Octave 3.8, jcobi/3

Average Error: 4.0 → 1.6

Time: 13.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \le 0.083446765035005532:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.77777777777777779, {0.1111111111111111}^{\frac{1}{3}} \cdot \left(\sqrt{0.5} \cdot \beta\right), \mathsf{fma}\left(1, {0.1111111111111111}^{\frac{1}{3}} \cdot \sqrt{0.5}, 0.75 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {0.1111111111111111}^{\frac{1}{3}}\right)\right)\right) - 0.125 \cdot \left({0.1111111111111111}^{\frac{1}{3}} \cdot \frac{\beta}{\sqrt{0.5}}\right)\right) - 2 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {\left( 1.52415790275872584 \cdot 10^{-4} \right)}^{\frac{1}{3}}\right)\right) \cdot \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r110845 = alpha;
        double r110846 = beta;
        double r110847 = r110845 + r110846;
        double r110848 = r110846 * r110845;
        double r110849 = r110847 + r110848;
        double r110850 = 1.0;
        double r110851 = r110849 + r110850;
        double r110852 = 2.0;
        double r110853 = r110852 * r110850;
        double r110854 = r110847 + r110853;
        double r110855 = r110851 / r110854;
        double r110856 = r110855 / r110854;
        double r110857 = r110854 + r110850;
        double r110858 = r110856 / r110857;
        return r110858;
}

↓

double f(double alpha, double beta) {
        double r110859 = alpha;
        double r110860 = beta;
        double r110861 = r110859 + r110860;
        double r110862 = r110860 * r110859;
        double r110863 = r110861 + r110862;
        double r110864 = 1.0;
        double r110865 = r110863 + r110864;
        double r110866 = 2.0;
        double r110867 = r110866 * r110864;
        double r110868 = r110861 + r110867;
        double r110869 = r110865 / r110868;
        double r110870 = r110869 / r110868;
        double r110871 = r110868 + r110864;
        double r110872 = r110870 / r110871;
        double r110873 = 0.08344676503500553;
        bool r110874 = r110872 <= r110873;
        double r110875 = 0.7777777777777778;
        double r110876 = 0.1111111111111111;
        double r110877 = 0.3333333333333333;
        double r110878 = pow(r110876, r110877);
        double r110879 = 0.5;
        double r110880 = sqrt(r110879);
        double r110881 = r110880 * r110860;
        double r110882 = r110878 * r110881;
        double r110883 = r110878 * r110880;
        double r110884 = 0.75;
        double r110885 = r110859 * r110880;
        double r110886 = r110885 * r110878;
        double r110887 = r110884 * r110886;
        double r110888 = fma(r110864, r110883, r110887);
        double r110889 = fma(r110875, r110882, r110888);
        double r110890 = 0.125;
        double r110891 = r110860 / r110880;
        double r110892 = r110878 * r110891;
        double r110893 = r110890 * r110892;
        double r110894 = r110889 - r110893;
        double r110895 = 0.00015241579027587258;
        double r110896 = pow(r110895, r110877);
        double r110897 = r110885 * r110896;
        double r110898 = r110866 * r110897;
        double r110899 = r110894 - r110898;
        double r110900 = 1.0;
        double r110901 = r110900 / r110868;
        double r110902 = sqrt(r110868);
        double r110903 = r110901 / r110902;
        double r110904 = cbrt(r110871);
        double r110905 = r110903 / r110904;
        double r110906 = r110899 * r110905;
        double r110907 = r110874 ? r110872 : r110906;
        return r110907;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)) < 0.08344676503500553

   1. Initial program 0.1

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

   if 0.08344676503500553 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0))

   1. Initial program 60.9

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

   3. Applied add-cube-cbrt 61.0

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

   4. Applied add-sqr-sqrt 60.9

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

   5. Applied div-inv 61.0

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

   6. Applied times-frac 60.9

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

   7. Applied times-frac 60.9

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

   8. Taylor expanded around 0 22.7

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

   9. Simplified 22.7

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \le 0.083446765035005532:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.77777777777777779, {0.1111111111111111}^{\frac{1}{3}} \cdot \left(\sqrt{0.5} \cdot \beta\right), \mathsf{fma}\left(1, {0.1111111111111111}^{\frac{1}{3}} \cdot \sqrt{0.5}, 0.75 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {0.1111111111111111}^{\frac{1}{3}}\right)\right)\right) - 0.125 \cdot \left({0.1111111111111111}^{\frac{1}{3}} \cdot \frac{\beta}{\sqrt{0.5}}\right)\right) - 2 \cdot \left(\left(\alpha \cdot \sqrt{0.5}\right) \cdot {\left( 1.52415790275872584 \cdot 10^{-4} \right)}^{\frac{1}{3}}\right)\right) \cdot \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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)))