math.sqrt on complex, real part

Average Error: 37.8 → 17.5

Time: 12.6s

Precision: 64

Math

\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.1214463413730795 \cdot 10^{+123}:\\ \;\;\;\;\left(\sqrt{2.0} \cdot \frac{\left|im\right|}{\sqrt{re \cdot -2}}\right) \cdot 0.5\\ \mathbf{elif}\;re \le -3.263660527236801 \cdot 10^{-218}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{\sqrt{2.0}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right) \cdot \sqrt{\sqrt{2.0}}\right)\\ \mathbf{elif}\;re \le 2.4935750622148784 \cdot 10^{-246}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + im\right) \cdot 2.0}\\ \mathbf{elif}\;re \le 9.701163352497263 \cdot 10^{+98}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]

C

double f(double re, double im) {
        double r3013309 = 0.5;
        double r3013310 = 2.0;
        double r3013311 = re;
        double r3013312 = r3013311 * r3013311;
        double r3013313 = im;
        double r3013314 = r3013313 * r3013313;
        double r3013315 = r3013312 + r3013314;
        double r3013316 = sqrt(r3013315);
        double r3013317 = r3013316 + r3013311;
        double r3013318 = r3013310 * r3013317;
        double r3013319 = sqrt(r3013318);
        double r3013320 = r3013309 * r3013319;
        return r3013320;
}

↓

double f(double re, double im) {
        double r3013321 = re;
        double r3013322 = -1.1214463413730795e+123;
        bool r3013323 = r3013321 <= r3013322;
        double r3013324 = 2.0;
        double r3013325 = sqrt(r3013324);
        double r3013326 = im;
        double r3013327 = fabs(r3013326);
        double r3013328 = -2.0;
        double r3013329 = r3013321 * r3013328;
        double r3013330 = sqrt(r3013329);
        double r3013331 = r3013327 / r3013330;
        double r3013332 = r3013325 * r3013331;
        double r3013333 = 0.5;
        double r3013334 = r3013332 * r3013333;
        double r3013335 = -3.263660527236801e-218;
        bool r3013336 = r3013321 <= r3013335;
        double r3013337 = sqrt(r3013325);
        double r3013338 = r3013321 * r3013321;
        double r3013339 = r3013326 * r3013326;
        double r3013340 = r3013338 + r3013339;
        double r3013341 = sqrt(r3013340);
        double r3013342 = r3013341 - r3013321;
        double r3013343 = sqrt(r3013342);
        double r3013344 = r3013327 / r3013343;
        double r3013345 = r3013337 * r3013344;
        double r3013346 = r3013345 * r3013337;
        double r3013347 = r3013333 * r3013346;
        double r3013348 = 2.4935750622148784e-246;
        bool r3013349 = r3013321 <= r3013348;
        double r3013350 = r3013321 + r3013326;
        double r3013351 = r3013350 * r3013324;
        double r3013352 = sqrt(r3013351);
        double r3013353 = r3013333 * r3013352;
        double r3013354 = 9.701163352497263e+98;
        bool r3013355 = r3013321 <= r3013354;
        double r3013356 = r3013341 + r3013321;
        double r3013357 = r3013356 * r3013324;
        double r3013358 = sqrt(r3013357);
        double r3013359 = r3013333 * r3013358;
        double r3013360 = r3013321 + r3013321;
        double r3013361 = r3013324 * r3013360;
        double r3013362 = sqrt(r3013361);
        double r3013363 = r3013362 * r3013333;
        double r3013364 = r3013355 ? r3013359 : r3013363;
        double r3013365 = r3013349 ? r3013353 : r3013364;
        double r3013366 = r3013336 ? r3013347 : r3013365;
        double r3013367 = r3013323 ? r3013334 : r3013366;
        return r3013367;
}

Error

Bits error versus re

Bits error versus im

Target

Original	37.8
Target	32.9
Herbie	17.5

\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

1. Split input into 5 regimes

2. if re < -1.1214463413730795e+123

   1. Initial program 60.9

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied flip-+ 60.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   4. Applied associate-*r/ 60.9

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   5. Applied sqrt-div 60.9

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   6. Simplified 45.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
   7. Using strategy rm

   8. Applied sqrt-prod 45.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0} \cdot \sqrt{im \cdot im + 0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

   9. Simplified 44.3

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0} \cdot \color{blue}{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
   10. Using strategy rm

   11. Applied *-un-lft-identity 44.3

       \[\leadsto 0.5 \cdot \frac{\sqrt{2.0} \cdot \left|im\right|}{\sqrt{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}}\]

   12. Applied sqrt-prod 44.3

       \[\leadsto 0.5 \cdot \frac{\sqrt{2.0} \cdot \left|im\right|}{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   13. Applied times-frac 44.3

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2.0}}{\sqrt{1}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\]

   14. Simplified 44.3

       \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{2.0}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]

   15. Taylor expanded around -inf 9.3

       \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\left|im\right|}{\sqrt{\color{blue}{-2 \cdot re}}}\right)\]

   if -1.1214463413730795e+123 < re < -3.263660527236801e-218

   1. Initial program 40.6

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied flip-+ 40.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   4. Applied associate-*r/ 40.5

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   5. Applied sqrt-div 40.6

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   6. Simplified 29.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
   7. Using strategy rm

   8. Applied sqrt-prod 29.7

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0} \cdot \sqrt{im \cdot im + 0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

   9. Simplified 18.1

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0} \cdot \color{blue}{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
   10. Using strategy rm

   11. Applied *-un-lft-identity 18.1

       \[\leadsto 0.5 \cdot \frac{\sqrt{2.0} \cdot \left|im\right|}{\sqrt{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}}\]

   12. Applied sqrt-prod 18.1

       \[\leadsto 0.5 \cdot \frac{\sqrt{2.0} \cdot \left|im\right|}{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   13. Applied times-frac 18.1

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2.0}}{\sqrt{1}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\]

   14. Simplified 18.1

       \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{2.0}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]
   15. Using strategy rm

   16. Applied add-sqr-sqrt 18.1

       \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{\sqrt{2.0} \cdot \sqrt{2.0}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]

   17. Applied sqrt-prod 18.2

       \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{\sqrt{2.0}} \cdot \sqrt{\sqrt{2.0}}\right)} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]

   18. Applied associate-*l* 18.2

       \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{2.0}} \cdot \left(\sqrt{\sqrt{2.0}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\right)}\]

   if -3.263660527236801e-218 < re < 2.4935750622148784e-246

   1. Initial program 29.9

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

   2. Taylor expanded around 0 32.8

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(re + im\right)}}\]

   if 2.4935750622148784e-246 < re < 9.701163352497263e+98

   1. Initial program 18.8

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

   if 9.701163352497263e+98 < re

   1. Initial program 49.8

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

   2. Taylor expanded around inf 10.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1214463413730795 \cdot 10^{+123}:\\ \;\;\;\;\left(\sqrt{2.0} \cdot \frac{\left|im\right|}{\sqrt{re \cdot -2}}\right) \cdot 0.5\\ \mathbf{elif}\;re \le -3.263660527236801 \cdot 10^{-218}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{\sqrt{2.0}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right) \cdot \sqrt{\sqrt{2.0}}\right)\\ \mathbf{elif}\;re \le 2.4935750622148784 \cdot 10^{-246}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + im\right) \cdot 2.0}\\ \mathbf{elif}\;re \le 9.701163352497263 \cdot 10^{+98}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))