Octave 3.8, jcobi/3

Average Error: 3.5 → 1.4

Time: 2.5m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 8.976072120634706026867062753663154221973 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{\alpha + \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}}{\sqrt{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\beta}^{2}}}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r358231 = alpha;
        double r358232 = beta;
        double r358233 = r358231 + r358232;
        double r358234 = r358232 * r358231;
        double r358235 = r358233 + r358234;
        double r358236 = 1.0;
        double r358237 = r358235 + r358236;
        double r358238 = 2.0;
        double r358239 = r358238 * r358236;
        double r358240 = r358233 + r358239;
        double r358241 = r358237 / r358240;
        double r358242 = r358241 / r358240;
        double r358243 = r358240 + r358236;
        double r358244 = r358242 / r358243;
        return r358244;
}

↓

double f(double alpha, double beta) {
        double r358245 = beta;
        double r358246 = 8.976072120634706e+140;
        bool r358247 = r358245 <= r358246;
        double r358248 = 1.0;
        double r358249 = alpha;
        double r358250 = r358245 + r358249;
        double r358251 = fma(r358245, r358249, r358250);
        double r358252 = r358248 + r358251;
        double r358253 = 2.0;
        double r358254 = fma(r358253, r358248, r358250);
        double r358255 = r358252 / r358254;
        double r358256 = r358255 / r358254;
        double r358257 = 3.0;
        double r358258 = r358257 + r358245;
        double r358259 = r358249 + r358258;
        double r358260 = r358256 / r358259;
        double r358261 = 1.0;
        double r358262 = sqrt(r358254);
        double r358263 = r358261 / r358262;
        double r358264 = r358263 / r358262;
        double r358265 = r358261 / r358249;
        double r358266 = r358261 / r358245;
        double r358267 = r358265 + r358266;
        double r358268 = 2.0;
        double r358269 = pow(r358245, r358268);
        double r358270 = r358261 / r358269;
        double r358271 = r358267 - r358270;
        double r358272 = r358264 / r358271;
        double r358273 = r358272 / r358254;
        double r358274 = r358247 ? r358260 : r358273;
        return r358274;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if beta < 8.976072120634706e+140

   1. Initial program 0.9

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

   2. Simplified 0.9

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

   3. Taylor expanded around 0 0.9

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

   4. Simplified 0.9

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

   if 8.976072120634706e+140 < beta

   1. Initial program 15.3

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

   2. Simplified 15.3

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

   3. Taylor expanded around 0 15.3

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

   4. Simplified 15.3

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

   6. Applied add-sqr-sqrt 15.4

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

   7. Applied add-sqr-sqrt 15.4

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

   8. Applied *-un-lft-identity 15.4

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

   9. Applied times-frac 15.4

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

   10. Applied times-frac 15.4

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

   11. Applied associate-/l* 15.4

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

   12. Simplified 15.4

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

   14. Applied associate-/r/ 15.4

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

   15. Applied associate-/r* 15.4

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

   16. Taylor expanded around inf 3.8

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.976072120634706026867062753663154221973 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right)}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}{\alpha + \left(3 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}}{\sqrt{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}}}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\beta}^{2}}}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))