Octave 3.8, jcobi/3

Average Error: 3.7 → 1.6

Time: 43.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)} \le 0.0840188988913027900995444952059187926352:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{\sqrt{2 \cdot 1 + \left(\beta + \alpha\right)}}}{\sqrt{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}} \cdot \frac{\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}{\sqrt{2 \cdot 1 + \left(\beta + \alpha\right)}}}{\sqrt{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\alpha}}{\alpha} + \left(1 - \frac{1}{\alpha}\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r4664485 = alpha;
        double r4664486 = beta;
        double r4664487 = r4664485 + r4664486;
        double r4664488 = r4664486 * r4664485;
        double r4664489 = r4664487 + r4664488;
        double r4664490 = 1.0;
        double r4664491 = r4664489 + r4664490;
        double r4664492 = 2.0;
        double r4664493 = r4664492 * r4664490;
        double r4664494 = r4664487 + r4664493;
        double r4664495 = r4664491 / r4664494;
        double r4664496 = r4664495 / r4664494;
        double r4664497 = r4664494 + r4664490;
        double r4664498 = r4664496 / r4664497;
        return r4664498;
}

↓

double f(double alpha, double beta) {
        double r4664499 = 1.0;
        double r4664500 = alpha;
        double r4664501 = beta;
        double r4664502 = r4664500 * r4664501;
        double r4664503 = r4664501 + r4664500;
        double r4664504 = r4664502 + r4664503;
        double r4664505 = r4664499 + r4664504;
        double r4664506 = 2.0;
        double r4664507 = r4664506 * r4664499;
        double r4664508 = r4664507 + r4664503;
        double r4664509 = r4664505 / r4664508;
        double r4664510 = r4664509 / r4664508;
        double r4664511 = r4664499 + r4664508;
        double r4664512 = r4664510 / r4664511;
        double r4664513 = 0.08401889889130279;
        bool r4664514 = r4664512 <= r4664513;
        double r4664515 = sqrt(r4664505);
        double r4664516 = r4664515 / r4664508;
        double r4664517 = sqrt(r4664508);
        double r4664518 = r4664516 / r4664517;
        double r4664519 = sqrt(r4664511);
        double r4664520 = r4664518 / r4664519;
        double r4664521 = r4664515 / r4664517;
        double r4664522 = r4664521 / r4664519;
        double r4664523 = r4664520 * r4664522;
        double r4664524 = r4664506 / r4664500;
        double r4664525 = r4664524 / r4664500;
        double r4664526 = 1.0;
        double r4664527 = r4664499 / r4664500;
        double r4664528 = r4664526 - r4664527;
        double r4664529 = r4664525 + r4664528;
        double r4664530 = r4664529 / r4664508;
        double r4664531 = r4664530 / r4664511;
        double r4664532 = r4664514 ? r4664523 : r4664531;
        return r4664532;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if (/ (/ (/ (+ (+ (+ 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)) < 0.08401889889130279

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.6

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

   4. Applied add-sqr-sqrt 0.3

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

   5. Applied *-un-lft-identity 0.3

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

   6. Applied add-sqr-sqrt 0.2

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

   7. Applied times-frac 0.3

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

   8. Applied times-frac 0.2

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

   9. Applied times-frac 0.2

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

   if 0.08401889889130279 < (/ (/ (/ (+ (+ (+ 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))

   1. Initial program 62.1

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

   2. Taylor expanded around inf 23.0

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

   3. Simplified 23.0

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)} \le 0.0840188988913027900995444952059187926352:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{\sqrt{2 \cdot 1 + \left(\beta + \alpha\right)}}}{\sqrt{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}} \cdot \frac{\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}{\sqrt{2 \cdot 1 + \left(\beta + \alpha\right)}}}{\sqrt{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\alpha}}{\alpha} + \left(1 - \frac{1}{\alpha}\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{1 + \left(2 \cdot 1 + \left(\beta + \alpha\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))