Average Error: 54.3 → 36.9
Time: 1.2m
Precision: 64
\[\alpha \gt -1.0 \land \beta \gt -1.0 \land i \gt 1.0\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.605441052792236 \cdot 10^{+216}:\\ \;\;\;\;\left(\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2.0 \cdot i} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2.0 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2.0 \cdot i\right)}\right) \cdot \frac{1}{\left(\left(\beta + \alpha\right) + 2.0 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 1.605441052792236 \cdot 10^{+216}:\\
\;\;\;\;\left(\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2.0 \cdot i} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2.0 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2.0 \cdot i\right)}\right) \cdot \frac{1}{\left(\left(\beta + \alpha\right) + 2.0 \cdot i\right) - \sqrt{1.0}}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r3423664 = i;
        double r3423665 = alpha;
        double r3423666 = beta;
        double r3423667 = r3423665 + r3423666;
        double r3423668 = r3423667 + r3423664;
        double r3423669 = r3423664 * r3423668;
        double r3423670 = r3423666 * r3423665;
        double r3423671 = r3423670 + r3423669;
        double r3423672 = r3423669 * r3423671;
        double r3423673 = 2.0;
        double r3423674 = r3423673 * r3423664;
        double r3423675 = r3423667 + r3423674;
        double r3423676 = r3423675 * r3423675;
        double r3423677 = r3423672 / r3423676;
        double r3423678 = 1.0;
        double r3423679 = r3423676 - r3423678;
        double r3423680 = r3423677 / r3423679;
        return r3423680;
}
↓
double f(double alpha, double beta, double i) {
        double r3423681 = beta;
        double r3423682 = 1.605441052792236e+216;
        bool r3423683 = r3423681 <= r3423682;
        double r3423684 = i;
        double r3423685 = alpha;
        double r3423686 = r3423681 + r3423685;
        double r3423687 = r3423684 + r3423686;
        double r3423688 = r3423684 * r3423687;
        double r3423689 = r3423685 * r3423681;
        double r3423690 = r3423688 + r3423689;
        double r3423691 = 2.0;
        double r3423692 = r3423691 * r3423684;
        double r3423693 = r3423686 + r3423692;
        double r3423694 = r3423690 / r3423693;
        double r3423695 = r3423688 / r3423693;
        double r3423696 = 1.0;
        double r3423697 = sqrt(r3423696);
        double r3423698 = r3423697 + r3423693;
        double r3423699 = r3423695 / r3423698;
        double r3423700 = r3423694 * r3423699;
        double r3423701 = 1.0;
        double r3423702 = r3423693 - r3423697;
        double r3423703 = r3423701 / r3423702;
        double r3423704 = r3423700 * r3423703;
        double r3423705 = 0.0;
        double r3423706 = r3423683 ? r3423704 : r3423705;
        return r3423706;
}

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 beta < 1.605441052792236e+216

    1. Initial program 53.3

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - 1.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt53.3

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

    4. Applied difference-of-squares53.3

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right)}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - \sqrt{1.0}\right)}}\]

    5. Applied times-frac38.7

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2.0 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2.0 \cdot i}}}{\left(\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) + \sqrt{1.0}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - \sqrt{1.0}\right)}\]

    6. Applied times-frac36.4

      \[\leadsto \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2.0 \cdot i}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2.0 \cdot i}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - \sqrt{1.0}}}\]
    7. Using strategy rm

    8. Applied div-inv36.4

      \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2.0 \cdot i}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) + \sqrt{1.0}} \cdot \color{blue}{\left(\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2.0 \cdot i} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - \sqrt{1.0}}\right)}\]

    9. Applied associate-*r*36.4

      \[\leadsto \color{blue}{\left(\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2.0 \cdot i}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2.0 \cdot i}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - \sqrt{1.0}}}\]

    if 1.605441052792236e+216 < beta

    1. Initial program 64.0

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2.0 \cdot i\right) - 1.0}\]

    2. Taylor expanded around inf 41.0

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

  4. Final simplification36.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.605441052792236 \cdot 10^{+216}:\\ \;\;\;\;\left(\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2.0 \cdot i} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2.0 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2.0 \cdot i\right)}\right) \cdot \frac{1}{\left(\left(\beta + \alpha\right) + 2.0 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))