
        double r1567264 = d;
        double r1567265 = r1567263 * r1567264;
        double r1567266 = r1567262 - r1567265;
        double r1567267 = r1567261 * r1567261;
        double r1567268 = r1567264 * r1567264;
        double r1567269 = r1567267 + r1567268;
        double r1567270 = r1567266 / r1567269;
        return r1567270;
}

↓

double f(double a, double b, double c, double d) {
        double r1567271 = c;
        double r1567272 = -1.0127401874663891e+151;
        bool r1567273 = r1567271 <= r1567272;
        double r1567274 = b;
        double r1567275 = d;
        double r1567276 = hypot(r1567275, r1567271);
        double r1567277 = r1567274 / r1567276;
        double r1567278 = -r1567277;
        double r1567279 = 1.6977188157374524e+71;
        bool r1567280 = r1567271 <= r1567279;
        double r1567281 = r1567274 * r1567271;
        double r1567282 = a;
        double r1567283 = r1567275 * r1567282;
        double r1567284 = r1567281 - r1567283;
        double r1567285 = r1567284 / r1567276;
        double r1567286 = r1567285 / r1567276;
        double r1567287 = r1567280 ? r1567286 : r1567277;
        double r1567288 = r1567273 ? r1567278 : r1567287;
        return r1567288;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.6
Target	0.4
Herbie	12.9

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

Derivation

1. Split input into 3 regimes

2. if c < -1.0127401874663891e+151

   1. Initial program 43.5

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

   2. Simplified 43.5

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

   4. Applied add-sqr-sqrt 43.5

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

   5. Applied associate-/r* 43.5

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

   7. Applied fma-udef 43.5

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

   8. Applied hypot-def 43.5

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

   9. Taylor expanded around -inf 13.0

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

   10. Simplified 13.0

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

   if -1.0127401874663891e+151 < c < 1.6977188157374524e+71

   1. Initial program 18.8

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

   2. Simplified 18.8

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

   4. Applied add-sqr-sqrt 18.8

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

   5. Applied associate-/r* 18.7

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

   7. Applied fma-udef 18.7

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

   8. Applied hypot-def 18.7

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

   10. Applied fma-udef 18.7

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

   11. Applied hypot-def 11.5

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

   if 1.6977188157374524e+71 < c

   1. Initial program 35.8

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

   2. Simplified 35.8

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

   4. Applied add-sqr-sqrt 35.8

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

   5. Applied associate-/r* 35.8

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

   7. Applied fma-udef 35.8

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

   8. Applied hypot-def 35.8

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

   9. Taylor expanded around inf 17.3

      \[\leadsto \frac{\color{blue}{b}}{\mathsf{hypot}\left(d, c\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.0127401874663891 \cdot 10^{+151}:\\ \;\;\;\;-\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 1.6977188157374524 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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))))