Average Error: 16.1 → 16.1
Time: 17.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\frac{e^{\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\log \left(1 + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\right)\right)}^{3}}\right)}^{3}}\right)}^{3}}}}{2}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\frac{e^{\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\log \left(1 + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\right)\right)}^{3}}\right)}^{3}}\right)}^{3}}}}{2}
double f(double alpha, double beta) {
        double r112486 = beta;
        double r112487 = alpha;
        double r112488 = r112486 - r112487;
        double r112489 = r112487 + r112486;
        double r112490 = 2.0;
        double r112491 = r112489 + r112490;
        double r112492 = r112488 / r112491;
        double r112493 = 1.0;
        double r112494 = r112492 + r112493;
        double r112495 = r112494 / r112490;
        return r112495;
}
↓
double f(double alpha, double beta) {
        double r112496 = 1.0;
        double r112497 = beta;
        double r112498 = alpha;
        double r112499 = r112497 - r112498;
        double r112500 = 2.0;
        double r112501 = r112497 + r112498;
        double r112502 = r112500 + r112501;
        double r112503 = r112499 / r112502;
        double r112504 = r112496 + r112503;
        double r112505 = log(r112504);
        double r112506 = 3.0;
        double r112507 = pow(r112505, r112506);
        double r112508 = cbrt(r112507);
        double r112509 = pow(r112508, r112506);
        double r112510 = cbrt(r112509);
        double r112511 = pow(r112510, r112506);
        double r112512 = cbrt(r112511);
        double r112513 = exp(r112512);
        double r112514 = r112513 / r112500;
        return r112514;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.1

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

  2. Simplified16.1

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

  4. Applied add-exp-log16.1

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

  5. Simplified16.1

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

  7. Applied add-cbrt-cube16.1

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

  8. Simplified16.1

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

  10. Applied add-cbrt-cube16.1

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

  11. Simplified16.1

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

  13. Applied add-cbrt-cube16.1

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

  14. Simplified16.1

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

  15. Final simplification16.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))