Average Error: 52.9 → 36.4
Time: 3.5m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.891349107737516 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \left(\frac{1}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.891349107737516 \cdot 10^{+164}:\\
\;\;\;\;\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)}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \left(\frac{1}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right)\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r15483246 = i;
        double r15483247 = alpha;
        double r15483248 = beta;
        double r15483249 = r15483247 + r15483248;
        double r15483250 = r15483249 + r15483246;
        double r15483251 = r15483246 * r15483250;
        double r15483252 = r15483248 * r15483247;
        double r15483253 = r15483252 + r15483251;
        double r15483254 = r15483251 * r15483253;
        double r15483255 = 2.0;
        double r15483256 = r15483255 * r15483246;
        double r15483257 = r15483249 + r15483256;
        double r15483258 = r15483257 * r15483257;
        double r15483259 = r15483254 / r15483258;
        double r15483260 = 1.0;
        double r15483261 = r15483258 - r15483260;
        double r15483262 = r15483259 / r15483261;
        return r15483262;
}
↓
double f(double alpha, double beta, double i) {
        double r15483263 = alpha;
        double r15483264 = 2.891349107737516e+164;
        bool r15483265 = r15483263 <= r15483264;
        double r15483266 = i;
        double r15483267 = beta;
        double r15483268 = r15483267 + r15483263;
        double r15483269 = r15483266 + r15483268;
        double r15483270 = r15483267 * r15483263;
        double r15483271 = fma(r15483269, r15483266, r15483270);
        double r15483272 = 2.0;
        double r15483273 = fma(r15483272, r15483266, r15483268);
        double r15483274 = r15483271 / r15483273;
        double r15483275 = 1.0;
        double r15483276 = sqrt(r15483275);
        double r15483277 = r15483276 + r15483273;
        double r15483278 = r15483274 / r15483277;
        double r15483279 = 1.0;
        double r15483280 = r15483273 - r15483276;
        double r15483281 = r15483279 / r15483280;
        double r15483282 = r15483266 * r15483269;
        double r15483283 = r15483282 / r15483273;
        double r15483284 = r15483281 * r15483283;
        double r15483285 = r15483278 * r15483284;
        double r15483286 = 0.0;
        double r15483287 = r15483265 ? r15483285 : r15483286;
        return r15483287;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2.891349107737516e+164

    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-frac34.2

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

    9. Applied div-inv34.2

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

    if 2.891349107737516e+164 < alpha

    1. Initial program 62.5

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

    2. Simplified62.5

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

    3. Taylor expanded around inf 47.8

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

  4. Final simplification36.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.891349107737516 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(i + \left(\beta + \alpha\right)\right), i, \left(\beta \cdot \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}}{\sqrt{1.0} + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \left(\frac{1}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right) - \sqrt{1.0}} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +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)))