Complex division, real part

Average Error: 25.9 → 13.0

Time: 14.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -3.4568057680652865 \cdot 10^{+145}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 4.1067418235703216 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, \left(d \cdot b\right)\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r2927560 = a;
        double r2927561 = c;
        double r2927562 = r2927560 * r2927561;
        double r2927563 = b;
        double r2927564 = d;
        double r2927565 = r2927563 * r2927564;
        double r2927566 = r2927562 + r2927565;
        double r2927567 = r2927561 * r2927561;
        double r2927568 = r2927564 * r2927564;
        double r2927569 = r2927567 + r2927568;
        double r2927570 = r2927566 / r2927569;
        return r2927570;
}

↓

double f(double a, double b, double c, double d) {
        double r2927571 = c;
        double r2927572 = -3.4568057680652865e+145;
        bool r2927573 = r2927571 <= r2927572;
        double r2927574 = a;
        double r2927575 = -r2927574;
        double r2927576 = d;
        double r2927577 = hypot(r2927576, r2927571);
        double r2927578 = r2927575 / r2927577;
        double r2927579 = 4.1067418235703216e+95;
        bool r2927580 = r2927571 <= r2927579;
        double r2927581 = b;
        double r2927582 = r2927576 * r2927581;
        double r2927583 = fma(r2927574, r2927571, r2927582);
        double r2927584 = r2927583 / r2927577;
        double r2927585 = r2927584 / r2927577;
        double r2927586 = r2927574 / r2927577;
        double r2927587 = r2927580 ? r2927585 : r2927586;
        double r2927588 = r2927573 ? r2927578 : r2927587;
        return r2927588;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.9
Target	0.4
Herbie	13.0

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

Derivation

1. Split input into 3 regimes

2. if c < -3.4568057680652865e+145

   1. Initial program 43.5

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

   2. Simplified 43.5

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

   4. Applied add-sqr-sqrt 43.5

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

   5. Applied associate-/r* 43.5

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

   7. Applied fma-udef 43.5

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

   8. Applied hypot-def 43.5

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

   10. Applied fma-udef 43.5

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

   11. Applied hypot-def 28.2

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

   12. Taylor expanded around -inf 14.0

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

   13. Simplified 14.0

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

   if -3.4568057680652865e+145 < c < 4.1067418235703216e+95

   1. Initial program 18.7

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

   2. Simplified 18.7

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

   4. Applied add-sqr-sqrt 18.7

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

   5. Applied associate-/r* 18.6

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

   7. Applied fma-udef 18.6

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

   8. Applied hypot-def 18.6

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

   10. Applied fma-udef 18.6

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

   11. Applied hypot-def 11.9

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

   if 4.1067418235703216e+95 < c

   1. Initial program 39.0

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

   2. Simplified 39.0

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

   4. Applied add-sqr-sqrt 39.0

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

   5. Applied associate-/r* 39.0

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

   7. Applied fma-udef 39.0

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

   8. Applied hypot-def 39.0

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

   10. Applied fma-udef 39.0

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

   11. Applied hypot-def 25.6

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

   12. Taylor expanded around inf 16.8

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

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -3.4568057680652865 \cdot 10^{+145}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 4.1067418235703216 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, \left(d \cdot b\right)\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +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))))