Complex division, real part

Average Error: 25.9 → 12.6

Time: 12.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -7.572549191493678695157155584196926396668 \cdot 10^{93}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(d, c\right)} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\\ \mathbf{elif}\;c \le 8.470980326725044930021835762231456125214 \cdot 10^{114}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r130397 = a;
        double r130398 = c;
        double r130399 = r130397 * r130398;
        double r130400 = b;
        double r130401 = d;
        double r130402 = r130400 * r130401;
        double r130403 = r130399 + r130402;
        double r130404 = r130398 * r130398;
        double r130405 = r130401 * r130401;
        double r130406 = r130404 + r130405;
        double r130407 = r130403 / r130406;
        return r130407;
}

↓

double f(double a, double b, double c, double d) {
        double r130408 = c;
        double r130409 = -7.572549191493679e+93;
        bool r130410 = r130408 <= r130409;
        double r130411 = a;
        double r130412 = -r130411;
        double r130413 = d;
        double r130414 = hypot(r130413, r130408);
        double r130415 = r130412 / r130414;
        double r130416 = 1.0;
        double r130417 = cbrt(r130416);
        double r130418 = r130417 * r130417;
        double r130419 = r130415 * r130418;
        double r130420 = 8.470980326725045e+114;
        bool r130421 = r130408 <= r130420;
        double r130422 = b;
        double r130423 = r130413 * r130422;
        double r130424 = fma(r130411, r130408, r130423);
        double r130425 = r130424 / r130414;
        double r130426 = r130425 / r130414;
        double r130427 = r130418 * r130426;
        double r130428 = r130411 / r130414;
        double r130429 = r130418 * r130428;
        double r130430 = r130421 ? r130427 : r130429;
        double r130431 = r130410 ? r130419 : r130430;
        return r130431;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.9
Target	0.4
Herbie	12.6

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

Derivation

1. Split input into 3 regimes

2. if c < -7.572549191493679e+93

   1. Initial program 41.0

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

   2. Simplified 41.0

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

   4. Applied add-sqr-sqrt 41.0

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

   5. Applied *-un-lft-identity 41.0

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

   6. Applied times-frac 41.0

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

   7. Simplified 41.0

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

   8. Simplified 28.0

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

   10. Applied *-un-lft-identity 28.0

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

   11. Applied add-cube-cbrt 28.0

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}\]

   12. Applied times-frac 28.0

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)}\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}\]

   13. Applied associate-*l* 28.0

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \left(\frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}\right)}\]

   14. Simplified 27.9

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}}\]

   15. Taylor expanded around -inf 17.6

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\color{blue}{-1 \cdot a}}{\mathsf{hypot}\left(d, c\right)}\]

   16. Simplified 17.6

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\color{blue}{-a}}{\mathsf{hypot}\left(d, c\right)}\]

   if -7.572549191493679e+93 < c < 8.470980326725045e+114

   1. Initial program 18.2

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

   2. Simplified 18.2

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

   4. Applied add-sqr-sqrt 18.2

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

   5. Applied *-un-lft-identity 18.2

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

   6. Applied times-frac 18.2

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

   7. Simplified 18.2

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

   8. Simplified 11.1

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

   10. Applied *-un-lft-identity 11.1

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

   11. Applied add-cube-cbrt 11.1

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}\]

   12. Applied times-frac 11.1

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)}\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}\]

   13. Applied associate-*l* 11.1

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \left(\frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}\right)}\]

   14. Simplified 11.0

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}}\]

   if 8.470980326725045e+114 < c

   1. Initial program 39.1

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

   2. Simplified 39.1

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

   4. Applied add-sqr-sqrt 39.1

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

   5. Applied *-un-lft-identity 39.1

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

   6. Applied times-frac 39.1

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

   7. Simplified 39.1

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

   8. Simplified 25.6

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

   10. Applied *-un-lft-identity 25.6

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

   11. Applied add-cube-cbrt 25.6

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}\]

   12. Applied times-frac 25.6

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)}\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}\]

   13. Applied associate-*l* 25.6

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \left(\frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}\right)}\]

   14. Simplified 25.5

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}}\]

   15. Taylor expanded around inf 13.4

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

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -7.572549191493678695157155584196926396668 \cdot 10^{93}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(d, c\right)} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\\ \mathbf{elif}\;c \le 8.470980326725044930021835762231456125214 \cdot 10^{114}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :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))))