Average Error: 4.0 → 2.5
Time: 23.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.5039397066266325 \cdot 10^{117}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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 3.5039397066266325 \cdot 10^{117}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 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}
double f(double alpha, double beta) {
        double r182489 = alpha;
        double r182490 = beta;
        double r182491 = r182489 + r182490;
        double r182492 = r182490 * r182489;
        double r182493 = r182491 + r182492;
        double r182494 = 1.0;
        double r182495 = r182493 + r182494;
        double r182496 = 2.0;
        double r182497 = r182496 * r182494;
        double r182498 = r182491 + r182497;
        double r182499 = r182495 / r182498;
        double r182500 = r182499 / r182498;
        double r182501 = r182498 + r182494;
        double r182502 = r182500 / r182501;
        return r182502;
}
↓
double f(double alpha, double beta) {
        double r182503 = alpha;
        double r182504 = 3.5039397066266325e+117;
        bool r182505 = r182503 <= r182504;
        double r182506 = beta;
        double r182507 = r182503 + r182506;
        double r182508 = r182506 * r182503;
        double r182509 = r182507 + r182508;
        double r182510 = 1.0;
        double r182511 = r182509 + r182510;
        double r182512 = sqrt(r182511);
        double r182513 = 2.0;
        double r182514 = r182513 * r182510;
        double r182515 = r182507 + r182514;
        double r182516 = sqrt(r182515);
        double r182517 = r182512 / r182516;
        double r182518 = r182515 / r182517;
        double r182519 = r182517 / r182518;
        double r182520 = r182515 + r182510;
        double r182521 = r182519 / r182520;
        double r182522 = 1.0;
        double r182523 = 2.0;
        double r182524 = r182506 / r182503;
        double r182525 = r182503 / r182506;
        double r182526 = r182524 + r182525;
        double r182527 = r182523 + r182526;
        double r182528 = r182522 / r182527;
        double r182529 = r182528 / r182520;
        double r182530 = r182505 ? r182521 : r182529;
        return r182530;
}

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 alpha < 3.5039397066266325e+117

    1. Initial program 1.0

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

    3. Applied add-sqr-sqrt1.6

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

    4. Applied add-sqr-sqrt1.5

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

    5. Applied times-frac1.5

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

    6. Applied associate-/l*1.0

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

    if 3.5039397066266325e+117 < alpha

    1. Initial program 15.7

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

    3. Applied *-un-lft-identity15.7

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

    4. Applied *-un-lft-identity15.7

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

    5. Applied times-frac15.7

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

    6. Applied associate-/l*15.7

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

    7. Taylor expanded around inf 8.1

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.5039397066266325 \cdot 10^{117}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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)))