Complex division, imag part

Average Error: 26.1 → 13.2

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.24989762348172646 \cdot 10^{107}:\\ \;\;\;\;{\left(\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;c \le 7.04631729015775065 \cdot 10^{128}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r93250 = b;
        double r93251 = c;
        double r93252 = r93250 * r93251;
        double r93253 = a;
        double r93254 = d;
        double r93255 = r93253 * r93254;
        double r93256 = r93252 - r93255;
        double r93257 = r93251 * r93251;
        double r93258 = r93254 * r93254;
        double r93259 = r93257 + r93258;
        double r93260 = r93256 / r93259;
        return r93260;
}

↓

double f(double a, double b, double c, double d) {
        double r93261 = c;
        double r93262 = -1.2498976234817265e+107;
        bool r93263 = r93261 <= r93262;
        double r93264 = -1.0;
        double r93265 = b;
        double r93266 = r93264 * r93265;
        double r93267 = d;
        double r93268 = hypot(r93261, r93267);
        double r93269 = r93266 / r93268;
        double r93270 = 1.0;
        double r93271 = pow(r93269, r93270);
        double r93272 = 7.04631729015775e+128;
        bool r93273 = r93261 <= r93272;
        double r93274 = r93270 / r93268;
        double r93275 = r93265 * r93261;
        double r93276 = a;
        double r93277 = r93276 * r93267;
        double r93278 = r93275 - r93277;
        double r93279 = r93278 / r93268;
        double r93280 = r93274 * r93279;
        double r93281 = pow(r93280, r93270);
        double r93282 = r93265 / r93268;
        double r93283 = pow(r93282, r93270);
        double r93284 = r93273 ? r93281 : r93283;
        double r93285 = r93263 ? r93271 : r93284;
        return r93285;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.1
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 3 regimes

2. if c < -1.2498976234817265e+107

   1. Initial program 40.3

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

   3. Applied add-sqr-sqrt 40.3

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

   4. Applied *-un-lft-identity 40.3

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

   5. Applied times-frac 40.3

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

   6. Simplified 40.3

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

   7. Simplified 27.0

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

   9. Applied pow1 27.0

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

   10. Applied pow1 27.0

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

   11. Applied pow-prod-down 27.0

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

   12. Simplified 26.9

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

   13. Taylor expanded around -inf 16.4

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

   if -1.2498976234817265e+107 < c < 7.04631729015775e+128

   1. Initial program 18.7

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

   3. Applied add-sqr-sqrt 18.7

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

   4. Applied *-un-lft-identity 18.7

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

   5. Applied times-frac 18.7

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

   6. Simplified 18.7

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

   7. Simplified 11.9

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

   9. Applied pow1 11.9

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

   10. Applied pow1 11.9

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

   11. Applied pow-prod-down 11.9

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

   12. Simplified 11.7

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

   14. Applied add-sqr-sqrt 12.0

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

   15. Applied add-sqr-sqrt 12.2

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

   16. Applied *-un-lft-identity 12.2

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

   17. Applied times-frac 12.3

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

   18. Applied times-frac 12.3

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

   19. Simplified 12.1

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

   20. Simplified 11.9

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

   if 7.04631729015775e+128 < c

   1. Initial program 42.0

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

   3. Applied add-sqr-sqrt 42.0

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

   4. Applied *-un-lft-identity 42.0

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

   5. Applied times-frac 42.0

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

   6. Simplified 42.0

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

   7. Simplified 28.1

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

   9. Applied pow1 28.1

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

   10. Applied pow1 28.1

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

   11. Applied pow-prod-down 28.1

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

   12. Simplified 28.0

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

   13. Taylor expanded around inf 15.0

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.24989762348172646 \cdot 10^{107}:\\ \;\;\;\;{\left(\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;c \le 7.04631729015775065 \cdot 10^{128}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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))))