Complex division, imag part

Average Error: 26.5 → 13.2

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -5.457377104622505852467015767494503842035 \cdot 10^{208}:\\ \;\;\;\;{\left(\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;c \le 1.098050593933298912479119802869378133859 \cdot 10^{-16}:\\ \;\;\;\;{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;c \le 1.368280955057290886281182593174139313509 \cdot 10^{154}:\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r98323 = b;
        double r98324 = c;
        double r98325 = r98323 * r98324;
        double r98326 = a;
        double r98327 = d;
        double r98328 = r98326 * r98327;
        double r98329 = r98325 - r98328;
        double r98330 = r98324 * r98324;
        double r98331 = r98327 * r98327;
        double r98332 = r98330 + r98331;
        double r98333 = r98329 / r98332;
        return r98333;
}

↓

double f(double a, double b, double c, double d) {
        double r98334 = c;
        double r98335 = -5.457377104622506e+208;
        bool r98336 = r98334 <= r98335;
        double r98337 = -1.0;
        double r98338 = b;
        double r98339 = r98337 * r98338;
        double r98340 = d;
        double r98341 = hypot(r98334, r98340);
        double r98342 = r98339 / r98341;
        double r98343 = 1.0;
        double r98344 = pow(r98342, r98343);
        double r98345 = 1.0980505939332989e-16;
        bool r98346 = r98334 <= r98345;
        double r98347 = r98338 * r98334;
        double r98348 = a;
        double r98349 = r98348 * r98340;
        double r98350 = r98347 - r98349;
        double r98351 = r98350 / r98341;
        double r98352 = r98351 / r98341;
        double r98353 = pow(r98352, r98343);
        double r98354 = 1.368280955057291e+154;
        bool r98355 = r98334 <= r98354;
        double r98356 = r98340 * r98340;
        double r98357 = fma(r98334, r98334, r98356);
        double r98358 = r98357 / r98334;
        double r98359 = r98338 / r98358;
        double r98360 = r98357 / r98340;
        double r98361 = r98348 / r98360;
        double r98362 = r98359 - r98361;
        double r98363 = r98338 / r98341;
        double r98364 = pow(r98363, r98343);
        double r98365 = r98355 ? r98362 : r98364;
        double r98366 = r98346 ? r98353 : r98365;
        double r98367 = r98336 ? r98344 : r98366;
        return r98367;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.5
Target	0.5
Herbie	13.2

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

Derivation

1. Split input into 4 regimes

2. if c < -5.457377104622506e+208

   1. Initial program 42.4

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

   3. Applied add-sqr-sqrt 42.4

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

   4. Applied *-un-lft-identity 42.4

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

   5. Applied times-frac 42.4

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

   6. Simplified 42.4

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

   7. Simplified 31.4

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

   9. Applied pow1 31.4

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

   10. Applied pow1 31.4

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

   11. Applied pow-prod-down 31.4

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

   12. Simplified 31.3

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

   13. Taylor expanded around -inf 10.1

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

   if -5.457377104622506e+208 < c < 1.0980505939332989e-16

   1. Initial program 22.0

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

   3. Applied add-sqr-sqrt 22.0

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

   4. Applied *-un-lft-identity 22.0

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

   5. Applied times-frac 22.0

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

   6. Simplified 22.0

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

   7. Simplified 13.6

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

   9. Applied pow1 13.6

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

   10. Applied pow1 13.6

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

   11. Applied pow-prod-down 13.6

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

   12. Simplified 13.5

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

   if 1.0980505939332989e-16 < c < 1.368280955057291e+154

   1. Initial program 18.9

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

   3. Applied div-sub 18.9

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

   4. Simplified 13.3

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

   5. Simplified 12.5

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

   if 1.368280955057291e+154 < c

   1. Initial program 45.0

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

   3. Applied add-sqr-sqrt 45.0

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

   4. Applied *-un-lft-identity 45.0

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

   5. Applied times-frac 45.0

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

   6. Simplified 45.0

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

   7. Simplified 27.9

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

   9. Applied pow1 27.9

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

   10. Applied pow1 27.9

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

   11. Applied pow-prod-down 27.9

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

   12. Simplified 27.9

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

   13. Taylor expanded around inf 14.6

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -5.457377104622505852467015767494503842035 \cdot 10^{208}:\\ \;\;\;\;{\left(\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;c \le 1.098050593933298912479119802869378133859 \cdot 10^{-16}:\\ \;\;\;\;{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;c \le 1.368280955057290886281182593174139313509 \cdot 10^{154}:\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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))))