Octave 3.8, jcobi/1

Average Error: 16.3 → 5.9

Time: 20.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 307451316.10967922210693359375:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta}{2 + \left(\alpha + \beta\right)}\right)\right) - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r88545 = beta;
        double r88546 = alpha;
        double r88547 = r88545 - r88546;
        double r88548 = r88546 + r88545;
        double r88549 = 2.0;
        double r88550 = r88548 + r88549;
        double r88551 = r88547 / r88550;
        double r88552 = 1.0;
        double r88553 = r88551 + r88552;
        double r88554 = r88553 / r88549;
        return r88554;
}

↓

double f(double alpha, double beta) {
        double r88555 = alpha;
        double r88556 = 307451316.1096792;
        bool r88557 = r88555 <= r88556;
        double r88558 = beta;
        double r88559 = 2.0;
        double r88560 = r88555 + r88558;
        double r88561 = r88559 + r88560;
        double r88562 = r88558 / r88561;
        double r88563 = expm1(r88562);
        double r88564 = log1p(r88563);
        double r88565 = r88555 / r88561;
        double r88566 = 1.0;
        double r88567 = r88565 - r88566;
        double r88568 = r88564 - r88567;
        double r88569 = r88568 / r88559;
        double r88570 = cbrt(r88558);
        double r88571 = r88560 + r88559;
        double r88572 = 0.3333333333333333;
        double r88573 = pow(r88571, r88572);
        double r88574 = r88570 / r88573;
        double r88575 = 3.0;
        double r88576 = pow(r88574, r88575);
        double r88577 = 4.0;
        double r88578 = 2.0;
        double r88579 = pow(r88555, r88578);
        double r88580 = r88577 / r88579;
        double r88581 = 8.0;
        double r88582 = pow(r88555, r88575);
        double r88583 = r88581 / r88582;
        double r88584 = r88580 - r88583;
        double r88585 = r88559 / r88555;
        double r88586 = r88584 - r88585;
        double r88587 = r88576 - r88586;
        double r88588 = r88587 / r88559;
        double r88589 = r88557 ? r88569 : r88588;
        return r88589;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 307451316.1096792

   1. Initial program 0.1

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

   3. Applied div-sub 0.1

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

   4. Applied associate-+l- 0.1

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 0.2

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

   7. Applied add-cube-cbrt 0.4

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

   8. Applied add-cube-cbrt 0.2

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

   9. Applied times-frac 0.2

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

   10. Applied prod-diff 0.2

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}}, \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}}, -\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}, \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}, \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)\right)}}{2}\]

   11. Simplified 0.1

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

   12. Simplified 0.1

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + \color{blue}{0}}{2}\]
   13. Using strategy rm

   14. Applied pow1/3 1.8

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]
   15. Using strategy rm

   16. Applied log1p-expm1-u 1.8

       \[\leadsto \frac{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3}\right)\right)} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   17. Simplified 0.1

       \[\leadsto \frac{\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\beta}{2 + \left(\alpha + \beta\right)}\right)}\right) - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   if 307451316.1096792 < alpha

   1. Initial program 49.9

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

   3. Applied div-sub 49.9

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

   4. Applied associate-+l- 48.3

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 48.4

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

   7. Applied add-cube-cbrt 48.5

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

   8. Applied add-cube-cbrt 48.4

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

   9. Applied times-frac 48.4

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

   10. Applied prod-diff 48.4

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}}, \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}}, -\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}, \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}, \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)\right)}}{2}\]

   11. Simplified 48.4

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

   12. Simplified 48.4

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + \color{blue}{0}}{2}\]
   13. Using strategy rm

   14. Applied pow1/3 49.3

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   15. Taylor expanded around inf 17.8

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}\right) + 0}{2}\]

   16. Simplified 17.8

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \color{blue}{\left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}\right) + 0}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 307451316.10967922210693359375:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\beta}{2 + \left(\alpha + \beta\right)}\right)\right) - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))