Complex division, imag part

Average Error: 26.5 → 4.8

Time: 3.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -2.2616143508082277 \cdot 10^{79} \lor \neg \left(c \le 1.03827084215452011 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}}{\mathsf{hypot}\left(c, d\right)} - \frac{\frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} - \frac{\frac{a}{1}}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{d}{\mathsf{hypot}\left(c, d\right)}}}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r114390 = b;
        double r114391 = c;
        double r114392 = r114390 * r114391;
        double r114393 = a;
        double r114394 = d;
        double r114395 = r114393 * r114394;
        double r114396 = r114392 - r114395;
        double r114397 = r114391 * r114391;
        double r114398 = r114394 * r114394;
        double r114399 = r114397 + r114398;
        double r114400 = r114396 / r114399;
        return r114400;
}

↓

double f(double a, double b, double c, double d) {
        double r114401 = c;
        double r114402 = -2.2616143508082277e+79;
        bool r114403 = r114401 <= r114402;
        double r114404 = 1.0382708421545201e-35;
        bool r114405 = r114401 <= r114404;
        double r114406 = !r114405;
        bool r114407 = r114403 || r114406;
        double r114408 = b;
        double r114409 = d;
        double r114410 = hypot(r114401, r114409);
        double r114411 = r114410 / r114401;
        double r114412 = r114408 / r114411;
        double r114413 = r114412 / r114410;
        double r114414 = a;
        double r114415 = r114414 * r114409;
        double r114416 = r114415 / r114410;
        double r114417 = r114416 / r114410;
        double r114418 = r114413 - r114417;
        double r114419 = r114408 * r114401;
        double r114420 = r114419 / r114410;
        double r114421 = r114420 / r114410;
        double r114422 = 1.0;
        double r114423 = r114414 / r114422;
        double r114424 = r114409 / r114410;
        double r114425 = r114410 / r114424;
        double r114426 = r114423 / r114425;
        double r114427 = r114421 - r114426;
        double r114428 = r114407 ? r114418 : r114427;
        return r114428;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.5
Target	0.4
Herbie	4.8

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

Derivation

1. Split input into 2 regimes

2. if c < -2.2616143508082277e+79 or 1.0382708421545201e-35 < c

   1. Initial program 33.6

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

   3. Applied add-sqr-sqrt 33.6

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

   4. Applied *-un-lft-identity 33.6

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

   5. Applied times-frac 33.6

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

   6. Simplified 33.6

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

   7. Simplified 23.3

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

   9. Applied associate-*r/ 23.2

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

   10. Simplified 23.2

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

   12. Applied div-sub 23.2

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

   13. Applied div-sub 23.2

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

   15. Applied associate-/l* 7.8

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

   if -2.2616143508082277e+79 < c < 1.0382708421545201e-35

   1. Initial program 19.6

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

   3. Applied add-sqr-sqrt 19.6

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

   4. Applied *-un-lft-identity 19.6

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

   5. Applied times-frac 19.6

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

   6. Simplified 19.6

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

   7. Simplified 11.8

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

   9. Applied associate-*r/ 11.7

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

   10. Simplified 11.7

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

   12. Applied div-sub 11.7

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

   13. Applied div-sub 11.7

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

   15. Applied *-un-lft-identity 11.7

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

   16. Applied times-frac 1.7

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

   17. Applied associate-/l* 1.9

       \[\leadsto \frac{\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} - \color{blue}{\frac{\frac{a}{1}}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{d}{\mathsf{hypot}\left(c, d\right)}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -2.2616143508082277 \cdot 10^{79} \lor \neg \left(c \le 1.03827084215452011 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}}{\mathsf{hypot}\left(c, d\right)} - \frac{\frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} - \frac{\frac{a}{1}}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{d}{\mathsf{hypot}\left(c, d\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

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

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