Average Error: 23.6 → 11.3
Time: 14.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.237473251530060772390463749885445358102 \cdot 10^{106}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) - 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]
\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 6.237473251530060772390463749885445358102 \cdot 10^{106}:\\
\;\;\;\;\frac{\log \left(e^{\frac{\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) - 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}\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}
double f(double alpha, double beta, double i) {
        double r104326 = alpha;
        double r104327 = beta;
        double r104328 = r104326 + r104327;
        double r104329 = r104327 - r104326;
        double r104330 = r104328 * r104329;
        double r104331 = 2.0;
        double r104332 = i;
        double r104333 = r104331 * r104332;
        double r104334 = r104328 + r104333;
        double r104335 = r104330 / r104334;
        double r104336 = r104334 + r104331;
        double r104337 = r104335 / r104336;
        double r104338 = 1.0;
        double r104339 = r104337 + r104338;
        double r104340 = r104339 / r104331;
        return r104340;
}
↓
double f(double alpha, double beta, double i) {
        double r104341 = alpha;
        double r104342 = 6.237473251530061e+106;
        bool r104343 = r104341 <= r104342;
        double r104344 = beta;
        double r104345 = r104341 + r104344;
        double r104346 = i;
        double r104347 = 2.0;
        double r104348 = fma(r104346, r104347, r104345);
        double r104349 = r104345 / r104348;
        double r104350 = r104344 - r104341;
        double r104351 = r104347 * r104346;
        double r104352 = r104345 - r104351;
        double r104353 = r104350 / r104352;
        double r104354 = r104349 * r104353;
        double r104355 = r104345 + r104351;
        double r104356 = r104355 + r104347;
        double r104357 = sqrt(r104356);
        double r104358 = r104354 / r104357;
        double r104359 = r104352 / r104357;
        double r104360 = r104358 * r104359;
        double r104361 = 1.0;
        double r104362 = r104360 + r104361;
        double r104363 = exp(r104362);
        double r104364 = log(r104363);
        double r104365 = r104364 / r104347;
        double r104366 = 1.0;
        double r104367 = r104366 / r104341;
        double r104368 = 8.0;
        double r104369 = 3.0;
        double r104370 = pow(r104341, r104369);
        double r104371 = r104366 / r104370;
        double r104372 = r104368 * r104371;
        double r104373 = 4.0;
        double r104374 = 2.0;
        double r104375 = pow(r104341, r104374);
        double r104376 = r104366 / r104375;
        double r104377 = r104373 * r104376;
        double r104378 = r104372 - r104377;
        double r104379 = fma(r104347, r104367, r104378);
        double r104380 = r104379 / r104347;
        double r104381 = r104343 ? r104365 : r104380;
        return r104381;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 6.237473251530061e+106

    1. Initial program 13.9

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

    3. Applied add-sqr-sqrt13.9

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

    4. Applied flip-+16.2

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

    5. Applied associate-/r/16.2

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

    6. Applied times-frac16.2

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

    7. Simplified3.3

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

    9. Applied add-log-exp3.3

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

    10. Applied add-log-exp3.4

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

    11. Applied sum-log3.4

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

    12. Simplified3.4

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

    if 6.237473251530061e+106 < alpha

    1. Initial program 60.1

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

    2. Taylor expanded around inf 41.1

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

    3. Simplified41.1

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.237473251530060772390463749885445358102 \cdot 10^{106}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) - 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot i}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +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))