Average Error: 52.8 → 35.4
Time: 4.7m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.0145625771446497 \cdot 10^{+217}:\\ \;\;\;\;\frac{\frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i} - \sqrt{\sqrt{1.0}}} \cdot \left(\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\sqrt{\sqrt{1.0}} + \sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 4.0145625771446497 \cdot 10^{+217}:\\
\;\;\;\;\frac{\frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i} - \sqrt{\sqrt{1.0}}} \cdot \left(\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\sqrt{\sqrt{1.0}} + \sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}\right)\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r32628292 = i;
        double r32628293 = alpha;
        double r32628294 = beta;
        double r32628295 = r32628293 + r32628294;
        double r32628296 = r32628295 + r32628292;
        double r32628297 = r32628292 * r32628296;
        double r32628298 = r32628294 * r32628293;
        double r32628299 = r32628298 + r32628297;
        double r32628300 = r32628297 * r32628299;
        double r32628301 = 2.0;
        double r32628302 = r32628301 * r32628292;
        double r32628303 = r32628295 + r32628302;
        double r32628304 = r32628303 * r32628303;
        double r32628305 = r32628300 / r32628304;
        double r32628306 = 1.0;
        double r32628307 = r32628304 - r32628306;
        double r32628308 = r32628305 / r32628307;
        return r32628308;
}
↓
double f(double alpha, double beta, double i) {
        double r32628309 = alpha;
        double r32628310 = 4.0145625771446497e+217;
        bool r32628311 = r32628309 <= r32628310;
        double r32628312 = i;
        double r32628313 = beta;
        double r32628314 = r32628309 + r32628313;
        double r32628315 = r32628312 + r32628314;
        double r32628316 = r32628312 * r32628315;
        double r32628317 = r32628313 * r32628309;
        double r32628318 = r32628316 + r32628317;
        double r32628319 = sqrt(r32628318);
        double r32628320 = 2.0;
        double r32628321 = r32628320 * r32628312;
        double r32628322 = r32628314 + r32628321;
        double r32628323 = r32628319 / r32628322;
        double r32628324 = sqrt(r32628322);
        double r32628325 = 1.0;
        double r32628326 = sqrt(r32628325);
        double r32628327 = sqrt(r32628326);
        double r32628328 = r32628324 - r32628327;
        double r32628329 = r32628323 / r32628328;
        double r32628330 = r32628316 / r32628322;
        double r32628331 = r32628326 + r32628322;
        double r32628332 = r32628330 / r32628331;
        double r32628333 = r32628327 + r32628324;
        double r32628334 = r32628319 / r32628333;
        double r32628335 = r32628332 * r32628334;
        double r32628336 = r32628329 * r32628335;
        double r32628337 = 0.0;
        double r32628338 = r32628311 ? r32628336 : r32628337;
        return r32628338;
}

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 < 4.0145625771446497e+217

    1. Initial program 51.7

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

    3. Applied add-sqr-sqrt51.7

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

    4. Applied difference-of-squares51.7

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

    5. Applied times-frac36.8

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

    6. Applied times-frac34.4

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

    8. Applied add-sqr-sqrt34.4

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

    9. Applied add-sqr-sqrt34.5

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

    10. Applied difference-of-squares34.5

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

    11. Applied *-un-lft-identity34.5

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

    12. Applied add-sqr-sqrt34.6

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

    13. Applied times-frac34.5

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

    14. Applied times-frac34.5

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

    15. Applied associate-*r*34.6

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

    if 4.0145625771446497e+217 < 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 43.2

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

  4. Final simplification35.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.0145625771446497 \cdot 10^{+217}:\\ \;\;\;\;\frac{\frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i} - \sqrt{\sqrt{1.0}}} \cdot \left(\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \beta \cdot \alpha}}{\sqrt{\sqrt{1.0}} + \sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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)))