Average Error: 3.8 → 2.5
Time: 1.3m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 8.004231800207308 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\beta + \alpha\right) + 2}}}{\frac{\left(\beta + \alpha\right) + 2}{\frac{\left(\left(\beta + \alpha\right) + \alpha \cdot \beta\right) + 1.0}{\sqrt{\left(\beta + \alpha\right) + 2}}}}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\beta + \alpha\right) + 2}}}{\left(\left(\sqrt{8} \cdot 1.0 + 6.0 \cdot \left(\alpha \cdot \sqrt{\frac{1}{8}}\right)\right) + \frac{\beta}{\sqrt{8}} \cdot 6.0\right) - \left(\left(\beta \cdot \sqrt{8}\right) \cdot 1.0 + 1.0 \cdot \left(\alpha \cdot \sqrt{8}\right)\right)}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]
\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 8.004231800207308 \cdot 10^{+201}:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\beta + \alpha\right) + 2}}}{\frac{\left(\beta + \alpha\right) + 2}{\frac{\left(\left(\beta + \alpha\right) + \alpha \cdot \beta\right) + 1.0}{\sqrt{\left(\beta + \alpha\right) + 2}}}}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\beta + \alpha\right) + 2}}}{\left(\left(\sqrt{8} \cdot 1.0 + 6.0 \cdot \left(\alpha \cdot \sqrt{\frac{1}{8}}\right)\right) + \frac{\beta}{\sqrt{8}} \cdot 6.0\right) - \left(\left(\beta \cdot \sqrt{8}\right) \cdot 1.0 + 1.0 \cdot \left(\alpha \cdot \sqrt{8}\right)\right)}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\

\end{array}
double f(double alpha, double beta) {
        double r2701518 = alpha;
        double r2701519 = beta;
        double r2701520 = r2701518 + r2701519;
        double r2701521 = r2701519 * r2701518;
        double r2701522 = r2701520 + r2701521;
        double r2701523 = 1.0;
        double r2701524 = r2701522 + r2701523;
        double r2701525 = 2.0;
        double r2701526 = 1.0;
        double r2701527 = r2701525 * r2701526;
        double r2701528 = r2701520 + r2701527;
        double r2701529 = r2701524 / r2701528;
        double r2701530 = r2701529 / r2701528;
        double r2701531 = r2701528 + r2701523;
        double r2701532 = r2701530 / r2701531;
        return r2701532;
}
↓
double f(double alpha, double beta) {
        double r2701533 = beta;
        double r2701534 = 8.004231800207308e+201;
        bool r2701535 = r2701533 <= r2701534;
        double r2701536 = 1.0;
        double r2701537 = alpha;
        double r2701538 = r2701533 + r2701537;
        double r2701539 = 2.0;
        double r2701540 = r2701538 + r2701539;
        double r2701541 = sqrt(r2701540);
        double r2701542 = r2701536 / r2701541;
        double r2701543 = r2701537 * r2701533;
        double r2701544 = r2701538 + r2701543;
        double r2701545 = 1.0;
        double r2701546 = r2701544 + r2701545;
        double r2701547 = r2701546 / r2701541;
        double r2701548 = r2701540 / r2701547;
        double r2701549 = r2701542 / r2701548;
        double r2701550 = r2701545 + r2701540;
        double r2701551 = r2701549 / r2701550;
        double r2701552 = 8.0;
        double r2701553 = sqrt(r2701552);
        double r2701554 = r2701553 * r2701545;
        double r2701555 = 6.0;
        double r2701556 = 0.125;
        double r2701557 = sqrt(r2701556);
        double r2701558 = r2701537 * r2701557;
        double r2701559 = r2701555 * r2701558;
        double r2701560 = r2701554 + r2701559;
        double r2701561 = r2701533 / r2701553;
        double r2701562 = r2701561 * r2701555;
        double r2701563 = r2701560 + r2701562;
        double r2701564 = r2701533 * r2701553;
        double r2701565 = r2701564 * r2701545;
        double r2701566 = r2701537 * r2701553;
        double r2701567 = r2701545 * r2701566;
        double r2701568 = r2701565 + r2701567;
        double r2701569 = r2701563 - r2701568;
        double r2701570 = r2701542 / r2701569;
        double r2701571 = r2701570 / r2701550;
        double r2701572 = r2701535 ? r2701551 : r2701571;
        return r2701572;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if beta < 8.004231800207308e+201

    1. Initial program 1.9

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

    3. Applied +-commutative1.9

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)} + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt2.4

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

    6. Applied *-un-lft-identity2.4

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

    7. Applied *-un-lft-identity2.4

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

    8. Applied distribute-lft-out2.4

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

    9. Applied times-frac2.4

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

    10. Applied associate-/l*2.0

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

    if 8.004231800207308e+201 < beta

    1. Initial program 19.0

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

    3. Applied +-commutative19.0

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)} + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt19.0

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

    6. Applied *-un-lft-identity19.0

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

    7. Applied *-un-lft-identity19.0

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

    8. Applied distribute-lft-out19.0

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

    9. Applied times-frac19.0

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

    10. Applied associate-/l*19.0

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

    11. Taylor expanded around 0 6.5

      \[\leadsto \frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\color{blue}{\left(6.0 \cdot \frac{\beta}{\sqrt{8}} + \left(6.0 \cdot \left(\sqrt{\frac{1}{8}} \cdot \alpha\right) + 1.0 \cdot \sqrt{8}\right)\right) - \left(1.0 \cdot \left(\sqrt{8} \cdot \alpha\right) + 1.0 \cdot \left(\sqrt{8} \cdot \beta\right)\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.004231800207308 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\beta + \alpha\right) + 2}}}{\frac{\left(\beta + \alpha\right) + 2}{\frac{\left(\left(\beta + \alpha\right) + \alpha \cdot \beta\right) + 1.0}{\sqrt{\left(\beta + \alpha\right) + 2}}}}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\left(\beta + \alpha\right) + 2}}}{\left(\left(\sqrt{8} \cdot 1.0 + 6.0 \cdot \left(\alpha \cdot \sqrt{\frac{1}{8}}\right)\right) + \frac{\beta}{\sqrt{8}} \cdot 6.0\right) - \left(\left(\beta \cdot \sqrt{8}\right) \cdot 1.0 + 1.0 \cdot \left(\alpha \cdot \sqrt{8}\right)\right)}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))