math.log/2 on complex, real part

Average Error: 32.2 → 18.7

Time: 14.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.21796865557664327 \cdot 10^{125}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -2.3495244626294698 \cdot 10^{-161}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \left({\left(\log base\right)}^{2} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}\\ \mathbf{elif}\;re \le -9.88855865063237161 \cdot 10^{-243}:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(-re\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 6.97757305283699572 \cdot 10^{98}:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r42173 = re;
        double r42174 = r42173 * r42173;
        double r42175 = im;
        double r42176 = r42175 * r42175;
        double r42177 = r42174 + r42176;
        double r42178 = sqrt(r42177);
        double r42179 = log(r42178);
        double r42180 = base;
        double r42181 = log(r42180);
        double r42182 = r42179 * r42181;
        double r42183 = atan2(r42175, r42173);
        double r42184 = 0.0;
        double r42185 = r42183 * r42184;
        double r42186 = r42182 + r42185;
        double r42187 = r42181 * r42181;
        double r42188 = r42184 * r42184;
        double r42189 = r42187 + r42188;
        double r42190 = r42186 / r42189;
        return r42190;
}

↓

double f(double re, double im, double base) {
        double r42191 = re;
        double r42192 = -1.2179686555766433e+125;
        bool r42193 = r42191 <= r42192;
        double r42194 = -r42191;
        double r42195 = log(r42194);
        double r42196 = base;
        double r42197 = log(r42196);
        double r42198 = r42195 * r42197;
        double r42199 = im;
        double r42200 = atan2(r42199, r42191);
        double r42201 = 0.0;
        double r42202 = r42200 * r42201;
        double r42203 = r42198 + r42202;
        double r42204 = r42197 * r42197;
        double r42205 = r42201 * r42201;
        double r42206 = r42204 + r42205;
        double r42207 = r42203 / r42206;
        double r42208 = -2.3495244626294698e-161;
        bool r42209 = r42191 <= r42208;
        double r42210 = r42191 * r42191;
        double r42211 = r42199 * r42199;
        double r42212 = r42210 + r42211;
        double r42213 = sqrt(r42212);
        double r42214 = log(r42213);
        double r42215 = 2.0;
        double r42216 = pow(r42197, r42215);
        double r42217 = r42216 * r42214;
        double r42218 = r42214 * r42217;
        double r42219 = r42202 * r42202;
        double r42220 = r42218 - r42219;
        double r42221 = r42205 + r42216;
        double r42222 = r42214 * r42197;
        double r42223 = r42222 - r42202;
        double r42224 = r42221 * r42223;
        double r42225 = r42220 / r42224;
        double r42226 = -9.888558650632372e-243;
        bool r42227 = r42191 <= r42226;
        double r42228 = r42197 * r42195;
        double r42229 = r42202 + r42228;
        double r42230 = sqrt(r42221);
        double r42231 = r42229 / r42230;
        double r42232 = sqrt(r42206);
        double r42233 = r42231 / r42232;
        double r42234 = 6.977573052836996e+98;
        bool r42235 = r42191 <= r42234;
        double r42236 = r42197 * r42214;
        double r42237 = r42202 + r42236;
        double r42238 = r42237 / r42230;
        double r42239 = r42238 / r42232;
        double r42240 = log(r42191);
        double r42241 = r42240 / r42197;
        double r42242 = r42235 ? r42239 : r42241;
        double r42243 = r42227 ? r42233 : r42242;
        double r42244 = r42209 ? r42225 : r42243;
        double r42245 = r42193 ? r42207 : r42244;
        return r42245;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -1.2179686555766433e+125

   1. Initial program 56.7

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

   2. Taylor expanded around -inf 7.9

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

   3. Simplified 7.9

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

   if -1.2179686555766433e+125 < re < -2.3495244626294698e-161

   1. Initial program 16.5

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

   3. Applied flip-+ 16.5

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

   4. Applied associate-/l/ 16.5

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

   5. Simplified 16.5

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

   7. Applied associate-*l* 16.6

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

   8. Simplified 16.5

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

   if -2.3495244626294698e-161 < re < -9.888558650632372e-243

   1. Initial program 34.0

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

   3. Applied add-sqr-sqrt 34.0

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

   4. Applied associate-/r* 33.9

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

   5. Simplified 33.9

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

   6. Taylor expanded around -inf 48.1

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

   7. Simplified 48.1

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

   if -9.888558650632372e-243 < re < 6.977573052836996e+98

   1. Initial program 23.5

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

   3. Applied add-sqr-sqrt 23.5

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

   4. Applied associate-/r* 23.4

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

   5. Simplified 23.4

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

   if 6.977573052836996e+98 < re

   1. Initial program 51.1

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

   3. Applied add-sqr-sqrt 51.1

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

   4. Applied associate-/r* 51.1

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

   5. Simplified 51.1

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

   6. Taylor expanded around 0 9.6

      \[\leadsto \color{blue}{\frac{\log re}{\log base}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.21796865557664327 \cdot 10^{125}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -2.3495244626294698 \cdot 10^{-161}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \left({\left(\log base\right)}^{2} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}\\ \mathbf{elif}\;re \le -9.88855865063237161 \cdot 10^{-243}:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(-re\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 6.97757305283699572 \cdot 10^{98}:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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))))