Average Error: 23.9 → 11.8
Time: 16.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}} \cdot \frac{1}{\sqrt[3]{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}\]
\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{\frac{\frac{\beta}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}} \cdot \frac{1}{\sqrt[3]{\alpha + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - \left(\frac{\frac{\alpha}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} - 1\right)}{2}
double f(double alpha, double beta, double i) {
        double r148751 = alpha;
        double r148752 = beta;
        double r148753 = r148751 + r148752;
        double r148754 = r148752 - r148751;
        double r148755 = r148753 * r148754;
        double r148756 = 2.0;
        double r148757 = i;
        double r148758 = r148756 * r148757;
        double r148759 = r148753 + r148758;
        double r148760 = r148755 / r148759;
        double r148761 = r148759 + r148756;
        double r148762 = r148760 / r148761;
        double r148763 = 1.0;
        double r148764 = r148762 + r148763;
        double r148765 = r148764 / r148756;
        return r148765;
}
↓
double f(double alpha, double beta, double i) {
        double r148766 = beta;
        double r148767 = i;
        double r148768 = 2.0;
        double r148769 = alpha;
        double r148770 = r148769 + r148766;
        double r148771 = fma(r148767, r148768, r148770);
        double r148772 = cbrt(r148770);
        double r148773 = r148772 * r148772;
        double r148774 = r148771 / r148773;
        double r148775 = 1.0;
        double r148776 = r148775 / r148772;
        double r148777 = r148774 * r148776;
        double r148778 = r148766 / r148777;
        double r148779 = r148768 * r148767;
        double r148780 = r148770 + r148779;
        double r148781 = r148780 + r148768;
        double r148782 = r148778 / r148781;
        double r148783 = r148771 / r148770;
        double r148784 = r148769 / r148783;
        double r148785 = r148784 / r148781;
        double r148786 = 1.0;
        double r148787 = r148785 - r148786;
        double r148788 = r148782 - r148787;
        double r148789 = r148788 / r148768;
        return r148789;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.9

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

  3. Applied *-un-lft-identity23.9

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

  4. Applied associate-/r*23.9

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

  5. Simplified12.0

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

  7. Applied div-sub12.0

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

  8. Applied div-sub12.0

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

  9. Applied associate-+l-11.7

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

  11. Applied div-inv11.7

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

  13. Applied add-cube-cbrt11.8

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

  14. Applied *-un-lft-identity11.8

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

  15. Applied times-frac11.8

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

  16. Applied associate-*r*11.8

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

  17. Simplified11.8

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

  18. Final simplification11.8

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

Reproduce

herbie shell --seed 2020060 +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))