Average Error: 52.4 → 35.5
Time: 22.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.718881236776059 \cdot 10^{+214}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}} \cdot \frac{1}{\sqrt{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \sqrt{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 4.718881236776059 \cdot 10^{+214}:\\
\;\;\;\;\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}} \cdot \frac{1}{\sqrt{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \sqrt{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}}}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta, double i) {
        double r1580533 = i;
        double r1580534 = alpha;
        double r1580535 = beta;
        double r1580536 = r1580534 + r1580535;
        double r1580537 = r1580536 + r1580533;
        double r1580538 = r1580533 * r1580537;
        double r1580539 = r1580535 * r1580534;
        double r1580540 = r1580539 + r1580538;
        double r1580541 = r1580538 * r1580540;
        double r1580542 = 2.0;
        double r1580543 = r1580542 * r1580533;
        double r1580544 = r1580536 + r1580543;
        double r1580545 = r1580544 * r1580544;
        double r1580546 = r1580541 / r1580545;
        double r1580547 = 1.0;
        double r1580548 = r1580545 - r1580547;
        double r1580549 = r1580546 / r1580548;
        return r1580549;
}
↓
double f(double alpha, double beta, double i) {
        double r1580550 = alpha;
        double r1580551 = 4.718881236776059e+214;
        bool r1580552 = r1580550 <= r1580551;
        double r1580553 = i;
        double r1580554 = beta;
        double r1580555 = r1580550 + r1580554;
        double r1580556 = r1580555 + r1580553;
        double r1580557 = r1580553 * r1580556;
        double r1580558 = r1580554 * r1580550;
        double r1580559 = r1580557 + r1580558;
        double r1580560 = 2.0;
        double r1580561 = r1580553 * r1580560;
        double r1580562 = r1580555 + r1580561;
        double r1580563 = r1580559 / r1580562;
        double r1580564 = 1.0;
        double r1580565 = sqrt(r1580564);
        double r1580566 = r1580562 - r1580565;
        double r1580567 = r1580563 / r1580566;
        double r1580568 = 1.0;
        double r1580569 = r1580565 + r1580562;
        double r1580570 = r1580557 / r1580562;
        double r1580571 = r1580569 / r1580570;
        double r1580572 = sqrt(r1580571);
        double r1580573 = r1580572 * r1580572;
        double r1580574 = r1580568 / r1580573;
        double r1580575 = r1580567 * r1580574;
        double r1580576 = 0.0;
        double r1580577 = r1580552 ? r1580575 : r1580576;
        return r1580577;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4.718881236776059e+214

    1. Initial program 51.3

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

    3. Applied add-sqr-sqrt51.3

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

    4. Applied difference-of-squares51.3

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

    5. Applied times-frac37.2

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

    6. Applied times-frac34.7

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}}\]
    7. Using strategy rm

    8. Applied clear-num34.7

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

    10. Applied add-sqr-sqrt34.7

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

    if 4.718881236776059e+214 < alpha

    1. Initial program 62.6

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]

    2. Taylor expanded around inf 42.7

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification35.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.718881236776059 \cdot 10^{+214}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \beta \cdot \alpha}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}} \cdot \frac{1}{\sqrt{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \sqrt{\frac{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))