Average Error: 24.4 → 12.6
Time: 37.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \sqrt[3]{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}, \left(\frac{\frac{\sqrt[3]{\sqrt[3]{\beta - \alpha}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\sqrt[3]{\sqrt[3]{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}} \cdot \left(\alpha + \beta\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta - \alpha}} \cdot \sqrt[3]{\sqrt[3]{\beta - \alpha}}}{\sqrt[3]{\sqrt[3]{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}} \cdot \sqrt[3]{\sqrt[3]{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}}, 1\right)}{2}\]
\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}
↓
\frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \sqrt[3]{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}, \left(\frac{\frac{\sqrt[3]{\sqrt[3]{\beta - \alpha}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\sqrt[3]{\sqrt[3]{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}} \cdot \left(\alpha + \beta\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta - \alpha}} \cdot \sqrt[3]{\sqrt[3]{\beta - \alpha}}}{\sqrt[3]{\sqrt[3]{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}} \cdot \sqrt[3]{\sqrt[3]{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}}, 1\right)}{2}
double f(double alpha, double beta, double i) {
        double r5973775 = alpha;
        double r5973776 = beta;
        double r5973777 = r5973775 + r5973776;
        double r5973778 = r5973776 - r5973775;
        double r5973779 = r5973777 * r5973778;
        double r5973780 = 2.0;
        double r5973781 = i;
        double r5973782 = r5973780 * r5973781;
        double r5973783 = r5973777 + r5973782;
        double r5973784 = r5973779 / r5973783;
        double r5973785 = r5973783 + r5973780;
        double r5973786 = r5973784 / r5973785;
        double r5973787 = 1.0;
        double r5973788 = r5973786 + r5973787;
        double r5973789 = r5973788 / r5973780;
        return r5973789;
}
↓
double f(double alpha, double beta, double i) {
        double r5973790 = beta;
        double r5973791 = alpha;
        double r5973792 = r5973790 - r5973791;
        double r5973793 = cbrt(r5973792);
        double r5973794 = r5973793 * r5973793;
        double r5973795 = 2.0;
        double r5973796 = i;
        double r5973797 = r5973791 + r5973790;
        double r5973798 = fma(r5973795, r5973796, r5973797);
        double r5973799 = r5973795 + r5973798;
        double r5973800 = cbrt(r5973799);
        double r5973801 = r5973800 * r5973800;
        double r5973802 = r5973794 / r5973801;
        double r5973803 = cbrt(r5973793);
        double r5973804 = r5973803 / r5973798;
        double r5973805 = cbrt(r5973800);
        double r5973806 = r5973804 / r5973805;
        double r5973807 = r5973806 * r5973797;
        double r5973808 = r5973803 * r5973803;
        double r5973809 = r5973805 * r5973805;
        double r5973810 = r5973808 / r5973809;
        double r5973811 = r5973807 * r5973810;
        double r5973812 = 1.0;
        double r5973813 = fma(r5973802, r5973811, r5973812);
        double r5973814 = r5973813 / r5973795;
        return r5973814;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 24.4

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

  2. Simplified12.7

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

  4. Applied fma-udef12.6

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

  6. Applied add-cube-cbrt12.8

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

  7. Applied *-un-lft-identity12.8

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

  8. Applied add-cube-cbrt12.6

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

  9. Applied times-frac12.6

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

  10. Applied times-frac12.6

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

  11. Applied associate-*l*12.6

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

  13. Applied fma-def12.6

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

  15. Applied add-cube-cbrt12.8

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

  16. Applied *-un-lft-identity12.8

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

  17. Applied add-cube-cbrt12.6

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

  18. Applied times-frac12.7

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

  19. Applied times-frac12.6

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

  20. Applied associate-*l*12.6

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

  21. Final simplification12.6

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))