Octave 3.8, jcobi/2

Average Error: 23.7 → 11.6

Time: 16.7s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.0312336573895782 \cdot 10^{+150}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\beta + \alpha, \frac{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right)}}{2.0 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}, 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r1759132 = alpha;
        double r1759133 = beta;
        double r1759134 = r1759132 + r1759133;
        double r1759135 = r1759133 - r1759132;
        double r1759136 = r1759134 * r1759135;
        double r1759137 = 2.0;
        double r1759138 = i;
        double r1759139 = r1759137 * r1759138;
        double r1759140 = r1759134 + r1759139;
        double r1759141 = r1759136 / r1759140;
        double r1759142 = 2.0;
        double r1759143 = r1759140 + r1759142;
        double r1759144 = r1759141 / r1759143;
        double r1759145 = 1.0;
        double r1759146 = r1759144 + r1759145;
        double r1759147 = r1759146 / r1759142;
        return r1759147;
}

↓

double f(double alpha, double beta, double i) {
        double r1759148 = alpha;
        double r1759149 = 1.0312336573895782e+150;
        bool r1759150 = r1759148 <= r1759149;
        double r1759151 = beta;
        double r1759152 = r1759151 + r1759148;
        double r1759153 = r1759151 - r1759148;
        double r1759154 = 2.0;
        double r1759155 = i;
        double r1759156 = fma(r1759154, r1759155, r1759152);
        double r1759157 = r1759153 / r1759156;
        double r1759158 = r1759157 * r1759157;
        double r1759159 = r1759157 * r1759158;
        double r1759160 = cbrt(r1759159);
        double r1759161 = 2.0;
        double r1759162 = r1759161 + r1759156;
        double r1759163 = r1759160 / r1759162;
        double r1759164 = 1.0;
        double r1759165 = fma(r1759152, r1759163, r1759164);
        double r1759166 = log(r1759165);
        double r1759167 = exp(r1759166);
        double r1759168 = r1759167 / r1759161;
        double r1759169 = 8.0;
        double r1759170 = r1759148 * r1759148;
        double r1759171 = r1759148 * r1759170;
        double r1759172 = r1759169 / r1759171;
        double r1759173 = r1759161 / r1759148;
        double r1759174 = 4.0;
        double r1759175 = r1759174 / r1759170;
        double r1759176 = r1759173 - r1759175;
        double r1759177 = r1759172 + r1759176;
        double r1759178 = r1759177 / r1759161;
        double r1759179 = r1759150 ? r1759168 : r1759178;
        return r1759179;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.0312336573895782e+150

   1. Initial program 15.6

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Simplified 15.6

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)} + 1.0}{2.0}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 15.6

      \[\leadsto \frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}} + 1.0}{2.0}\]

   5. Applied times-frac 12.9

      \[\leadsto \frac{\color{blue}{\frac{\beta + \alpha}{1} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}} + 1.0}{2.0}\]

   6. Applied fma-def 12.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\beta + \alpha}{1}, \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}, 1.0\right)}}{2.0}\]
   7. Using strategy rm

   8. Applied add-exp-log 12.9

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\beta + \alpha}{1}, \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}, 1.0\right)\right)}}}{2.0}\]

   9. Simplified 12.9

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) \cdot \left(\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0\right)}, 1.0\right)\right)}}}{2.0}\]
   10. Using strategy rm

   11. Applied associate-/r* 5.5

       \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\alpha + \beta, \color{blue}{\frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0}}, 1.0\right)\right)}}{2.0}\]
   12. Using strategy rm

   13. Applied add-cbrt-cube 5.5

       \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\alpha + \beta, \frac{\color{blue}{\sqrt[3]{\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0}, 1.0\right)\right)}}{2.0}\]

   if 1.0312336573895782e+150 < alpha

   1. Initial program 62.9

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Simplified 62.2

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)} + 1.0}{2.0}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 62.2

      \[\leadsto \frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}} + 1.0}{2.0}\]

   5. Applied times-frac 53.3

      \[\leadsto \frac{\color{blue}{\frac{\beta + \alpha}{1} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}} + 1.0}{2.0}\]

   6. Applied fma-def 53.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\beta + \alpha}{1}, \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}, 1.0\right)}}{2.0}\]
   7. Using strategy rm

   8. Applied add-exp-log 53.3

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\beta + \alpha}{1}, \frac{\beta - \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)}, 1.0\right)\right)}}}{2.0}\]

   9. Simplified 53.3

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) \cdot \left(\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0\right)}, 1.0\right)\right)}}}{2.0}\]

   10. Taylor expanded around inf 41.3

       \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

   11. Simplified 41.3

       \[\leadsto \frac{\color{blue}{\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.0312336573895782 \cdot 10^{+150}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(\beta + \alpha, \frac{\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right)}}{2.0 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}, 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))