Complex division, real part

Average Error: 26.4 → 12.8

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -2.9459980091776752 \cdot 10^{158}:\\ \;\;\;\;\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{elif}\;c \le -86223460658051.5:\\ \;\;\;\;\left(\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} \cdot c\right) \cdot a + \left(\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} \cdot d\right) \cdot b\\ \mathbf{elif}\;c \le 2.2413672752256256 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r100511 = a;
        double r100512 = c;
        double r100513 = r100511 * r100512;
        double r100514 = b;
        double r100515 = d;
        double r100516 = r100514 * r100515;
        double r100517 = r100513 + r100516;
        double r100518 = r100512 * r100512;
        double r100519 = r100515 * r100515;
        double r100520 = r100518 + r100519;
        double r100521 = r100517 / r100520;
        return r100521;
}

↓

double f(double a, double b, double c, double d) {
        double r100522 = c;
        double r100523 = -2.945998009177675e+158;
        bool r100524 = r100522 <= r100523;
        double r100525 = -1.0;
        double r100526 = a;
        double r100527 = r100525 * r100526;
        double r100528 = d;
        double r100529 = hypot(r100522, r100528);
        double r100530 = 1.0;
        double r100531 = r100529 * r100530;
        double r100532 = r100527 / r100531;
        double r100533 = -86223460658051.5;
        bool r100534 = r100522 <= r100533;
        double r100535 = r100530 / r100529;
        double r100536 = r100535 / r100529;
        double r100537 = r100536 * r100522;
        double r100538 = r100537 * r100526;
        double r100539 = r100536 * r100528;
        double r100540 = b;
        double r100541 = r100539 * r100540;
        double r100542 = r100538 + r100541;
        double r100543 = 2.2413672752256256e+125;
        bool r100544 = r100522 <= r100543;
        double r100545 = r100540 * r100528;
        double r100546 = fma(r100526, r100522, r100545);
        double r100547 = r100546 / r100529;
        double r100548 = r100547 / r100531;
        double r100549 = r100526 / r100531;
        double r100550 = r100544 ? r100548 : r100549;
        double r100551 = r100534 ? r100542 : r100550;
        double r100552 = r100524 ? r100532 : r100551;
        return r100552;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.4
Target	0.5
Herbie	12.8

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

Derivation

1. Split input into 4 regimes

2. if c < -2.945998009177675e+158

   1. Initial program 45.8

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 45.8

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   4. Applied *-un-lft-identity 45.8

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

   5. Applied times-frac 45.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

   6. Simplified 45.8

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

   7. Simplified 29.5

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

   9. Applied associate-*r/ 29.5

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

   10. Simplified 29.5

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

   11. Taylor expanded around -inf 12.9

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

   if -2.945998009177675e+158 < c < -86223460658051.5

   1. Initial program 22.3

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 22.3

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   4. Applied *-un-lft-identity 22.3

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

   5. Applied times-frac 22.3

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

   6. Simplified 22.3

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

   7. Simplified 17.2

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

   9. Applied *-un-lft-identity 17.2

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

   10. Applied times-frac 17.3

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

   11. Simplified 17.3

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

   13. Applied fma-udef 17.3

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

   14. Applied distribute-lft-in 17.3

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

   15. Applied distribute-lft-in 17.3

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

   16. Simplified 12.0

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

   17. Simplified 15.5

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

   if -86223460658051.5 < c < 2.2413672752256256e+125

   1. Initial program 18.7

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 18.7

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   4. Applied *-un-lft-identity 18.7

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

   5. Applied times-frac 18.7

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

   6. Simplified 18.7

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

   7. Simplified 11.9

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

   9. Applied associate-*r/ 11.8

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

   10. Simplified 11.7

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

   if 2.2413672752256256e+125 < c

   1. Initial program 42.7

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 42.7

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   4. Applied *-un-lft-identity 42.7

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

   5. Applied times-frac 42.7

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

   6. Simplified 42.7

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

   7. Simplified 28.1

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

   9. Applied associate-*r/ 28.1

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

   10. Simplified 28.1

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

   11. Taylor expanded around inf 14.9

       \[\leadsto \frac{\color{blue}{a}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -2.9459980091776752 \cdot 10^{158}:\\ \;\;\;\;\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{elif}\;c \le -86223460658051.5:\\ \;\;\;\;\left(\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} \cdot c\right) \cdot a + \left(\frac{\frac{1}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} \cdot d\right) \cdot b\\ \mathbf{elif}\;c \le 2.2413672752256256 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :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))))