Average Error: 54.1 → 37.2
Time: 14.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 3.7225611126883413 \cdot 10^{87}:\\ \;\;\;\;\frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \mathbf{elif}\;i \le 6.72619285808465442 \cdot 10^{153}:\\ \;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;i \le 3.7225611126883413 \cdot 10^{87}:\\
\;\;\;\;\frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \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}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\

\mathbf{elif}\;i \le 6.72619285808465442 \cdot 10^{153}:\\
\;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r174272 = i;
        double r174273 = alpha;
        double r174274 = beta;
        double r174275 = r174273 + r174274;
        double r174276 = r174275 + r174272;
        double r174277 = r174272 * r174276;
        double r174278 = r174274 * r174273;
        double r174279 = r174278 + r174277;
        double r174280 = r174277 * r174279;
        double r174281 = 2.0;
        double r174282 = r174281 * r174272;
        double r174283 = r174275 + r174282;
        double r174284 = r174283 * r174283;
        double r174285 = r174280 / r174284;
        double r174286 = 1.0;
        double r174287 = r174284 - r174286;
        double r174288 = r174285 / r174287;
        return r174288;
}
↓
double f(double alpha, double beta, double i) {
        double r174289 = i;
        double r174290 = 3.7225611126883413e+87;
        bool r174291 = r174289 <= r174290;
        double r174292 = alpha;
        double r174293 = beta;
        double r174294 = r174292 + r174293;
        double r174295 = r174294 + r174289;
        double r174296 = r174289 * r174295;
        double r174297 = 2.0;
        double r174298 = fma(r174297, r174289, r174294);
        double r174299 = r174296 / r174298;
        double r174300 = fma(r174293, r174292, r174296);
        double r174301 = r174300 / r174298;
        double r174302 = r174299 * r174301;
        double r174303 = 1.0;
        double r174304 = sqrt(r174303);
        double r174305 = r174298 + r174304;
        double r174306 = r174302 / r174305;
        double r174307 = r174297 * r174289;
        double r174308 = r174294 + r174307;
        double r174309 = r174308 - r174304;
        double r174310 = r174306 / r174309;
        double r174311 = 6.726192858084654e+153;
        bool r174312 = r174289 <= r174311;
        double r174313 = 0.25;
        double r174314 = 2.0;
        double r174315 = pow(r174289, r174314);
        double r174316 = r174313 * r174315;
        double r174317 = r174308 * r174308;
        double r174318 = r174317 - r174303;
        double r174319 = r174316 / r174318;
        double r174320 = 0.0;
        double r174321 = r174312 ? r174319 : r174320;
        double r174322 = r174291 ? r174310 : r174321;
        return r174322;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if i < 3.7225611126883413e+87

    1. Initial program 29.1

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

    3. Applied times-frac12.0

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

    4. Simplified12.0

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

    6. Applied add-sqr-sqrt12.0

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

    7. Applied difference-of-squares12.0

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

    8. Applied associate-/r*7.6

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

    9. Simplified7.6

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

    if 3.7225611126883413e+87 < i < 6.726192858084654e+153

    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 18.4

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]

    if 6.726192858084654e+153 < i

    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. Using strategy rm

    3. Applied times-frac64.0

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

    4. Simplified64.0

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

    5. Taylor expanded around inf 61.9

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

  4. Final simplification37.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3.7225611126883413 \cdot 10^{87}:\\ \;\;\;\;\frac{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + \sqrt{1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \mathbf{elif}\;i \le 6.72619285808465442 \cdot 10^{153}:\\ \;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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)))