Octave 3.8, jcobi/3

Average Error: 3.8 → 2.6

Time: 7.2m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.549858114861534202911911997089848239365 \cdot 10^{127}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)}}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{\sqrt{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt[3]{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt[3]{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \cdot \frac{\left({\alpha}^{\frac{1}{12}} + {\alpha}^{\frac{1}{12}} \cdot \beta\right) - 1.75 \cdot \left({\left(\frac{1}{{\alpha}^{11}}\right)}^{\frac{1}{12}} \cdot \beta\right)}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r435479 = alpha;
        double r435480 = beta;
        double r435481 = r435479 + r435480;
        double r435482 = r435480 * r435479;
        double r435483 = r435481 + r435482;
        double r435484 = 1.0;
        double r435485 = r435483 + r435484;
        double r435486 = 2.0;
        double r435487 = r435486 * r435484;
        double r435488 = r435481 + r435487;
        double r435489 = r435485 / r435488;
        double r435490 = r435489 / r435488;
        double r435491 = r435488 + r435484;
        double r435492 = r435490 / r435491;
        return r435492;
}

↓

double f(double alpha, double beta) {
        double r435493 = alpha;
        double r435494 = 3.549858114861534e+127;
        bool r435495 = r435493 <= r435494;
        double r435496 = beta;
        double r435497 = r435493 + r435496;
        double r435498 = r435496 * r435493;
        double r435499 = r435497 + r435498;
        double r435500 = 1.0;
        double r435501 = r435499 + r435500;
        double r435502 = r435497 * r435497;
        double r435503 = 2.0;
        double r435504 = r435503 * r435500;
        double r435505 = r435504 * r435504;
        double r435506 = r435502 - r435505;
        double r435507 = r435501 / r435506;
        double r435508 = 1.0;
        double r435509 = r435507 / r435508;
        double r435510 = r435497 + r435504;
        double r435511 = r435510 + r435500;
        double r435512 = r435497 - r435504;
        double r435513 = r435512 / r435510;
        double r435514 = r435511 / r435513;
        double r435515 = r435509 / r435514;
        double r435516 = sqrt(r435510);
        double r435517 = sqrt(r435516);
        double r435518 = r435508 / r435517;
        double r435519 = cbrt(r435516);
        double r435520 = r435519 * r435519;
        double r435521 = r435518 / r435520;
        double r435522 = r435521 / r435516;
        double r435523 = sqrt(r435511);
        double r435524 = r435522 / r435523;
        double r435525 = 0.08333333333333333;
        double r435526 = pow(r435493, r435525);
        double r435527 = r435526 * r435496;
        double r435528 = r435526 + r435527;
        double r435529 = 1.75;
        double r435530 = 11.0;
        double r435531 = pow(r435493, r435530);
        double r435532 = r435508 / r435531;
        double r435533 = pow(r435532, r435525);
        double r435534 = r435533 * r435496;
        double r435535 = r435529 * r435534;
        double r435536 = r435528 - r435535;
        double r435537 = r435536 / r435523;
        double r435538 = r435524 * r435537;
        double r435539 = r435495 ? r435515 : r435538;
        return r435539;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 3.549858114861534e+127

   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. Using strategy rm

   3. Applied *-un-lft-identity 0.9

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

   4. Applied flip-+ 1.5

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

   5. Applied associate-/r/ 1.5

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

   6. Applied times-frac 1.5

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

   7. Applied associate-/l* 1.5

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

   if 3.549858114861534e+127 < alpha

   1. Initial program 15.0

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

   3. Applied add-sqr-sqrt 15.0

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

   4. Applied associate-/r* 15.0

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

   6. Applied add-sqr-sqrt 15.1

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

   7. Applied add-sqr-sqrt 15.1

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

   8. Applied add-cube-cbrt 15.1

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

   9. Applied add-sqr-sqrt 15.1

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

   10. Applied sqrt-prod 15.1

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

   11. Applied *-un-lft-identity 15.1

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

   12. Applied times-frac 15.2

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

   13. Applied times-frac 15.2

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

   14. Applied times-frac 15.1

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

   15. Applied times-frac 15.2

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

   16. Taylor expanded around inf 6.9

       \[\leadsto \frac{\frac{\frac{\frac{1}{\sqrt{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt[3]{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt[3]{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \cdot \frac{\color{blue}{\left({\alpha}^{\frac{1}{12}} + {\alpha}^{\frac{1}{12}} \cdot \beta\right) - 1.75 \cdot \left({\left(\frac{1}{{\alpha}^{11}}\right)}^{\frac{1}{12}} \cdot \beta\right)}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.549858114861534202911911997089848239365 \cdot 10^{127}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)}}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{\sqrt{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt[3]{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt[3]{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \cdot \frac{\left({\alpha}^{\frac{1}{12}} + {\alpha}^{\frac{1}{12}} \cdot \beta\right) - 1.75 \cdot \left({\left(\frac{1}{{\alpha}^{11}}\right)}^{\frac{1}{12}} \cdot \beta\right)}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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)))