Octave 3.8, jcobi/2

Average Error: 23.7 → 11.6

Time: 19.2s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.0312336573895782 \cdot 10^{+150}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\beta + \alpha, \frac{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right)}}{2.0 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}, 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r1836340 = alpha;
        double r1836341 = beta;
        double r1836342 = r1836340 + r1836341;
        double r1836343 = r1836341 - r1836340;
        double r1836344 = r1836342 * r1836343;
        double r1836345 = 2.0;
        double r1836346 = i;
        double r1836347 = r1836345 * r1836346;
        double r1836348 = r1836342 + r1836347;
        double r1836349 = r1836344 / r1836348;
        double r1836350 = 2.0;
        double r1836351 = r1836348 + r1836350;
        double r1836352 = r1836349 / r1836351;
        double r1836353 = 1.0;
        double r1836354 = r1836352 + r1836353;
        double r1836355 = r1836354 / r1836350;
        return r1836355;
}

↓

double f(double alpha, double beta, double i) {
        double r1836356 = alpha;
        double r1836357 = 1.0312336573895782e+150;
        bool r1836358 = r1836356 <= r1836357;
        double r1836359 = beta;
        double r1836360 = r1836359 + r1836356;
        double r1836361 = r1836359 - r1836356;
        double r1836362 = 2.0;
        double r1836363 = i;
        double r1836364 = fma(r1836362, r1836363, r1836360);
        double r1836365 = r1836361 / r1836364;
        double r1836366 = r1836365 * r1836365;
        double r1836367 = r1836365 * r1836366;
        double r1836368 = cbrt(r1836367);
        double r1836369 = 2.0;
        double r1836370 = r1836369 + r1836364;
        double r1836371 = r1836368 / r1836370;
        double r1836372 = 1.0;
        double r1836373 = fma(r1836360, r1836371, r1836372);
        double r1836374 = log(r1836373);
        double r1836375 = exp(r1836374);
        double r1836376 = r1836375 / r1836369;
        double r1836377 = 8.0;
        double r1836378 = r1836356 * r1836356;
        double r1836379 = r1836377 / r1836378;
        double r1836380 = r1836379 / r1836356;
        double r1836381 = 4.0;
        double r1836382 = r1836381 / r1836378;
        double r1836383 = r1836380 - r1836382;
        double r1836384 = r1836369 / r1836356;
        double r1836385 = r1836383 + r1836384;
        double r1836386 = r1836385 / r1836369;
        double r1836387 = r1836358 ? r1836376 : r1836386;
        return r1836387;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.0312336573895782e+150

   1. Initial program 15.6

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Simplified 15.6

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)} + 1.0}{2.0}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 15.6

      \[\leadsto \frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}} + 1.0}{2.0}\]

   5. Applied times-frac 12.9

      \[\leadsto \frac{\color{blue}{\frac{\beta + \alpha}{1} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}} + 1.0}{2.0}\]

   6. Applied fma-def 12.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\beta + \alpha}{1}, \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}, 1.0\right)}}{2.0}\]
   7. Using strategy rm

   8. Applied add-exp-log 12.9

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\beta + \alpha}{1}, \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}, 1.0\right)\right)}}}{2.0}\]

   9. Simplified 12.9

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) \cdot \left(\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0\right)}, 1.0\right)\right)}}}{2.0}\]
   10. Using strategy rm

   11. Applied associate-/r* 5.5

       \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\alpha + \beta, \color{blue}{\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0}}, 1.0\right)\right)}}{2.0}\]
   12. Using strategy rm

   13. Applied add-cbrt-cube 5.5

       \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\alpha + \beta, \frac{\color{blue}{\sqrt[3]{\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0}, 1.0\right)\right)}}{2.0}\]

   if 1.0312336573895782e+150 < alpha

   1. Initial program 62.9

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Simplified 62.2

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)} + 1.0}{2.0}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 62.2

      \[\leadsto \frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}} + 1.0}{2.0}\]

   5. Applied times-frac 53.3

      \[\leadsto \frac{\color{blue}{\frac{\beta + \alpha}{1} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}} + 1.0}{2.0}\]

   6. Applied fma-def 53.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\beta + \alpha}{1}, \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}, 1.0\right)}}{2.0}\]
   7. Using strategy rm

   8. Applied add-exp-log 53.3

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\beta + \alpha}{1}, \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}, 1.0\right)\right)}}}{2.0}\]

   9. Simplified 53.3

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) \cdot \left(\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0\right)}, 1.0\right)\right)}}}{2.0}\]
   10. Using strategy rm

   11. Applied associate-/r* 48.7

       \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\alpha + \beta, \color{blue}{\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0}}, 1.0\right)\right)}}{2.0}\]

   12. Taylor expanded around -inf 41.3

       \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

   13. Simplified 41.3

       \[\leadsto \frac{\color{blue}{\left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.0312336573895782 \cdot 10^{+150}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\beta + \alpha, \frac{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right)}}{2.0 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}, 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))