Average Error: 54.0 → 38.5
Time: 10.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 2.134998786884618 \cdot 10^{147}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(2 \cdot i\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right) + \left(\alpha + \beta\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 2.134998786884618 \cdot 10^{147}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(2 \cdot i\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right) + \left(\alpha + \beta\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r80378 = i;
        double r80379 = alpha;
        double r80380 = beta;
        double r80381 = r80379 + r80380;
        double r80382 = r80381 + r80378;
        double r80383 = r80378 * r80382;
        double r80384 = r80380 * r80379;
        double r80385 = r80384 + r80383;
        double r80386 = r80383 * r80385;
        double r80387 = 2.0;
        double r80388 = r80387 * r80378;
        double r80389 = r80381 + r80388;
        double r80390 = r80389 * r80389;
        double r80391 = r80386 / r80390;
        double r80392 = 1.0;
        double r80393 = r80390 - r80392;
        double r80394 = r80391 / r80393;
        return r80394;
}
↓
double f(double alpha, double beta, double i) {
        double r80395 = beta;
        double r80396 = 2.134998786884618e+147;
        bool r80397 = r80395 <= r80396;
        double r80398 = alpha;
        double r80399 = i;
        double r80400 = r80398 + r80395;
        double r80401 = r80400 + r80399;
        double r80402 = r80399 * r80401;
        double r80403 = fma(r80395, r80398, r80402);
        double r80404 = 2.0;
        double r80405 = r80404 * r80399;
        double r80406 = fma(r80404, r80399, r80400);
        double r80407 = r80405 * r80406;
        double r80408 = r80400 * r80406;
        double r80409 = r80407 + r80408;
        double r80410 = r80403 / r80409;
        double r80411 = 1.0;
        double r80412 = -r80411;
        double r80413 = fma(r80406, r80406, r80412);
        double r80414 = r80402 / r80413;
        double r80415 = r80410 * r80414;
        double r80416 = 0.0;
        double r80417 = r80397 ? r80415 : r80416;
        return r80417;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if beta < 2.134998786884618e+147

    1. Initial program 51.9

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

    2. Simplified53.1

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

    4. Applied times-frac36.2

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

    6. Applied fma-udef36.2

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

    7. Applied distribute-lft-in36.2

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

    8. Simplified36.2

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

    9. Simplified36.2

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

    if 2.134998786884618e+147 < 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. Simplified64.0

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

    4. Applied times-frac57.6

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

    5. Taylor expanded around inf 49.6

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

  4. Final simplification38.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 2.134998786884618 \cdot 10^{147}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(2 \cdot i\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right) + \left(\alpha + \beta\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))