Complex division, imag part

Average Error: 26.4 → 12.6

Time: 5.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -5.3927648088213951 \cdot 10^{132}:\\ \;\;\;\;\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le -1.2987370292207596 \cdot 10^{-142}:\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \mathbf{elif}\;c \le 6.1765680479198497 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.45382927677037052 \cdot 10^{189}:\\ \;\;\;\;\frac{\frac{c}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{b}} - \frac{d}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{a}}}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r116432 = b;
        double r116433 = c;
        double r116434 = r116432 * r116433;
        double r116435 = a;
        double r116436 = d;
        double r116437 = r116435 * r116436;
        double r116438 = r116434 - r116437;
        double r116439 = r116433 * r116433;
        double r116440 = r116436 * r116436;
        double r116441 = r116439 + r116440;
        double r116442 = r116438 / r116441;
        return r116442;
}

↓

double f(double a, double b, double c, double d) {
        double r116443 = c;
        double r116444 = -5.392764808821395e+132;
        bool r116445 = r116443 <= r116444;
        double r116446 = -1.0;
        double r116447 = b;
        double r116448 = r116446 * r116447;
        double r116449 = d;
        double r116450 = hypot(r116443, r116449);
        double r116451 = r116448 / r116450;
        double r116452 = -1.2987370292207596e-142;
        bool r116453 = r116443 <= r116452;
        double r116454 = r116449 * r116449;
        double r116455 = fma(r116443, r116443, r116454);
        double r116456 = r116455 / r116443;
        double r116457 = r116447 / r116456;
        double r116458 = a;
        double r116459 = r116455 / r116449;
        double r116460 = r116458 / r116459;
        double r116461 = r116457 - r116460;
        double r116462 = 6.17656804791985e-128;
        bool r116463 = r116443 <= r116462;
        double r116464 = r116447 * r116443;
        double r116465 = r116458 * r116449;
        double r116466 = r116464 - r116465;
        double r116467 = r116466 / r116450;
        double r116468 = r116467 / r116450;
        double r116469 = 1.4538292767703705e+189;
        bool r116470 = r116443 <= r116469;
        double r116471 = sqrt(r116450);
        double r116472 = 3.0;
        double r116473 = pow(r116471, r116472);
        double r116474 = r116473 / r116447;
        double r116475 = r116443 / r116474;
        double r116476 = r116473 / r116458;
        double r116477 = r116449 / r116476;
        double r116478 = r116475 - r116477;
        double r116479 = r116478 / r116471;
        double r116480 = r116447 / r116450;
        double r116481 = r116470 ? r116479 : r116480;
        double r116482 = r116463 ? r116468 : r116481;
        double r116483 = r116453 ? r116461 : r116482;
        double r116484 = r116445 ? r116451 : r116483;
        return r116484;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.4
Target	0.5
Herbie	12.6

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

Derivation

1. Split input into 5 regimes

2. if c < -5.392764808821395e+132

   1. Initial program 44.3

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

   3. Applied add-sqr-sqrt 44.3

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

   4. Applied *-un-lft-identity 44.3

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

   5. Applied times-frac 44.3

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

   6. Simplified 44.3

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

   7. Simplified 29.0

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

   9. Applied associate-*r/ 28.9

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

   10. Simplified 28.9

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

   11. Taylor expanded around -inf 14.7

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

   if -5.392764808821395e+132 < c < -1.2987370292207596e-142

   1. Initial program 16.4

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

   3. Applied div-sub 16.4

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

   4. Simplified 13.8

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

   5. Simplified 12.1

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

   if -1.2987370292207596e-142 < c < 6.17656804791985e-128

   1. Initial program 22.6

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

   3. Applied add-sqr-sqrt 22.6

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

   4. Applied *-un-lft-identity 22.6

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

   5. Applied times-frac 22.6

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

   6. Simplified 22.6

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

   7. Simplified 12.4

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

   9. Applied associate-*r/ 12.4

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

   10. Simplified 12.2

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

   if 6.17656804791985e-128 < c < 1.4538292767703705e+189

   1. Initial program 21.0

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

   3. Applied add-sqr-sqrt 21.0

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

   4. Applied *-un-lft-identity 21.0

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

   5. Applied times-frac 21.0

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

   6. Simplified 21.0

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

   7. Simplified 13.2

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

   9. Applied associate-*r/ 13.2

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

   10. Simplified 13.1

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

   12. Applied add-sqr-sqrt 13.3

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

   13. Applied associate-/r* 13.4

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

   15. Applied div-sub 13.4

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

   16. Applied div-sub 13.4

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

   17. Simplified 11.9

       \[\leadsto \frac{\color{blue}{\frac{c}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{b}}} - \frac{\frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\sqrt{\mathsf{hypot}\left(c, d\right)}}}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\]

   18. Simplified 12.2

       \[\leadsto \frac{\frac{c}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{b}} - \color{blue}{\frac{d}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{a}}}}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\]

   if 1.4538292767703705e+189 < c

   1. Initial program 44.5

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

   3. Applied add-sqr-sqrt 44.5

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

   4. Applied *-un-lft-identity 44.5

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

   5. Applied times-frac 44.5

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

   6. Simplified 44.5

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

   7. Simplified 32.0

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

   9. Applied associate-*r/ 32.0

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

   10. Simplified 32.0

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

   11. Taylor expanded around inf 12.6

       \[\leadsto \frac{\color{blue}{b}}{\mathsf{hypot}\left(c, d\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -5.3927648088213951 \cdot 10^{132}:\\ \;\;\;\;\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le -1.2987370292207596 \cdot 10^{-142}:\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \mathbf{elif}\;c \le 6.1765680479198497 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.45382927677037052 \cdot 10^{189}:\\ \;\;\;\;\frac{\frac{c}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{b}} - \frac{d}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{a}}}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))