Average Error: 54.1 → 36.2
Time: 42.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 4.07062621862513012 \cdot 10^{207}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}} \cdot \frac{1}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, \sqrt{1}\right)}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 4.07062621862513012 \cdot 10^{207}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}} \cdot \frac{1}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, \sqrt{1}\right)}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r139294 = i;
        double r139295 = alpha;
        double r139296 = beta;
        double r139297 = r139295 + r139296;
        double r139298 = r139297 + r139294;
        double r139299 = r139294 * r139298;
        double r139300 = r139296 * r139295;
        double r139301 = r139300 + r139299;
        double r139302 = r139299 * r139301;
        double r139303 = 2.0;
        double r139304 = r139303 * r139294;
        double r139305 = r139297 + r139304;
        double r139306 = r139305 * r139305;
        double r139307 = r139302 / r139306;
        double r139308 = 1.0;
        double r139309 = r139306 - r139308;
        double r139310 = r139307 / r139309;
        return r139310;
}
↓
double f(double alpha, double beta, double i) {
        double r139311 = beta;
        double r139312 = 4.07062621862513e+207;
        bool r139313 = r139311 <= r139312;
        double r139314 = alpha;
        double r139315 = i;
        double r139316 = r139314 + r139311;
        double r139317 = r139316 + r139315;
        double r139318 = r139315 * r139317;
        double r139319 = fma(r139311, r139314, r139318);
        double r139320 = sqrt(r139319);
        double r139321 = 2.0;
        double r139322 = fma(r139321, r139315, r139316);
        double r139323 = 1.0;
        double r139324 = sqrt(r139323);
        double r139325 = r139322 - r139324;
        double r139326 = r139320 / r139322;
        double r139327 = r139325 / r139326;
        double r139328 = r139320 / r139327;
        double r139329 = 1.0;
        double r139330 = fma(r139315, r139321, r139324);
        double r139331 = r139316 + r139330;
        double r139332 = r139318 / r139322;
        double r139333 = r139331 / r139332;
        double r139334 = r139329 / r139333;
        double r139335 = r139328 * r139334;
        double r139336 = 0.0;
        double r139337 = r139313 ? r139335 : r139336;
        return r139337;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if beta < 4.07062621862513e+207

    1. Initial program 52.8

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

    3. Applied add-sqr-sqrt52.8

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

    4. Applied difference-of-squares52.8

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

    5. Applied times-frac37.6

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

    6. Applied times-frac35.3

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

    7. Simplified35.3

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

    8. Simplified35.3

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

    10. Applied clear-num35.3

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

    12. Applied *-un-lft-identity35.3

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

    13. Applied add-sqr-sqrt35.3

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

    14. Applied times-frac35.3

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

    15. Applied associate-/l*35.3

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

    if 4.07062621862513e+207 < 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 44.1

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

  4. Final simplification36.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 4.07062621862513012 \cdot 10^{207}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}} \cdot \frac{1}{\frac{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, \sqrt{1}\right)}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(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)))