Average Error: 54.0 → 47.2
Time: 15.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 4.14491162512083 \cdot 10^{96}:\\ \;\;\;\;\frac{i \cdot 1}{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 4.14491162512083 \cdot 10^{96}:\\
\;\;\;\;\frac{i \cdot 1}{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r119213 = i;
        double r119214 = alpha;
        double r119215 = beta;
        double r119216 = r119214 + r119215;
        double r119217 = r119216 + r119213;
        double r119218 = r119213 * r119217;
        double r119219 = r119215 * r119214;
        double r119220 = r119219 + r119218;
        double r119221 = r119218 * r119220;
        double r119222 = 2.0;
        double r119223 = r119222 * r119213;
        double r119224 = r119216 + r119223;
        double r119225 = r119224 * r119224;
        double r119226 = r119221 / r119225;
        double r119227 = 1.0;
        double r119228 = r119225 - r119227;
        double r119229 = r119226 / r119228;
        return r119229;
}
↓
double f(double alpha, double beta, double i) {
        double r119230 = beta;
        double r119231 = 4.144911625120832e+96;
        bool r119232 = r119230 <= r119231;
        double r119233 = i;
        double r119234 = 1.0;
        double r119235 = r119233 * r119234;
        double r119236 = alpha;
        double r119237 = r119236 + r119230;
        double r119238 = 2.0;
        double r119239 = r119238 * r119233;
        double r119240 = r119237 + r119239;
        double r119241 = 3.0;
        double r119242 = pow(r119240, r119241);
        double r119243 = fma(r119233, r119238, r119237);
        double r119244 = 1.0;
        double r119245 = -r119244;
        double r119246 = r119243 * r119245;
        double r119247 = r119242 + r119246;
        double r119248 = r119235 / r119247;
        double r119249 = r119237 + r119233;
        double r119250 = r119233 * r119249;
        double r119251 = fma(r119230, r119236, r119250);
        double r119252 = r119243 / r119251;
        double r119253 = r119249 / r119252;
        double r119254 = r119248 * r119253;
        double r119255 = 0.0;
        double r119256 = r119232 ? r119254 : r119255;
        return r119256;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if beta < 4.144911625120832e+96

    1. Initial program 51.4

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

    2. Simplified50.6

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

    4. Applied *-un-lft-identity50.6

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

    5. Applied times-frac45.8

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

    6. Applied times-frac45.8

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

    7. Simplified45.8

      \[\leadsto \color{blue}{\frac{i \cdot 1}{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    if 4.144911625120832e+96 < beta

    1. Initial program 62.6

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

    2. Simplified59.7

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

    3. Taylor expanded around inf 52.1

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

  4. Final simplification47.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 4.14491162512083 \cdot 10^{96}:\\ \;\;\;\;\frac{i \cdot 1}{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :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)))