Octave 3.8, jcobi/2

Average Error: 24.0 → 11.1

Time: 9.5s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.74962969817274088 \cdot 10^{176}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot i} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \le 8.4222384033691505 \cdot 10^{217}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \mathbf{elif}\;\alpha \le 2.724310899095564 \cdot 10^{225}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r93268 = alpha;
        double r93269 = beta;
        double r93270 = r93268 + r93269;
        double r93271 = r93269 - r93268;
        double r93272 = r93270 * r93271;
        double r93273 = 2.0;
        double r93274 = i;
        double r93275 = r93273 * r93274;
        double r93276 = r93270 + r93275;
        double r93277 = r93272 / r93276;
        double r93278 = r93276 + r93273;
        double r93279 = r93277 / r93278;
        double r93280 = 1.0;
        double r93281 = r93279 + r93280;
        double r93282 = r93281 / r93273;
        return r93282;
}

↓

double f(double alpha, double beta, double i) {
        double r93283 = alpha;
        double r93284 = 6.749629698172741e+176;
        bool r93285 = r93283 <= r93284;
        double r93286 = beta;
        double r93287 = r93283 + r93286;
        double r93288 = 1.0;
        double r93289 = r93287 / r93288;
        double r93290 = r93289 / r93288;
        double r93291 = r93286 - r93283;
        double r93292 = i;
        double r93293 = 2.0;
        double r93294 = fma(r93292, r93293, r93287);
        double r93295 = r93291 / r93294;
        double r93296 = r93293 * r93292;
        double r93297 = r93287 - r93296;
        double r93298 = r93295 / r93297;
        double r93299 = r93287 + r93296;
        double r93300 = r93299 + r93293;
        double r93301 = r93297 / r93300;
        double r93302 = r93298 * r93301;
        double r93303 = 1.0;
        double r93304 = fma(r93290, r93302, r93303);
        double r93305 = log(r93304);
        double r93306 = exp(r93305);
        double r93307 = r93306 / r93293;
        double r93308 = 8.42223840336915e+217;
        bool r93309 = r93283 <= r93308;
        double r93310 = r93288 / r93283;
        double r93311 = 8.0;
        double r93312 = 3.0;
        double r93313 = pow(r93283, r93312);
        double r93314 = r93288 / r93313;
        double r93315 = r93311 * r93314;
        double r93316 = 4.0;
        double r93317 = 2.0;
        double r93318 = pow(r93283, r93317);
        double r93319 = r93288 / r93318;
        double r93320 = r93316 * r93319;
        double r93321 = r93315 - r93320;
        double r93322 = fma(r93293, r93310, r93321);
        double r93323 = r93322 / r93293;
        double r93324 = 2.7243108990955635e+225;
        bool r93325 = r93283 <= r93324;
        double r93326 = sqrt(r93300);
        double r93327 = r93289 / r93326;
        double r93328 = r93291 / r93299;
        double r93329 = r93328 / r93326;
        double r93330 = fma(r93327, r93329, r93303);
        double r93331 = log(r93330);
        double r93332 = exp(r93331);
        double r93333 = r93332 / r93293;
        double r93334 = r93325 ? r93333 : r93323;
        double r93335 = r93309 ? r93323 : r93334;
        double r93336 = r93285 ? r93307 : r93335;
        return r93336;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 6.749629698172741e+176

   1. Initial program 17.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 17.2

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2}\]

   4. Applied *-un-lft-identity 17.2

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)} + 1}{2}\]

   5. Applied times-frac 6.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)} + 1}{2}\]

   6. Applied times-frac 6.0

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\]

   7. Applied fma-def 6.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 6.0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}}, 1\right)}{2}\]

   10. Applied flip-+ 17.5

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}{\left(\alpha + \beta\right) - 2 \cdot i}}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}, 1\right)}{2}\]

   11. Applied associate-/r/ 17.5

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\color{blue}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot i\right)}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}, 1\right)}{2}\]

   12. Applied times-frac 17.5

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \color{blue}{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{1} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, 1\right)}{2}\]

   13. Simplified 6.0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \color{blue}{\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot i}} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2}\]
   14. Using strategy rm

   15. Applied add-exp-log 6.1

       \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot i} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}}}{2}\]

   if 6.749629698172741e+176 < alpha < 8.42223840336915e+217 or 2.7243108990955635e+225 < alpha

   1. Initial program 64.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

   2. Taylor expanded around inf 40.8

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   3. Simplified 40.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}}{2}\]

   if 8.42223840336915e+217 < alpha < 2.7243108990955635e+225

   1. Initial program 64.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 64.0

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}} + 1}{2}\]

   4. Applied *-un-lft-identity 64.0

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\]

   5. Applied times-frac 45.7

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\]

   6. Applied times-frac 45.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}} + 1}{2}\]

   7. Applied fma-def 45.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, 1\right)}}{2}\]
   8. Using strategy rm

   9. Applied add-exp-log 46.4

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, 1\right)\right)}}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.74962969817274088 \cdot 10^{176}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot i} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \le 8.4222384033691505 \cdot 10^{217}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \mathbf{elif}\;\alpha \le 2.724310899095564 \cdot 10^{225}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))