Average Error: 52.5 → 35.4
Time: 3.7m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.5101402696953266 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}{i}}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) + \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;\alpha \le 2.5101402696953266 \cdot 10^{+189}:\\
\;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}{i}}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) + \sqrt{1.0}}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r25188326 = i;
        double r25188327 = alpha;
        double r25188328 = beta;
        double r25188329 = r25188327 + r25188328;
        double r25188330 = r25188329 + r25188326;
        double r25188331 = r25188326 * r25188330;
        double r25188332 = r25188328 * r25188327;
        double r25188333 = r25188332 + r25188331;
        double r25188334 = r25188331 * r25188333;
        double r25188335 = 2.0;
        double r25188336 = r25188335 * r25188326;
        double r25188337 = r25188329 + r25188336;
        double r25188338 = r25188337 * r25188337;
        double r25188339 = r25188334 / r25188338;
        double r25188340 = 1.0;
        double r25188341 = r25188338 - r25188340;
        double r25188342 = r25188339 / r25188341;
        return r25188342;
}
↓
double f(double alpha, double beta, double i) {
        double r25188343 = alpha;
        double r25188344 = 2.5101402696953266e+189;
        bool r25188345 = r25188343 <= r25188344;
        double r25188346 = i;
        double r25188347 = beta;
        double r25188348 = r25188347 + r25188343;
        double r25188349 = r25188346 + r25188348;
        double r25188350 = r25188347 * r25188343;
        double r25188351 = fma(r25188349, r25188346, r25188350);
        double r25188352 = 2.0;
        double r25188353 = fma(r25188352, r25188346, r25188348);
        double r25188354 = r25188351 / r25188353;
        double r25188355 = r25188353 / r25188346;
        double r25188356 = r25188349 / r25188355;
        double r25188357 = r25188354 * r25188356;
        double r25188358 = 1.0;
        double r25188359 = sqrt(r25188358);
        double r25188360 = r25188353 - r25188359;
        double r25188361 = r25188357 / r25188360;
        double r25188362 = r25188353 + r25188359;
        double r25188363 = r25188361 / r25188362;
        double r25188364 = 0.0;
        double r25188365 = r25188345 ? r25188363 : r25188364;
        return r25188365;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2.5101402696953266e+189

    1. Initial program 51.0

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

    2. Simplified51.0

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

    4. Applied add-sqr-sqrt51.0

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

    5. Applied difference-of-squares51.0

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

    6. Applied times-frac35.8

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

    7. Applied times-frac33.7

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

    9. Applied associate-*l/33.7

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

    11. Applied associate-*r/33.7

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

    13. Applied associate-/l*33.7

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

    if 2.5101402696953266e+189 < 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. Simplified62.6

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

    4. Applied add-sqr-sqrt62.6

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

    5. Applied difference-of-squares62.6

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

    6. Applied times-frac56.8

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

    7. Applied times-frac53.6

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

    8. Taylor expanded around -inf 47.0

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

  4. Final simplification35.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.5101402696953266 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\frac{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}{i}}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}}}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) + \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(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)))