math.sqrt on complex, real part

Average Error: 38.6 → 12.1

Time: 23.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.805185896538098623594323820769745101779 \cdot 10^{55}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-1}{re} \cdot \frac{1}{{\left(\frac{-1}{im}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)}^{\frac{1}{2}}\\ \end{array}\]

C

double f(double re, double im) {
        double r201089 = 0.5;
        double r201090 = 2.0;
        double r201091 = re;
        double r201092 = r201091 * r201091;
        double r201093 = im;
        double r201094 = r201093 * r201093;
        double r201095 = r201092 + r201094;
        double r201096 = sqrt(r201095);
        double r201097 = r201096 + r201091;
        double r201098 = r201090 * r201097;
        double r201099 = sqrt(r201098);
        double r201100 = r201089 * r201099;
        return r201100;
}

↓

double f(double re, double im) {
        double r201101 = re;
        double r201102 = -3.8051858965380986e+55;
        bool r201103 = r201101 <= r201102;
        double r201104 = 0.5;
        double r201105 = -1.0;
        double r201106 = r201105 / r201101;
        double r201107 = 1.0;
        double r201108 = im;
        double r201109 = r201105 / r201108;
        double r201110 = 2.0;
        double r201111 = pow(r201109, r201110);
        double r201112 = r201107 / r201111;
        double r201113 = r201106 * r201112;
        double r201114 = sqrt(r201113);
        double r201115 = r201104 * r201114;
        double r201116 = hypot(r201101, r201108);
        double r201117 = r201101 + r201116;
        double r201118 = 2.0;
        double r201119 = r201117 * r201118;
        double r201120 = 0.5;
        double r201121 = pow(r201119, r201120);
        double r201122 = r201104 * r201121;
        double r201123 = r201103 ? r201115 : r201122;
        return r201123;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.6
Target	33.7
Herbie	12.1

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

Derivation

1. Split input into 2 regimes

2. if re < -3.8051858965380986e+55

   1. Initial program 58.8

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

   2. Simplified 39.7

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

   4. Applied sqrt-prod 39.8

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

   6. Applied add-sqr-sqrt 39.8

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

   7. Applied sqrt-prod 39.8

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

   8. Applied associate-*r* 39.8

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

   10. Applied pow1/2 39.8

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

   11. Applied pow1/2 39.8

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

   12. Applied pow1/2 39.8

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

   13. Applied pow-prod-down 39.8

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

   14. Applied pow-prod-down 39.8

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

   15. Simplified 39.7

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

   16. Taylor expanded around -inf 39.7

       \[\leadsto 0.5 \cdot \color{blue}{e^{\frac{1}{2} \cdot \left(\left(\log \left(\frac{-1}{re}\right) + \log 1\right) - 2 \cdot \log \left(\frac{-1}{im}\right)\right)}}\]

   17. Simplified 33.4

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

   if -3.8051858965380986e+55 < re

   1. Initial program 33.3

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

   2. Simplified 6.5

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

   4. Applied sqrt-prod 6.9

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

   6. Applied add-sqr-sqrt 6.9

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

   7. Applied sqrt-prod 7.0

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

   8. Applied associate-*r* 6.9

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

   10. Applied pow1/2 6.9

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

   11. Applied pow1/2 6.9

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

   12. Applied pow1/2 6.9

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

   13. Applied pow-prod-down 6.7

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

   14. Applied pow-prod-down 6.9

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

   15. Simplified 6.5

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

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.805185896538098623594323820769745101779 \cdot 10^{55}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-1}{re} \cdot \frac{1}{{\left(\frac{-1}{im}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))))