Complex division, real part

Average Error: 25.6 → 12.6

Time: 13.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.264168133681399 \cdot 10^{+152}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 1.4672736653528514 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r3836220 = a;
        double r3836221 = c;
        double r3836222 = r3836220 * r3836221;
        double r3836223 = b;
        double r3836224 = d;
        double r3836225 = r3836223 * r3836224;
        double r3836226 = r3836222 + r3836225;
        double r3836227 = r3836221 * r3836221;
        double r3836228 = r3836224 * r3836224;
        double r3836229 = r3836227 + r3836228;
        double r3836230 = r3836226 / r3836229;
        return r3836230;
}

↓

double f(double a, double b, double c, double d) {
        double r3836231 = c;
        double r3836232 = -1.264168133681399e+152;
        bool r3836233 = r3836231 <= r3836232;
        double r3836234 = a;
        double r3836235 = -r3836234;
        double r3836236 = d;
        double r3836237 = hypot(r3836236, r3836231);
        double r3836238 = r3836235 / r3836237;
        double r3836239 = 1.4672736653528514e+187;
        bool r3836240 = r3836231 <= r3836239;
        double r3836241 = b;
        double r3836242 = r3836236 * r3836241;
        double r3836243 = fma(r3836234, r3836231, r3836242);
        double r3836244 = r3836243 / r3836237;
        double r3836245 = r3836244 / r3836237;
        double r3836246 = r3836234 / r3836237;
        double r3836247 = r3836240 ? r3836245 : r3836246;
        double r3836248 = r3836233 ? r3836238 : r3836247;
        return r3836248;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.6
Target	0.6
Herbie	12.6

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

Derivation

1. Split input into 3 regimes

2. if c < -1.264168133681399e+152

   1. Initial program 43.7

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

   2. Simplified 43.7

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

   4. Applied add-sqr-sqrt 43.7

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

   5. Applied associate-/r* 43.7

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

   7. Applied fma-udef 43.7

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

   8. Applied hypot-def 43.7

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

   10. Applied fma-udef 43.7

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

   11. Applied hypot-def 28.2

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

   12. Taylor expanded around -inf 13.6

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

   13. Simplified 13.6

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

   if -1.264168133681399e+152 < c < 1.4672736653528514e+187

   1. Initial program 19.9

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

   2. Simplified 19.9

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

   4. Applied add-sqr-sqrt 19.9

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

   5. Applied associate-/r* 19.8

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

   7. Applied fma-udef 19.8

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

   8. Applied hypot-def 19.8

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

   10. Applied fma-udef 19.8

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

   11. Applied hypot-def 12.6

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

   if 1.4672736653528514e+187 < c

   1. Initial program 42.9

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

   2. Simplified 42.9

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

   4. Applied add-sqr-sqrt 42.9

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

   5. Applied associate-/r* 42.9

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

   7. Applied fma-udef 42.9

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

   8. Applied hypot-def 42.9

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

   9. Taylor expanded around inf 11.6

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

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.264168133681399 \cdot 10^{+152}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 1.4672736653528514 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, real part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))