Octave 3.8, jcobi/1

Average Error: 16.2 → 8.5

Time: 1.1m

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 7863989155318432839393331734927900672:\\ \;\;\;\;\frac{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} + \left(1 \cdot 1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot 1\right)\right) \cdot \beta - \left(\log \left(e^{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)}\right) \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)}}\right) - {1}^{3}\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}{2 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} + \left(1 \cdot 1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot 1\right)\right)\right)}\\ \mathbf{elif}\;\alpha \le 1.526450071771057118621392850450502817566 \cdot 10^{113}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \le 1.143721273478083214922895355787158286969 \cdot 10^{167}:\\ \;\;\;\;\frac{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} + \left(1 \cdot 1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot 1\right)\right) \cdot \beta - \left(\log \left(e^{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)}\right) \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)}}\right) - {1}^{3}\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}{2 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} + \left(1 \cdot 1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r5697276 = beta;
        double r5697277 = alpha;
        double r5697278 = r5697276 - r5697277;
        double r5697279 = r5697277 + r5697276;
        double r5697280 = 2.0;
        double r5697281 = r5697279 + r5697280;
        double r5697282 = r5697278 / r5697281;
        double r5697283 = 1.0;
        double r5697284 = r5697282 + r5697283;
        double r5697285 = r5697284 / r5697280;
        return r5697285;
}

↓

double f(double alpha, double beta) {
        double r5697286 = alpha;
        double r5697287 = 7.863989155318433e+36;
        bool r5697288 = r5697286 <= r5697287;
        double r5697289 = 2.0;
        double r5697290 = beta;
        double r5697291 = r5697290 + r5697286;
        double r5697292 = r5697289 + r5697291;
        double r5697293 = r5697286 / r5697292;
        double r5697294 = r5697293 * r5697293;
        double r5697295 = 1.0;
        double r5697296 = r5697295 * r5697295;
        double r5697297 = r5697293 * r5697295;
        double r5697298 = r5697296 + r5697297;
        double r5697299 = r5697294 + r5697298;
        double r5697300 = r5697299 * r5697290;
        double r5697301 = r5697294 * r5697293;
        double r5697302 = exp(r5697301);
        double r5697303 = log(r5697302);
        double r5697304 = 3.0;
        double r5697305 = pow(r5697295, r5697304);
        double r5697306 = r5697303 - r5697305;
        double r5697307 = r5697306 * r5697292;
        double r5697308 = r5697300 - r5697307;
        double r5697309 = r5697292 * r5697299;
        double r5697310 = r5697289 * r5697309;
        double r5697311 = r5697308 / r5697310;
        double r5697312 = 1.5264500717710571e+113;
        bool r5697313 = r5697286 <= r5697312;
        double r5697314 = r5697290 / r5697292;
        double r5697315 = 4.0;
        double r5697316 = r5697286 * r5697286;
        double r5697317 = r5697315 / r5697316;
        double r5697318 = r5697289 / r5697286;
        double r5697319 = r5697317 - r5697318;
        double r5697320 = 8.0;
        double r5697321 = r5697316 * r5697286;
        double r5697322 = r5697320 / r5697321;
        double r5697323 = r5697319 - r5697322;
        double r5697324 = r5697314 - r5697323;
        double r5697325 = r5697324 / r5697289;
        double r5697326 = 1.1437212734780832e+167;
        bool r5697327 = r5697286 <= r5697326;
        double r5697328 = r5697327 ? r5697311 : r5697325;
        double r5697329 = r5697313 ? r5697325 : r5697328;
        double r5697330 = r5697288 ? r5697311 : r5697329;
        return r5697330;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 7.863989155318433e+36 or 1.5264500717710571e+113 < alpha < 1.1437212734780832e+167

   1. Initial program 5.4

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

   3. Applied div-sub 5.4

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

   4. Applied associate-+l- 5.3

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

   6. Applied flip3-- 5.3

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

   7. Applied frac-sub 5.3

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

   8. Applied associate-/l/ 5.3

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

   10. Applied add-log-exp 5.3

       \[\leadsto \frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\color{blue}{\log \left(e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}}\right)} - {1}^{3}\right)}{2 \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right)\right)}\]

   11. Simplified 5.3

       \[\leadsto \frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\log \color{blue}{\left(e^{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right)} - {1}^{3}\right)}{2 \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right)\right)}\]

   if 7.863989155318433e+36 < alpha < 1.5264500717710571e+113 or 1.1437212734780832e+167 < alpha

   1. Initial program 51.0

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

   3. Applied div-sub 51.0

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

   4. Applied associate-+l- 49.2

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

   5. Taylor expanded around inf 18.7

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

   6. Simplified 18.7

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7863989155318432839393331734927900672:\\ \;\;\;\;\frac{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} + \left(1 \cdot 1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot 1\right)\right) \cdot \beta - \left(\log \left(e^{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)}\right) \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)}}\right) - {1}^{3}\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}{2 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} + \left(1 \cdot 1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot 1\right)\right)\right)}\\ \mathbf{elif}\;\alpha \le 1.526450071771057118621392850450502817566 \cdot 10^{113}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \le 1.143721273478083214922895355787158286969 \cdot 10^{167}:\\ \;\;\;\;\frac{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} + \left(1 \cdot 1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot 1\right)\right) \cdot \beta - \left(\log \left(e^{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)}\right) \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)}}\right) - {1}^{3}\right) \cdot \left(2 + \left(\beta + \alpha\right)\right)}{2 \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} + \left(1 \cdot 1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))