math.sqrt on complex, real part

Average Error: 38.7 → 24.7

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.37964942532106859 \cdot 10^{83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{-2 \cdot re}{im}}}\\ \mathbf{elif}\;re \le -283.23902669347274:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le -1.448710066223221 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{e^{\log \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}{im}}}\\ \mathbf{elif}\;re \le -1.984730296439969 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le 1.573305619337669 \cdot 10^{-228}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{elif}\;re \le 3.9315901045752092 \cdot 10^{-191}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot im}}{\sqrt{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le 5.32364720038125515 \cdot 10^{39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r290076 = 0.5;
        double r290077 = 2.0;
        double r290078 = re;
        double r290079 = r290078 * r290078;
        double r290080 = im;
        double r290081 = r290080 * r290080;
        double r290082 = r290079 + r290081;
        double r290083 = sqrt(r290082);
        double r290084 = r290083 + r290078;
        double r290085 = r290077 * r290084;
        double r290086 = sqrt(r290085);
        double r290087 = r290076 * r290086;
        return r290087;
}

↓

double f(double re, double im) {
        double r290088 = re;
        double r290089 = -4.3796494253210686e+83;
        bool r290090 = r290088 <= r290089;
        double r290091 = 0.5;
        double r290092 = 2.0;
        double r290093 = im;
        double r290094 = -2.0;
        double r290095 = r290094 * r290088;
        double r290096 = r290095 / r290093;
        double r290097 = r290093 / r290096;
        double r290098 = r290092 * r290097;
        double r290099 = sqrt(r290098);
        double r290100 = r290091 * r290099;
        double r290101 = -283.23902669347274;
        bool r290102 = r290088 <= r290101;
        double r290103 = r290093 - r290088;
        double r290104 = r290103 / r290093;
        double r290105 = r290093 / r290104;
        double r290106 = r290092 * r290105;
        double r290107 = sqrt(r290106);
        double r290108 = r290091 * r290107;
        double r290109 = -1.448710066223221e-194;
        bool r290110 = r290088 <= r290109;
        double r290111 = r290088 * r290088;
        double r290112 = r290093 * r290093;
        double r290113 = r290111 + r290112;
        double r290114 = sqrt(r290113);
        double r290115 = r290114 - r290088;
        double r290116 = log(r290115);
        double r290117 = exp(r290116);
        double r290118 = r290117 / r290093;
        double r290119 = r290093 / r290118;
        double r290120 = r290092 * r290119;
        double r290121 = sqrt(r290120);
        double r290122 = r290091 * r290121;
        double r290123 = -1.984730296439969e-262;
        bool r290124 = r290088 <= r290123;
        double r290125 = 1.573305619337669e-228;
        bool r290126 = r290088 <= r290125;
        double r290127 = r290114 + r290088;
        double r290128 = r290092 * r290127;
        double r290129 = sqrt(r290128);
        double r290130 = r290091 * r290129;
        double r290131 = 3.931590104575209e-191;
        bool r290132 = r290088 <= r290131;
        double r290133 = r290092 * r290093;
        double r290134 = sqrt(r290133);
        double r290135 = sqrt(r290104);
        double r290136 = r290134 / r290135;
        double r290137 = r290091 * r290136;
        double r290138 = 5.323647200381255e+39;
        bool r290139 = r290088 <= r290138;
        double r290140 = 2.0;
        double r290141 = r290140 * r290088;
        double r290142 = r290092 * r290141;
        double r290143 = sqrt(r290142);
        double r290144 = r290091 * r290143;
        double r290145 = r290139 ? r290130 : r290144;
        double r290146 = r290132 ? r290137 : r290145;
        double r290147 = r290126 ? r290130 : r290146;
        double r290148 = r290124 ? r290108 : r290147;
        double r290149 = r290110 ? r290122 : r290148;
        double r290150 = r290102 ? r290108 : r290149;
        double r290151 = r290090 ? r290100 : r290150;
        return r290151;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.7
Target	33.7
Herbie	24.7

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

Derivation

1. Split input into 6 regimes

2. if re < -4.3796494253210686e+83

   1. Initial program 60.2

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

   3. Applied flip-+ 60.2

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

   4. Simplified 44.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
   5. Using strategy rm

   6. Applied unpow2 44.4

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

   7. Applied associate-/l* 44.1

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

   8. Taylor expanded around -inf 26.4

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

   if -4.3796494253210686e+83 < re < -283.23902669347274 or -1.448710066223221e-194 < re < -1.984730296439969e-262

   1. Initial program 41.0

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

   3. Applied flip-+ 40.7

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

   4. Simplified 31.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
   5. Using strategy rm

   6. Applied unpow2 31.4

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

   7. Applied associate-/l* 30.4

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

   8. Taylor expanded around 0 39.2

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

   if -283.23902669347274 < re < -1.448710066223221e-194

   1. Initial program 37.6

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

   3. Applied flip-+ 37.5

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

   4. Simplified 31.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
   5. Using strategy rm

   6. Applied unpow2 31.3

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

   7. Applied associate-/l* 27.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{im}}}}\]
   8. Using strategy rm

   9. Applied add-exp-log 29.4

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

   if -1.984730296439969e-262 < re < 1.573305619337669e-228 or 3.931590104575209e-191 < re < 5.323647200381255e+39

   1. Initial program 22.3

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

   if 1.573305619337669e-228 < re < 3.931590104575209e-191

   1. Initial program 30.3

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

   3. Applied flip-+ 32.4

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

   4. Simplified 32.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
   5. Using strategy rm

   6. Applied unpow2 32.4

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

   7. Applied associate-/l* 32.3

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

   8. Taylor expanded around 0 34.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\color{blue}{im} - re}{im}}}\]
   9. Using strategy rm

   10. Applied associate-*r/ 34.8

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

   11. Applied sqrt-div 35.0

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

   if 5.323647200381255e+39 < re

   1. Initial program 43.9

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

   3. Applied flip-+ 61.5

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

   4. Simplified 60.8

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

   5. Taylor expanded around 0 13.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 24.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.37964942532106859 \cdot 10^{83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{-2 \cdot re}{im}}}\\ \mathbf{elif}\;re \le -283.23902669347274:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le -1.448710066223221 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{e^{\log \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}{im}}}\\ \mathbf{elif}\;re \le -1.984730296439969 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le 1.573305619337669 \cdot 10^{-228}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{elif}\;re \le 3.9315901045752092 \cdot 10^{-191}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot im}}{\sqrt{\frac{im - re}{im}}}\\ \mathbf{elif}\;re \le 5.32364720038125515 \cdot 10^{39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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)))))