Average Error: 23.9 → 12.8
Time: 23.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{e^{\log \left(\sqrt[3]{\mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, 1.0\right) \cdot \left(\mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, 1.0\right) \cdot \mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, 1.0\right)\right)}\right)}}{2.0}\]
\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) + 2.0} + 1.0}{2.0}
↓
\frac{e^{\log \left(\sqrt[3]{\mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, 1.0\right) \cdot \left(\mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, 1.0\right) \cdot \mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, 1.0\right)\right)}\right)}}{2.0}
double f(double alpha, double beta, double i) {
        double r4388649 = alpha;
        double r4388650 = beta;
        double r4388651 = r4388649 + r4388650;
        double r4388652 = r4388650 - r4388649;
        double r4388653 = r4388651 * r4388652;
        double r4388654 = 2.0;
        double r4388655 = i;
        double r4388656 = r4388654 * r4388655;
        double r4388657 = r4388651 + r4388656;
        double r4388658 = r4388653 / r4388657;
        double r4388659 = 2.0;
        double r4388660 = r4388657 + r4388659;
        double r4388661 = r4388658 / r4388660;
        double r4388662 = 1.0;
        double r4388663 = r4388661 + r4388662;
        double r4388664 = r4388663 / r4388659;
        return r4388664;
}
↓
double f(double alpha, double beta, double i) {
        double r4388665 = beta;
        double r4388666 = alpha;
        double r4388667 = r4388665 - r4388666;
        double r4388668 = i;
        double r4388669 = 2.0;
        double r4388670 = r4388666 + r4388665;
        double r4388671 = fma(r4388668, r4388669, r4388670);
        double r4388672 = r4388667 / r4388671;
        double r4388673 = 2.0;
        double r4388674 = fma(r4388669, r4388668, r4388673);
        double r4388675 = r4388670 + r4388674;
        double r4388676 = sqrt(r4388675);
        double r4388677 = r4388672 / r4388676;
        double r4388678 = r4388670 / r4388676;
        double r4388679 = 1.0;
        double r4388680 = fma(r4388677, r4388678, r4388679);
        double r4388681 = r4388680 * r4388680;
        double r4388682 = r4388680 * r4388681;
        double r4388683 = cbrt(r4388682);
        double r4388684 = log(r4388683);
        double r4388685 = exp(r4388684);
        double r4388686 = r4388685 / r4388673;
        return r4388686;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.9

    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

  2. Simplified19.8

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}, \beta + \alpha, 1.0\right)}{2.0}}\]
  3. Using strategy rm

  4. Applied add-exp-log19.7

    \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}, \beta + \alpha, 1.0\right)\right)}}}{2.0}\]

  5. Simplified12.7

    \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}, 1.0\right)\right)}}}{2.0}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt12.8

    \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta + \alpha}{\color{blue}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}, 1.0\right)\right)}}{2.0}\]

  8. Applied associate-/r*12.8

    \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \color{blue}{\frac{\frac{\beta + \alpha}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}, 1.0\right)\right)}}{2.0}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt12.8

    \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\frac{\beta + \alpha}{\sqrt{\color{blue}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}}}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}, 1.0\right)\right)}}{2.0}\]

  11. Applied sqrt-prod12.8

    \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\frac{\beta + \alpha}{\color{blue}{\sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}} \cdot \sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}}}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}, 1.0\right)\right)}}{2.0}\]

  12. Applied add-cube-cbrt13.0

    \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta + \alpha} \cdot \sqrt[3]{\beta + \alpha}\right) \cdot \sqrt[3]{\beta + \alpha}}}{\sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}} \cdot \sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}, 1.0\right)\right)}}{2.0}\]

  13. Applied times-frac13.0

    \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\color{blue}{\frac{\sqrt[3]{\beta + \alpha} \cdot \sqrt[3]{\beta + \alpha}}{\sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}}}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}, 1.0\right)\right)}}{2.0}\]
  14. Using strategy rm

  15. Applied add-cbrt-cube12.9

    \[\leadsto \frac{e^{\log \color{blue}{\left(\sqrt[3]{\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\frac{\sqrt[3]{\beta + \alpha} \cdot \sqrt[3]{\beta + \alpha}}{\sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}, 1.0\right) \cdot \mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\frac{\sqrt[3]{\beta + \alpha} \cdot \sqrt[3]{\beta + \alpha}}{\sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}, 1.0\right)\right) \cdot \mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\frac{\sqrt[3]{\beta + \alpha} \cdot \sqrt[3]{\beta + \alpha}}{\sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\sqrt{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}}}{\sqrt{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}, 1.0\right)}\right)}}}{2.0}\]

  16. Simplified12.8

    \[\leadsto \frac{e^{\log \left(\sqrt[3]{\color{blue}{\left(\mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\sqrt{\mathsf{fma}\left(2, i, 2.0\right) + \left(\alpha + \beta\right)}}, \frac{\alpha + \beta}{\sqrt{\mathsf{fma}\left(2, i, 2.0\right) + \left(\alpha + \beta\right)}}, 1.0\right) \cdot \mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\sqrt{\mathsf{fma}\left(2, i, 2.0\right) + \left(\alpha + \beta\right)}}, \frac{\alpha + \beta}{\sqrt{\mathsf{fma}\left(2, i, 2.0\right) + \left(\alpha + \beta\right)}}, 1.0\right)\right) \cdot \mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\sqrt{\mathsf{fma}\left(2, i, 2.0\right) + \left(\alpha + \beta\right)}}, \frac{\alpha + \beta}{\sqrt{\mathsf{fma}\left(2, i, 2.0\right) + \left(\alpha + \beta\right)}}, 1.0\right)}}\right)}}{2.0}\]

  17. Final simplification12.8

    \[\leadsto \frac{e^{\log \left(\sqrt[3]{\mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, 1.0\right) \cdot \left(\mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, 1.0\right) \cdot \mathsf{fma}\left(\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, \frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2.0\right)}}, 1.0\right)\right)}\right)}}{2.0}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))