Octave 3.8, jcobi/1

Average Error: 16.5 → 6.4

Time: 11.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 121320078.19471414387226104736328125:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\frac{\frac{\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}}{\sqrt{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}}}}{\sqrt{1}}, \frac{\frac{\sqrt[3]{\alpha}}{\sqrt{\sqrt[3]{\left(\alpha + \beta\right) + 2}}}}{\sqrt{\left(\alpha + \beta\right) + 2}}, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r170147 = beta;
        double r170148 = alpha;
        double r170149 = r170147 - r170148;
        double r170150 = r170148 + r170147;
        double r170151 = 2.0;
        double r170152 = r170150 + r170151;
        double r170153 = r170149 / r170152;
        double r170154 = 1.0;
        double r170155 = r170153 + r170154;
        double r170156 = r170155 / r170151;
        return r170156;
}

↓

double f(double alpha, double beta) {
        double r170157 = alpha;
        double r170158 = 121320078.19471414;
        bool r170159 = r170157 <= r170158;
        double r170160 = beta;
        double r170161 = r170157 + r170160;
        double r170162 = 2.0;
        double r170163 = r170161 + r170162;
        double r170164 = r170160 / r170163;
        double r170165 = cbrt(r170157);
        double r170166 = r170165 * r170165;
        double r170167 = cbrt(r170163);
        double r170168 = r170167 * r170167;
        double r170169 = sqrt(r170168);
        double r170170 = r170166 / r170169;
        double r170171 = 1.0;
        double r170172 = sqrt(r170171);
        double r170173 = r170170 / r170172;
        double r170174 = sqrt(r170167);
        double r170175 = r170165 / r170174;
        double r170176 = sqrt(r170163);
        double r170177 = r170175 / r170176;
        double r170178 = 1.0;
        double r170179 = -r170178;
        double r170180 = fma(r170173, r170177, r170179);
        double r170181 = r170164 - r170180;
        double r170182 = r170181 / r170162;
        double r170183 = 4.0;
        double r170184 = 2.0;
        double r170185 = pow(r170157, r170184);
        double r170186 = r170171 / r170185;
        double r170187 = r170171 / r170157;
        double r170188 = 8.0;
        double r170189 = 3.0;
        double r170190 = pow(r170157, r170189);
        double r170191 = r170171 / r170190;
        double r170192 = r170188 * r170191;
        double r170193 = fma(r170162, r170187, r170192);
        double r170194 = -r170193;
        double r170195 = fma(r170183, r170186, r170194);
        double r170196 = r170164 - r170195;
        double r170197 = r170196 / r170162;
        double r170198 = r170159 ? r170182 : r170197;
        return r170198;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 121320078.19471414

   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-sqr-sqrt 0.1

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

   7. Applied associate-/r* 0.1

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

   9. Applied *-un-lft-identity 0.1

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

   10. Applied sqrt-prod 0.1

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

   11. Applied add-cube-cbrt 0.1

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

   12. Applied sqrt-prod 0.1

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

   13. Applied add-cube-cbrt 0.1

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

   14. Applied times-frac 0.1

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

   15. Applied times-frac 0.1

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

   16. Applied fma-neg 0.1

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

   if 121320078.19471414 < alpha

   1. Initial program 49.3

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

   3. Applied div-sub 49.2

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

   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. Using strategy rm

   6. Applied add-sqr-sqrt 48.9

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

   7. Applied associate-/r* 48.9

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

   8. Taylor expanded around inf 18.9

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

   9. Simplified 18.9

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

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 121320078.19471414387226104736328125:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(\frac{\frac{\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}}{\sqrt{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}}}}{\sqrt{1}}, \frac{\frac{\sqrt[3]{\alpha}}{\sqrt{\sqrt[3]{\left(\alpha + \beta\right) + 2}}}}{\sqrt{\left(\alpha + \beta\right) + 2}}, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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))