Octave 3.8, jcobi/1

Average Error: 15.9 → 5.9

Time: 25.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1845090834.4376380443572998046875:\\ \;\;\;\;e^{\log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1, {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{2}\right), \beta, -\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}{\left(2 \cdot \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r82824 = beta;
        double r82825 = alpha;
        double r82826 = r82824 - r82825;
        double r82827 = r82825 + r82824;
        double r82828 = 2.0;
        double r82829 = r82827 + r82828;
        double r82830 = r82826 / r82829;
        double r82831 = 1.0;
        double r82832 = r82830 + r82831;
        double r82833 = r82832 / r82828;
        return r82833;
}

↓

double f(double alpha, double beta) {
        double r82834 = alpha;
        double r82835 = 1845090834.437638;
        bool r82836 = r82834 <= r82835;
        double r82837 = 1.0;
        double r82838 = beta;
        double r82839 = r82834 + r82838;
        double r82840 = 2.0;
        double r82841 = r82839 + r82840;
        double r82842 = r82834 / r82841;
        double r82843 = r82842 + r82837;
        double r82844 = r82840 + r82839;
        double r82845 = r82834 / r82844;
        double r82846 = 2.0;
        double r82847 = pow(r82845, r82846);
        double r82848 = fma(r82837, r82843, r82847);
        double r82849 = r82842 * r82843;
        double r82850 = fma(r82837, r82837, r82849);
        double r82851 = r82842 - r82837;
        double r82852 = r82850 * r82851;
        double r82853 = r82841 * r82852;
        double r82854 = -r82853;
        double r82855 = fma(r82848, r82838, r82854);
        double r82856 = r82840 * r82850;
        double r82857 = r82856 * r82844;
        double r82858 = r82855 / r82857;
        double r82859 = log(r82858);
        double r82860 = exp(r82859);
        double r82861 = r82838 / r82841;
        double r82862 = 4.0;
        double r82863 = r82834 * r82834;
        double r82864 = r82862 / r82863;
        double r82865 = r82840 / r82834;
        double r82866 = 8.0;
        double r82867 = 3.0;
        double r82868 = pow(r82834, r82867);
        double r82869 = r82866 / r82868;
        double r82870 = r82865 + r82869;
        double r82871 = r82864 - r82870;
        double r82872 = r82861 - r82871;
        double r82873 = r82872 / r82840;
        double r82874 = r82836 ? r82860 : r82873;
        return r82874;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 1845090834.437638

   1. Initial program 0.1

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

   3. Applied div-sub 0.1

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

   4. Applied associate-+l- 0.1

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

   6. Applied flip3-- 0.1

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

   7. Applied frac-sub 0.1

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

   8. Applied associate-/l/ 0.1

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

   9. Simplified 0.1

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

   11. Applied add-exp-log 2.1

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

   12. Applied add-exp-log 2.1

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

   13. Applied add-exp-log 2.1

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

   14. Applied prod-exp 2.1

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

   15. Applied prod-exp 2.1

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

   16. Applied add-exp-log 0.7

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

   17. Applied div-exp 0.7

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

   18. Simplified 0.1

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

   20. Applied difference-cubes 0.1

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

   21. Simplified 0.1

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

   if 1845090834.437638 < alpha

   1. Initial program 49.2

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

   3. Applied div-sub 49.1

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

   4. Applied associate-+l- 47.7

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

   5. Taylor expanded around inf 18.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

   6. Simplified 18.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1845090834.4376380443572998046875:\\ \;\;\;\;e^{\log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1, {\left(\frac{\alpha}{2 + \left(\alpha + \beta\right)}\right)}^{2}\right), \beta, -\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}{\left(2 \cdot \mathsf{fma}\left(1, 1, \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))