math.sqrt on complex, real part

Average Error: 38.6 → 12.0

Time: 4.4s

Precision: 64

Math

\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.3362437626551582 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{elif}\;re \le -2.8979168285597056 \cdot 10^{45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, {\left(-1 \cdot re\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}, \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \mathbf{elif}\;re \le -3.39459795979939742 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r220337 = 0.5;
        double r220338 = 2.0;
        double r220339 = re;
        double r220340 = r220339 * r220339;
        double r220341 = im;
        double r220342 = r220341 * r220341;
        double r220343 = r220340 + r220342;
        double r220344 = sqrt(r220343);
        double r220345 = r220344 + r220339;
        double r220346 = r220338 * r220345;
        double r220347 = sqrt(r220346);
        double r220348 = r220337 * r220347;
        return r220348;
}

↓

double f(double re, double im) {
        double r220349 = re;
        double r220350 = -4.336243762655158e+86;
        bool r220351 = r220349 <= r220350;
        double r220352 = 0.5;
        double r220353 = 2.0;
        double r220354 = im;
        double r220355 = 2.0;
        double r220356 = pow(r220354, r220355);
        double r220357 = hypot(r220349, r220354);
        double r220358 = r220357 - r220349;
        double r220359 = r220356 / r220358;
        double r220360 = r220353 * r220359;
        double r220361 = sqrt(r220360);
        double r220362 = r220352 * r220361;
        double r220363 = -2.8979168285597056e+45;
        bool r220364 = r220349 <= r220363;
        double r220365 = 1.0;
        double r220366 = cbrt(r220349);
        double r220367 = r220366 * r220366;
        double r220368 = -1.0;
        double r220369 = r220368 * r220349;
        double r220370 = 0.3333333333333333;
        double r220371 = pow(r220369, r220370);
        double r220372 = cbrt(r220368);
        double r220373 = r220371 * r220372;
        double r220374 = fma(r220367, r220373, r220357);
        double r220375 = r220365 * r220374;
        double r220376 = r220353 * r220375;
        double r220377 = sqrt(r220376);
        double r220378 = r220352 * r220377;
        double r220379 = -3.3945979597993974e-89;
        bool r220380 = r220349 <= r220379;
        double r220381 = r220349 + r220357;
        double r220382 = r220365 * r220381;
        double r220383 = r220353 * r220382;
        double r220384 = sqrt(r220383);
        double r220385 = r220352 * r220384;
        double r220386 = r220380 ? r220362 : r220385;
        double r220387 = r220364 ? r220378 : r220386;
        double r220388 = r220351 ? r220362 : r220387;
        return r220388;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.6
Target	33.8
Herbie	12.0

\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if re < -4.336243762655158e+86 or -2.8979168285597056e+45 < re < -3.3945979597993974e-89

   1. Initial program 53.6

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

   3. Applied flip-+ 53.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   4. Simplified 39.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]

   5. Simplified 31.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{\mathsf{hypot}\left(re, im\right) - re}}}\]

   if -4.336243762655158e+86 < re < -2.8979168285597056e+45

   1. Initial program 49.1

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

   3. Applied *-un-lft-identity 49.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + \color{blue}{1 \cdot re}\right)}\]

   4. Applied *-un-lft-identity 49.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \sqrt{re \cdot re + im \cdot im}} + 1 \cdot re\right)}\]

   5. Applied distribute-lft-out 49.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)}}\]

   6. Simplified 33.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 37.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}} + \mathsf{hypot}\left(re, im\right)\right)\right)}\]

   9. Applied fma-def 37.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right)}\right)}\]

   10. Taylor expanded around -inf 36.8

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, \color{blue}{{\left(-1 \cdot re\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}}, \mathsf{hypot}\left(re, im\right)\right)\right)}\]

   if -3.3945979597993974e-89 < re

   1. Initial program 31.7

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

   3. Applied *-un-lft-identity 31.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + \color{blue}{1 \cdot re}\right)}\]

   4. Applied *-un-lft-identity 31.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \sqrt{re \cdot re + im \cdot im}} + 1 \cdot re\right)}\]

   5. Applied distribute-lft-out 31.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)}}\]

   6. Simplified 2.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.3362437626551582 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{elif}\;re \le -2.8979168285597056 \cdot 10^{45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, {\left(-1 \cdot re\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}, \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \mathbf{elif}\;re \le -3.39459795979939742 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))