Octave 3.8, jcobi/1

Average Error: 16.3 → 5.9

Time: 20.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 307451316.10967922210693359375:\\ \;\;\;\;\frac{\frac{\beta}{\left(2 + \alpha\right) + \beta} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r72072 = beta;
        double r72073 = alpha;
        double r72074 = r72072 - r72073;
        double r72075 = r72073 + r72072;
        double r72076 = 2.0;
        double r72077 = r72075 + r72076;
        double r72078 = r72074 / r72077;
        double r72079 = 1.0;
        double r72080 = r72078 + r72079;
        double r72081 = r72080 / r72076;
        return r72081;
}

↓

double f(double alpha, double beta) {
        double r72082 = alpha;
        double r72083 = 307451316.1096792;
        bool r72084 = r72082 <= r72083;
        double r72085 = beta;
        double r72086 = 2.0;
        double r72087 = r72086 + r72082;
        double r72088 = r72087 + r72085;
        double r72089 = r72085 / r72088;
        double r72090 = r72082 + r72085;
        double r72091 = r72086 + r72090;
        double r72092 = r72082 / r72091;
        double r72093 = 1.0;
        double r72094 = r72092 - r72093;
        double r72095 = r72089 - r72094;
        double r72096 = r72095 / r72086;
        double r72097 = cbrt(r72085);
        double r72098 = r72090 + r72086;
        double r72099 = 0.3333333333333333;
        double r72100 = pow(r72098, r72099);
        double r72101 = r72097 / r72100;
        double r72102 = 3.0;
        double r72103 = pow(r72101, r72102);
        double r72104 = 4.0;
        double r72105 = 2.0;
        double r72106 = pow(r72082, r72105);
        double r72107 = r72104 / r72106;
        double r72108 = 8.0;
        double r72109 = pow(r72082, r72102);
        double r72110 = r72108 / r72109;
        double r72111 = r72107 - r72110;
        double r72112 = r72086 / r72082;
        double r72113 = r72111 - r72112;
        double r72114 = r72103 - r72113;
        double r72115 = r72114 / r72086;
        double r72116 = r72084 ? r72096 : r72115;
        return r72116;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 307451316.1096792

   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 add-cube-cbrt 0.2

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

   7. Applied add-cube-cbrt 0.4

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

   8. Applied add-cube-cbrt 0.2

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

   9. Applied times-frac 0.2

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

   10. Applied prod-diff 0.2

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

   11. Simplified 0.1

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

   12. Simplified 0.1

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + \color{blue}{0}}{2}\]
   13. Using strategy rm

   14. Applied pow1/3 1.8

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]
   15. Using strategy rm

   16. Applied cube-div 1.8

       \[\leadsto \frac{\left(\color{blue}{\frac{{\left(\sqrt[3]{\beta}\right)}^{3}}{{\left({\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}\right)}^{3}}} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   17. Simplified 1.8

       \[\leadsto \frac{\left(\frac{\color{blue}{\beta}}{{\left({\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}\right)}^{3}} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   18. Simplified 0.1

       \[\leadsto \frac{\left(\frac{\beta}{\color{blue}{\left(2 + \alpha\right) + \beta}} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   if 307451316.1096792 < alpha

   1. Initial program 49.9

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

   3. Applied div-sub 49.9

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

   4. Applied associate-+l- 48.3

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

   6. Applied add-cube-cbrt 48.4

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

   7. Applied add-cube-cbrt 48.5

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

   8. Applied add-cube-cbrt 48.4

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

   9. Applied times-frac 48.4

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

   10. Applied prod-diff 48.4

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

   11. Simplified 48.4

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

   12. Simplified 48.4

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + \color{blue}{0}}{2}\]
   13. Using strategy rm

   14. Applied pow1/3 49.3

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}}\right)}^{3} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)\right) + 0}{2}\]

   15. Taylor expanded around inf 17.8

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}\right) + 0}{2}\]

   16. Simplified 17.8

       \[\leadsto \frac{\left({\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \color{blue}{\left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}\right) + 0}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 307451316.10967922210693359375:\\ \;\;\;\;\frac{\frac{\beta}{\left(2 + \alpha\right) + \beta} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\beta}}{{\left(\left(\alpha + \beta\right) + 2\right)}^{\frac{1}{3}}}\right)}^{3} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))