Octave 3.8, jcobi/3

Average Error: 3.9 → 1.7

Time: 1.4m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 1.460679046884028130600225706603739922752 \cdot 10^{143}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r239341 = alpha;
        double r239342 = beta;
        double r239343 = r239341 + r239342;
        double r239344 = r239342 * r239341;
        double r239345 = r239343 + r239344;
        double r239346 = 1.0;
        double r239347 = r239345 + r239346;
        double r239348 = 2.0;
        double r239349 = r239348 * r239346;
        double r239350 = r239343 + r239349;
        double r239351 = r239347 / r239350;
        double r239352 = r239351 / r239350;
        double r239353 = r239350 + r239346;
        double r239354 = r239352 / r239353;
        return r239354;
}

↓

double f(double alpha, double beta) {
        double r239355 = beta;
        double r239356 = 1.4606790468840281e+143;
        bool r239357 = r239355 <= r239356;
        double r239358 = 1.0;
        double r239359 = alpha;
        double r239360 = r239359 + r239355;
        double r239361 = 2.0;
        double r239362 = 1.0;
        double r239363 = r239361 * r239362;
        double r239364 = r239360 + r239363;
        double r239365 = sqrt(r239364);
        double r239366 = r239355 * r239359;
        double r239367 = r239360 + r239366;
        double r239368 = r239367 + r239362;
        double r239369 = r239368 / r239364;
        double r239370 = sqrt(r239369);
        double r239371 = r239365 / r239370;
        double r239372 = r239358 / r239371;
        double r239373 = r239364 + r239362;
        double r239374 = r239372 / r239373;
        double r239375 = r239374 * r239372;
        double r239376 = 2.0;
        double r239377 = r239355 / r239359;
        double r239378 = r239359 / r239355;
        double r239379 = r239377 + r239378;
        double r239380 = r239376 + r239379;
        double r239381 = r239358 / r239380;
        double r239382 = r239381 / r239373;
        double r239383 = r239357 ? r239375 : r239382;
        return r239383;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if beta < 1.4606790468840281e+143

   1. Initial program 1.1

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

   3. Applied *-un-lft-identity 1.1

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

   4. Applied *-un-lft-identity 1.1

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

   5. Applied times-frac 1.1

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

   6. Applied associate-/l* 1.1

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

   8. Applied *-un-lft-identity 1.1

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

   9. Applied add-sqr-sqrt 1.9

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

   10. Applied add-sqr-sqrt 1.2

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

   11. Applied times-frac 1.2

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

   12. Applied add-cube-cbrt 1.2

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

   13. Applied *-un-lft-identity 1.2

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

   14. Applied times-frac 1.2

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

   15. Applied times-frac 1.3

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

   16. Applied times-frac 1.3

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

   if 1.4606790468840281e+143 < beta

   1. Initial program 16.8

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

   3. Applied *-un-lft-identity 16.8

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

   4. Applied *-un-lft-identity 16.8

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

   5. Applied times-frac 16.8

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

   6. Applied associate-/l* 16.8

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

   7. Taylor expanded around inf 3.6

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.460679046884028130600225706603739922752 \cdot 10^{143}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))