Complex division, real part

Average Error: 26.1 → 13.0

Time: 16.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.1443562208296078 \cdot 10^{+200}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 1.3614475866611724 \cdot 10^{+128}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, \left(d \cdot b\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r3417366 = a;
        double r3417367 = c;
        double r3417368 = r3417366 * r3417367;
        double r3417369 = b;
        double r3417370 = d;
        double r3417371 = r3417369 * r3417370;
        double r3417372 = r3417368 + r3417371;
        double r3417373 = r3417367 * r3417367;
        double r3417374 = r3417370 * r3417370;
        double r3417375 = r3417373 + r3417374;
        double r3417376 = r3417372 / r3417375;
        return r3417376;
}

↓

double f(double a, double b, double c, double d) {
        double r3417377 = c;
        double r3417378 = -1.1443562208296078e+200;
        bool r3417379 = r3417377 <= r3417378;
        double r3417380 = a;
        double r3417381 = -r3417380;
        double r3417382 = d;
        double r3417383 = hypot(r3417382, r3417377);
        double r3417384 = r3417381 / r3417383;
        double r3417385 = 1.3614475866611724e+128;
        bool r3417386 = r3417377 <= r3417385;
        double r3417387 = b;
        double r3417388 = r3417382 * r3417387;
        double r3417389 = fma(r3417380, r3417377, r3417388);
        double r3417390 = 1.0;
        double r3417391 = r3417390 / r3417383;
        double r3417392 = r3417389 * r3417391;
        double r3417393 = r3417392 / r3417383;
        double r3417394 = r3417380 / r3417383;
        double r3417395 = r3417386 ? r3417393 : r3417394;
        double r3417396 = r3417379 ? r3417384 : r3417395;
        return r3417396;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.1
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 < -1.1443562208296078e+200

   1. Initial program 41.6

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

   2. Simplified 41.6

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

   4. Applied add-sqr-sqrt 41.6

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

   5. Applied associate-/r* 41.6

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

   7. Applied fma-udef 41.6

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

   8. Applied hypot-def 41.6

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

   10. Applied fma-udef 41.6

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

   11. Applied hypot-def 28.5

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

   12. Taylor expanded around -inf 10.3

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

   13. Simplified 10.3

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

   if -1.1443562208296078e+200 < c < 1.3614475866611724e+128

   1. Initial program 20.9

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

   2. Simplified 20.9

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

   4. Applied add-sqr-sqrt 20.9

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

   5. Applied associate-/r* 20.8

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

   7. Applied fma-udef 20.8

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

   8. Applied hypot-def 20.8

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

   10. Applied fma-udef 20.8

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

   11. Applied hypot-def 12.9

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

   13. Applied div-inv 13.0

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

   if 1.3614475866611724e+128 < c

   1. Initial program 42.4

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

   2. Simplified 42.4

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

   4. Applied add-sqr-sqrt 42.4

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

   5. Applied associate-/r* 42.4

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

   7. Applied fma-udef 42.4

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

   8. Applied hypot-def 42.4

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

   10. Applied fma-udef 42.4

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

   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 14.6

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

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.1443562208296078 \cdot 10^{+200}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 1.3614475866611724 \cdot 10^{+128}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, \left(d \cdot b\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +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))))