Complex division, real part

Average Error: 26.8 → 13.5

Time: 46.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le -9.494881135626707 \cdot 10^{+123}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \le 1.66807717135982 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r4881401 = a;
        double r4881402 = c;
        double r4881403 = r4881401 * r4881402;
        double r4881404 = b;
        double r4881405 = d;
        double r4881406 = r4881404 * r4881405;
        double r4881407 = r4881403 + r4881406;
        double r4881408 = r4881402 * r4881402;
        double r4881409 = r4881405 * r4881405;
        double r4881410 = r4881408 + r4881409;
        double r4881411 = r4881407 / r4881410;
        return r4881411;
}

↓

double f(double a, double b, double c, double d) {
        double r4881412 = d;
        double r4881413 = -9.494881135626707e+123;
        bool r4881414 = r4881412 <= r4881413;
        double r4881415 = b;
        double r4881416 = -r4881415;
        double r4881417 = c;
        double r4881418 = hypot(r4881412, r4881417);
        double r4881419 = r4881416 / r4881418;
        double r4881420 = 1.66807717135982e+77;
        bool r4881421 = r4881412 <= r4881420;
        double r4881422 = a;
        double r4881423 = r4881417 * r4881422;
        double r4881424 = fma(r4881412, r4881415, r4881423);
        double r4881425 = r4881424 / r4881418;
        double r4881426 = r4881425 / r4881418;
        double r4881427 = r4881415 / r4881418;
        double r4881428 = r4881421 ? r4881426 : r4881427;
        double r4881429 = r4881414 ? r4881419 : r4881428;
        return r4881429;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.8
Target	0.5
Herbie	13.5

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

Derivation

1. Split input into 3 regimes

2. if d < -9.494881135626707e+123

   1. Initial program 42.8

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

   2. Simplified 42.8

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

   4. Applied add-sqr-sqrt 42.8

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

   5. Applied associate-/r* 42.8

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

   7. Applied clear-num 42.9

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

   8. Simplified 30.4

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

   10. Applied associate-/r* 29.8

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

   11. Simplified 29.8

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

   12. Taylor expanded around -inf 15.4

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

   13. Simplified 15.4

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

   if -9.494881135626707e+123 < d < 1.66807717135982e+77

   1. Initial program 19.0

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

   2. Simplified 19.0

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

   4. Applied add-sqr-sqrt 19.0

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

   5. Applied associate-/r* 18.9

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

   7. Applied clear-num 19.2

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

   8. Simplified 12.3

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

   10. Applied associate-/r* 11.9

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

   11. Simplified 11.8

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

   if 1.66807717135982e+77 < d

   1. Initial program 38.9

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

   2. Simplified 38.9

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

   4. Applied add-sqr-sqrt 38.9

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

   5. Applied associate-/r* 38.9

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

   7. Applied clear-num 39.0

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

   8. Simplified 26.8

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

   10. Applied associate-/r* 26.3

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

   11. Simplified 26.3

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

   12. Taylor expanded around inf 17.4

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

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -9.494881135626707 \cdot 10^{+123}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \le 1.66807717135982 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +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))))