Octave 3.8, jcobi/3

Average Error: 3.4 → 2.1

Time: 10.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.9395704295709712 \cdot 10^{178}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}} \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.77777777777777779, {0.1111111111111111}^{\frac{1}{3}} \cdot \left(\sqrt{0.5} \cdot \beta\right), \mathsf{fma}\left(1, {0.1111111111111111}^{\frac{1}{3}} \cdot \sqrt{0.5}, 0.75 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {0.1111111111111111}^{\frac{1}{3}}\right)\right)\right) - 0.125 \cdot \left({0.1111111111111111}^{\frac{1}{3}} \cdot \frac{\beta}{\sqrt{0.5}}\right)\right) - 2 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {\left( 1.52415790275872584 \cdot 10^{-4} \right)}^{\frac{1}{3}}\right)\right) \cdot \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r111665 = alpha;
        double r111666 = beta;
        double r111667 = r111665 + r111666;
        double r111668 = r111666 * r111665;
        double r111669 = r111667 + r111668;
        double r111670 = 1.0;
        double r111671 = r111669 + r111670;
        double r111672 = 2.0;
        double r111673 = r111672 * r111670;
        double r111674 = r111667 + r111673;
        double r111675 = r111671 / r111674;
        double r111676 = r111675 / r111674;
        double r111677 = r111674 + r111670;
        double r111678 = r111676 / r111677;
        return r111678;
}

↓

double f(double alpha, double beta) {
        double r111679 = alpha;
        double r111680 = 1.939570429570971e+178;
        bool r111681 = r111679 <= r111680;
        double r111682 = beta;
        double r111683 = r111679 + r111682;
        double r111684 = r111682 * r111679;
        double r111685 = r111683 + r111684;
        double r111686 = 1.0;
        double r111687 = r111685 + r111686;
        double r111688 = sqrt(r111687);
        double r111689 = 2.0;
        double r111690 = r111689 * r111686;
        double r111691 = r111683 + r111690;
        double r111692 = sqrt(r111691);
        double r111693 = r111688 / r111692;
        double r111694 = r111693 / r111692;
        double r111695 = fma(r111689, r111686, r111686);
        double r111696 = r111683 + r111695;
        double r111697 = r111696 / r111688;
        double r111698 = fma(r111686, r111689, r111683);
        double r111699 = r111697 * r111698;
        double r111700 = r111694 / r111699;
        double r111701 = 0.7777777777777778;
        double r111702 = 0.1111111111111111;
        double r111703 = 0.3333333333333333;
        double r111704 = pow(r111702, r111703);
        double r111705 = 0.5;
        double r111706 = sqrt(r111705);
        double r111707 = r111706 * r111682;
        double r111708 = r111704 * r111707;
        double r111709 = r111704 * r111706;
        double r111710 = 0.75;
        double r111711 = r111679 * r111706;
        double r111712 = r111711 * r111704;
        double r111713 = r111710 * r111712;
        double r111714 = fma(r111686, r111709, r111713);
        double r111715 = fma(r111701, r111708, r111714);
        double r111716 = 0.125;
        double r111717 = r111682 / r111706;
        double r111718 = r111704 * r111717;
        double r111719 = r111716 * r111718;
        double r111720 = r111715 - r111719;
        double r111721 = 0.00015241579027587258;
        double r111722 = pow(r111721, r111703);
        double r111723 = r111711 * r111722;
        double r111724 = r111689 * r111723;
        double r111725 = r111720 - r111724;
        double r111726 = 1.0;
        double r111727 = r111726 / r111691;
        double r111728 = r111727 / r111692;
        double r111729 = r111691 + r111686;
        double r111730 = cbrt(r111729);
        double r111731 = r111728 / r111730;
        double r111732 = r111725 * r111731;
        double r111733 = r111681 ? r111700 : r111732;
        return r111733;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 1.939570429570971e+178

   1. Initial program 1.4

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

   3. Applied add-sqr-sqrt 2.0

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

   4. Applied add-sqr-sqrt 2.4

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

   5. Applied add-sqr-sqrt 2.3

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

   6. Applied times-frac 2.3

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

   7. Applied times-frac 2.1

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

   8. Applied associate-/l* 2.1

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

   9. Simplified 1.5

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

   if 1.939570429570971e+178 < alpha

   1. Initial program 15.2

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

   3. Applied add-cube-cbrt 15.2

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

   4. Applied add-sqr-sqrt 15.2

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

   5. Applied div-inv 15.2

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

   6. Applied times-frac 15.2

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

   7. Applied times-frac 15.9

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

   8. Taylor expanded around 0 5.8

      \[\leadsto \color{blue}{\left(\left(0.77777777777777779 \cdot \left({0.1111111111111111}^{\frac{1}{3}} \cdot \left(\sqrt{0.5} \cdot \beta\right)\right) + \left(1 \cdot \left({0.1111111111111111}^{\frac{1}{3}} \cdot \sqrt{0.5}\right) + 0.75 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {0.1111111111111111}^{\frac{1}{3}}\right)\right)\right) - \left(0.125 \cdot \left({0.1111111111111111}^{\frac{1}{3}} \cdot \frac{\beta}{\sqrt{0.5}}\right) + 2 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {\left( 1.52415790275872584 \cdot 10^{-4} \right)}^{\frac{1}{3}}\right)\right)\right)} \cdot \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\]

   9. Simplified 5.8

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(0.77777777777777779, {0.1111111111111111}^{\frac{1}{3}} \cdot \left(\sqrt{0.5} \cdot \beta\right), \mathsf{fma}\left(1, {0.1111111111111111}^{\frac{1}{3}} \cdot \sqrt{0.5}, 0.75 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {0.1111111111111111}^{\frac{1}{3}}\right)\right)\right) - 0.125 \cdot \left({0.1111111111111111}^{\frac{1}{3}} \cdot \frac{\beta}{\sqrt{0.5}}\right)\right) - 2 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {\left( 1.52415790275872584 \cdot 10^{-4} \right)}^{\frac{1}{3}}\right)\right)} \cdot \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.9395704295709712 \cdot 10^{178}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, 1, 1\right)}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}} \cdot \mathsf{fma}\left(1, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.77777777777777779, {0.1111111111111111}^{\frac{1}{3}} \cdot \left(\sqrt{0.5} \cdot \beta\right), \mathsf{fma}\left(1, {0.1111111111111111}^{\frac{1}{3}} \cdot \sqrt{0.5}, 0.75 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {0.1111111111111111}^{\frac{1}{3}}\right)\right)\right) - 0.125 \cdot \left({0.1111111111111111}^{\frac{1}{3}} \cdot \frac{\beta}{\sqrt{0.5}}\right)\right) - 2 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {\left( 1.52415790275872584 \cdot 10^{-4} \right)}^{\frac{1}{3}}\right)\right) \cdot \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))