Average Error: 52.7 → 35.7
Time: 37.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.447602491634548 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 \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}\;\beta \le 1.447602491634548 \cdot 10^{+178}:\\
\;\;\;\;\frac{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r5444923 = i;
        double r5444924 = alpha;
        double r5444925 = beta;
        double r5444926 = r5444924 + r5444925;
        double r5444927 = r5444926 + r5444923;
        double r5444928 = r5444923 * r5444927;
        double r5444929 = r5444925 * r5444924;
        double r5444930 = r5444929 + r5444928;
        double r5444931 = r5444928 * r5444930;
        double r5444932 = 2.0;
        double r5444933 = r5444932 * r5444923;
        double r5444934 = r5444926 + r5444933;
        double r5444935 = r5444934 * r5444934;
        double r5444936 = r5444931 / r5444935;
        double r5444937 = 1.0;
        double r5444938 = r5444935 - r5444937;
        double r5444939 = r5444936 / r5444938;
        return r5444939;
}
↓
double f(double alpha, double beta, double i) {
        double r5444940 = beta;
        double r5444941 = 1.447602491634548e+178;
        bool r5444942 = r5444940 <= r5444941;
        double r5444943 = i;
        double r5444944 = alpha;
        double r5444945 = r5444940 + r5444944;
        double r5444946 = r5444943 + r5444945;
        double r5444947 = r5444943 * r5444946;
        double r5444948 = r5444944 * r5444940;
        double r5444949 = r5444947 + r5444948;
        double r5444950 = 2.0;
        double r5444951 = r5444950 * r5444943;
        double r5444952 = r5444945 + r5444951;
        double r5444953 = r5444949 / r5444952;
        double r5444954 = r5444947 / r5444952;
        double r5444955 = r5444953 * r5444954;
        double r5444956 = 1.0;
        double r5444957 = sqrt(r5444956);
        double r5444958 = r5444957 + r5444952;
        double r5444959 = r5444955 / r5444958;
        double r5444960 = r5444952 - r5444957;
        double r5444961 = r5444959 / r5444960;
        double r5444962 = 0.0;
        double r5444963 = r5444942 ? r5444961 : r5444962;
        return r5444963;
}

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 < 1.447602491634548e+178

    1. Initial program 51.1

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

    3. Applied add-sqr-sqrt51.1

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

    4. Applied difference-of-squares51.1

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

    5. Applied times-frac35.8

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

    6. Applied times-frac34.0

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

    8. Applied associate-*r/34.0

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

    10. Applied associate-*l/34.0

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

    if 1.447602491634548e+178 < beta

    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 45.9

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

  4. Final simplification35.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.447602491634548 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \alpha \cdot \beta}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))