Octave 3.8, jcobi/3

Average Error: 3.7 → 2.4

Time: 3.1m

Precision: 64

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

Math

\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 6.253981382542825 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\frac{\frac{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt{\frac{\frac{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{1.0 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}} \cdot \frac{\frac{\sqrt[3]{1.0 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{1.0 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r5012195 = alpha;
        double r5012196 = beta;
        double r5012197 = r5012195 + r5012196;
        double r5012198 = r5012196 * r5012195;
        double r5012199 = r5012197 + r5012198;
        double r5012200 = 1.0;
        double r5012201 = r5012199 + r5012200;
        double r5012202 = 2.0;
        double r5012203 = 1.0;
        double r5012204 = r5012202 * r5012203;
        double r5012205 = r5012197 + r5012204;
        double r5012206 = r5012201 / r5012205;
        double r5012207 = r5012206 / r5012205;
        double r5012208 = r5012205 + r5012200;
        double r5012209 = r5012207 / r5012208;
        return r5012209;
}

↓

double f(double alpha, double beta) {
        double r5012210 = beta;
        double r5012211 = 6.253981382542825e+184;
        bool r5012212 = r5012210 <= r5012211;
        double r5012213 = 1.0;
        double r5012214 = alpha;
        double r5012215 = fma(r5012210, r5012214, r5012214);
        double r5012216 = r5012215 + r5012210;
        double r5012217 = r5012213 + r5012216;
        double r5012218 = 2.0;
        double r5012219 = r5012210 + r5012214;
        double r5012220 = r5012218 + r5012219;
        double r5012221 = r5012217 / r5012220;
        double r5012222 = r5012221 / r5012220;
        double r5012223 = sqrt(r5012222);
        double r5012224 = r5012220 + r5012213;
        double r5012225 = r5012223 / r5012224;
        double r5012226 = r5012223 * r5012225;
        double r5012227 = r5012213 + r5012219;
        double r5012228 = cbrt(r5012227);
        double r5012229 = r5012228 / r5012220;
        double r5012230 = sqrt(r5012220);
        double r5012231 = r5012229 / r5012230;
        double r5012232 = cbrt(r5012224);
        double r5012233 = r5012231 / r5012232;
        double r5012234 = r5012228 * r5012228;
        double r5012235 = r5012234 / r5012230;
        double r5012236 = r5012232 * r5012232;
        double r5012237 = r5012235 / r5012236;
        double r5012238 = r5012233 * r5012237;
        double r5012239 = r5012212 ? r5012226 : r5012238;
        return r5012239;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if beta < 6.253981382542825e+184

   1. Initial program 1.7

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

   2. Simplified 1.7

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

   4. Applied *-un-lft-identity 1.7

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

   5. Applied *-un-lft-identity 1.7

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

   6. Applied *-un-lft-identity 1.7

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

   7. Applied distribute-lft-out 1.7

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

   8. Applied distribute-lft-out 1.7

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

   9. Applied add-sqr-sqrt 1.8

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

   10. Applied times-frac 1.8

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

   if 6.253981382542825e+184 < beta

   1. Initial program 16.9

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

   2. Simplified 16.9

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

   3. Taylor expanded around 0 6.9

      \[\leadsto \frac{\frac{\frac{\color{blue}{1.0 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 6.9

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\beta + \alpha\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\color{blue}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}}\]

   6. Applied add-sqr-sqrt 6.9

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\beta + \alpha\right)}{2 + \left(\beta + \alpha\right)}}{\color{blue}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

   7. Applied *-un-lft-identity 6.9

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\beta + \alpha\right)}{2 + \color{blue}{1 \cdot \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

   8. Applied *-un-lft-identity 6.9

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\beta + \alpha\right)}{\color{blue}{1 \cdot 2} + 1 \cdot \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

   9. Applied distribute-lft-out 6.9

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\beta + \alpha\right)}{\color{blue}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

   10. Applied add-cube-cbrt 6.9

       \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{1.0 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{1.0 + \left(\beta + \alpha\right)}\right) \cdot \sqrt[3]{1.0 + \left(\beta + \alpha\right)}}}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

   11. Applied times-frac 6.9

       \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{1.0 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{1.0 + \left(\beta + \alpha\right)}}{1} \cdot \frac{\sqrt[3]{1.0 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{\sqrt{2 + \left(\beta + \alpha\right)} \cdot \sqrt{2 + \left(\beta + \alpha\right)}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

   12. Applied times-frac 6.9

       \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{1.0 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{1.0 + \left(\beta + \alpha\right)}}{1}}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\frac{\sqrt[3]{1.0 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}}{\left(\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\right) \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\]

   13. Applied times-frac 6.9

       \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt[3]{1.0 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{1.0 + \left(\beta + \alpha\right)}}{1}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}} \cdot \frac{\frac{\frac{\sqrt[3]{1.0 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 6.253981382542825 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\frac{\frac{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt{\frac{\frac{1.0 + \left(\mathsf{fma}\left(\beta, \alpha, \alpha\right) + \beta\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{1.0 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}} \cdot \frac{\frac{\sqrt[3]{1.0 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{1.0 + \left(\beta + \alpha\right)}}{\sqrt{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \cdot \sqrt[3]{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))