Average Error: 54.4 → 37.2
Time: 1.2m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 6.237470943579032829354147639761347668239 \cdot 10^{185}:\\ \;\;\;\;\frac{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1}}{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 6.237470943579032829354147639761347668239 \cdot 10^{185}:\\
\;\;\;\;\frac{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1}}{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r5488032 = i;
        double r5488033 = alpha;
        double r5488034 = beta;
        double r5488035 = r5488033 + r5488034;
        double r5488036 = r5488035 + r5488032;
        double r5488037 = r5488032 * r5488036;
        double r5488038 = r5488034 * r5488033;
        double r5488039 = r5488038 + r5488037;
        double r5488040 = r5488037 * r5488039;
        double r5488041 = 2.0;
        double r5488042 = r5488041 * r5488032;
        double r5488043 = r5488035 + r5488042;
        double r5488044 = r5488043 * r5488043;
        double r5488045 = r5488040 / r5488044;
        double r5488046 = 1.0;
        double r5488047 = r5488044 - r5488046;
        double r5488048 = r5488045 / r5488047;
        return r5488048;
}
↓
double f(double alpha, double beta, double i) {
        double r5488049 = beta;
        double r5488050 = 6.237470943579033e+185;
        bool r5488051 = r5488049 <= r5488050;
        double r5488052 = i;
        double r5488053 = alpha;
        double r5488054 = r5488049 + r5488053;
        double r5488055 = r5488052 + r5488054;
        double r5488056 = r5488052 * r5488055;
        double r5488057 = 2.0;
        double r5488058 = r5488057 * r5488052;
        double r5488059 = r5488054 + r5488058;
        double r5488060 = r5488056 / r5488059;
        double r5488061 = 1.0;
        double r5488062 = sqrt(r5488061);
        double r5488063 = r5488062 + r5488059;
        double r5488064 = r5488060 / r5488063;
        double r5488065 = r5488059 - r5488062;
        double r5488066 = r5488053 * r5488049;
        double r5488067 = r5488066 + r5488056;
        double r5488068 = r5488067 / r5488059;
        double r5488069 = r5488065 / r5488068;
        double r5488070 = r5488064 / r5488069;
        double r5488071 = 0.0;
        double r5488072 = r5488051 ? r5488070 : r5488071;
        return r5488072;
}

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 < 6.237470943579033e+185

    1. Initial program 52.9

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

    3. Applied add-sqr-sqrt52.9

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

    4. Applied difference-of-squares52.9

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

    5. Applied times-frac37.9

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

    6. Applied times-frac36.0

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

    8. Applied associate-*r/36.0

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

    10. Applied associate-/l*36.0

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

    if 6.237470943579033e+185 < beta

    1. Initial program 64.0

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]

    2. Taylor expanded around inf 45.4

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

  4. Final simplification37.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 6.237470943579032829354147639761347668239 \cdot 10^{185}:\\ \;\;\;\;\frac{\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{1} + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) - \sqrt{1}}{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2 \cdot i}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))