math.log/2 on complex, real part

Average Error: 31.6 → 18.1

Time: 9.1s

Precision: 64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.97688285622801022 \cdot 10^{55}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -6.632251258880961 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)}^{3} + {\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}^{3}}{\log base \cdot \log base + 0.0 \cdot 0.0}}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) + \left(\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) - \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)\right)}\\ \mathbf{elif}\;re \le -5.9140865977584641 \cdot 10^{-298}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 9.527329133926756 \cdot 10^{137}:\\ \;\;\;\;\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r42258 = re;
        double r42259 = r42258 * r42258;
        double r42260 = im;
        double r42261 = r42260 * r42260;
        double r42262 = r42259 + r42261;
        double r42263 = sqrt(r42262);
        double r42264 = log(r42263);
        double r42265 = base;
        double r42266 = log(r42265);
        double r42267 = r42264 * r42266;
        double r42268 = atan2(r42260, r42258);
        double r42269 = 0.0;
        double r42270 = r42268 * r42269;
        double r42271 = r42267 + r42270;
        double r42272 = r42266 * r42266;
        double r42273 = r42269 * r42269;
        double r42274 = r42272 + r42273;
        double r42275 = r42271 / r42274;
        return r42275;
}

↓

double f(double re, double im, double base) {
        double r42276 = re;
        double r42277 = -1.9768828562280102e+55;
        bool r42278 = r42276 <= r42277;
        double r42279 = -1.0;
        double r42280 = r42279 * r42276;
        double r42281 = log(r42280);
        double r42282 = base;
        double r42283 = log(r42282);
        double r42284 = r42281 * r42283;
        double r42285 = im;
        double r42286 = atan2(r42285, r42276);
        double r42287 = 0.0;
        double r42288 = r42286 * r42287;
        double r42289 = r42284 + r42288;
        double r42290 = r42283 * r42283;
        double r42291 = r42287 * r42287;
        double r42292 = r42290 + r42291;
        double r42293 = sqrt(r42292);
        double r42294 = r42289 / r42293;
        double r42295 = r42294 / r42293;
        double r42296 = -6.632251258880961e-245;
        bool r42297 = r42276 <= r42296;
        double r42298 = r42276 * r42276;
        double r42299 = r42285 * r42285;
        double r42300 = r42298 + r42299;
        double r42301 = sqrt(r42300);
        double r42302 = log(r42301);
        double r42303 = r42302 * r42283;
        double r42304 = 3.0;
        double r42305 = pow(r42303, r42304);
        double r42306 = pow(r42288, r42304);
        double r42307 = r42305 + r42306;
        double r42308 = r42307 / r42292;
        double r42309 = r42303 * r42303;
        double r42310 = r42288 * r42288;
        double r42311 = r42303 * r42288;
        double r42312 = r42310 - r42311;
        double r42313 = r42309 + r42312;
        double r42314 = r42308 / r42313;
        double r42315 = -5.914086597758464e-298;
        bool r42316 = r42276 <= r42315;
        double r42317 = log(r42285);
        double r42318 = r42317 / r42283;
        double r42319 = 9.527329133926756e+137;
        bool r42320 = r42276 <= r42319;
        double r42321 = r42303 + r42288;
        double r42322 = 1.0;
        double r42323 = 2.0;
        double r42324 = cbrt(r42282);
        double r42325 = log(r42324);
        double r42326 = r42323 * r42325;
        double r42327 = r42283 * r42326;
        double r42328 = r42283 * r42325;
        double r42329 = r42327 + r42328;
        double r42330 = r42329 + r42291;
        double r42331 = r42322 / r42330;
        double r42332 = r42321 * r42331;
        double r42333 = log(r42276);
        double r42334 = r42333 * r42283;
        double r42335 = r42334 + r42288;
        double r42336 = r42335 / r42293;
        double r42337 = r42336 / r42293;
        double r42338 = r42320 ? r42332 : r42337;
        double r42339 = r42316 ? r42318 : r42338;
        double r42340 = r42297 ? r42314 : r42339;
        double r42341 = r42278 ? r42295 : r42340;
        return r42341;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -1.9768828562280102e+55

   1. Initial program 44.6

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 44.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   4. Applied associate-/r* 44.6

      \[\leadsto \color{blue}{\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   5. Taylor expanded around -inf 11.5

      \[\leadsto \frac{\frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   if -1.9768828562280102e+55 < re < -6.632251258880961e-245

   1. Initial program 20.6

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied div-inv 20.6

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
   4. Using strategy rm

   5. Applied flip3-+ 20.7

      \[\leadsto \color{blue}{\frac{{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)}^{3} + {\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}^{3}}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) + \left(\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) - \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)\right)}} \cdot \frac{1}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

   6. Applied associate-*l/ 20.7

      \[\leadsto \color{blue}{\frac{\left({\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)}^{3} + {\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}^{3}\right) \cdot \frac{1}{\log base \cdot \log base + 0.0 \cdot 0.0}}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) + \left(\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) - \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)\right)}}\]

   7. Simplified 20.7

      \[\leadsto \frac{\color{blue}{\frac{{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)}^{3} + {\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}^{3}}{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) + \left(\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) - \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)\right)}\]

   if -6.632251258880961e-245 < re < -5.914086597758464e-298

   1. Initial program 30.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

   2. Taylor expanded around 0 35.7

      \[\leadsto \color{blue}{\frac{\log im}{\log base}}\]

   if -5.914086597758464e-298 < re < 9.527329133926756e+137

   1. Initial program 21.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied div-inv 21.7

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 21.7

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\log base \cdot \log \color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)} + 0.0 \cdot 0.0}\]

   6. Applied log-prod 21.7

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\log base \cdot \color{blue}{\left(\log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

   7. Applied distribute-lft-in 21.8

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\color{blue}{\left(\log base \cdot \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

   8. Simplified 21.8

      \[\leadsto \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\left(\color{blue}{\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right)} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

   if 9.527329133926756e+137 < re

   1. Initial program 59.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 59.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   4. Applied associate-/r* 59.4

      \[\leadsto \color{blue}{\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   5. Taylor expanded around inf 7.8

      \[\leadsto \frac{\frac{\log \color{blue}{re} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.97688285622801022 \cdot 10^{55}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -6.632251258880961 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)}^{3} + {\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}^{3}}{\log base \cdot \log base + 0.0 \cdot 0.0}}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) + \left(\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) - \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)\right)}\\ \mathbf{elif}\;re \le -5.9140865977584641 \cdot 10^{-298}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 9.527329133926756 \cdot 10^{137}:\\ \;\;\;\;\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))