Octave 3.8, jcobi/2

Average Error: 23.5 → 11.8

Time: 16.9s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.841782349532883 \cdot 10^{+51}:\\ \;\;\;\;\frac{e^{\log \left(\left(\beta - \alpha\right) \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) + 1.0\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.8551287992761453 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 8.621431291668078 \cdot 10^{+174}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(1, \beta, \left(-\sqrt{\alpha}\right) \cdot \sqrt{\alpha}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) + \left(\mathsf{fma}\left(-\sqrt{\alpha}, \sqrt{\alpha}, \sqrt{\alpha} \cdot \sqrt{\alpha}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) + 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r1599215 = alpha;
        double r1599216 = beta;
        double r1599217 = r1599215 + r1599216;
        double r1599218 = r1599216 - r1599215;
        double r1599219 = r1599217 * r1599218;
        double r1599220 = 2.0;
        double r1599221 = i;
        double r1599222 = r1599220 * r1599221;
        double r1599223 = r1599217 + r1599222;
        double r1599224 = r1599219 / r1599223;
        double r1599225 = 2.0;
        double r1599226 = r1599223 + r1599225;
        double r1599227 = r1599224 / r1599226;
        double r1599228 = 1.0;
        double r1599229 = r1599227 + r1599228;
        double r1599230 = r1599229 / r1599225;
        return r1599230;
}

↓

double f(double alpha, double beta, double i) {
        double r1599231 = alpha;
        double r1599232 = 5.841782349532883e+51;
        bool r1599233 = r1599231 <= r1599232;
        double r1599234 = beta;
        double r1599235 = r1599234 - r1599231;
        double r1599236 = 1.0;
        double r1599237 = i;
        double r1599238 = 2.0;
        double r1599239 = r1599231 + r1599234;
        double r1599240 = fma(r1599237, r1599238, r1599239);
        double r1599241 = r1599236 / r1599240;
        double r1599242 = 2.0;
        double r1599243 = r1599242 + r1599240;
        double r1599244 = r1599239 / r1599243;
        double r1599245 = r1599241 * r1599244;
        double r1599246 = r1599235 * r1599245;
        double r1599247 = 1.0;
        double r1599248 = r1599246 + r1599247;
        double r1599249 = log(r1599248);
        double r1599250 = exp(r1599249);
        double r1599251 = r1599250 / r1599242;
        double r1599252 = 1.8551287992761453e+84;
        bool r1599253 = r1599231 <= r1599252;
        double r1599254 = 8.0;
        double r1599255 = r1599231 * r1599231;
        double r1599256 = r1599254 / r1599255;
        double r1599257 = r1599256 / r1599231;
        double r1599258 = 4.0;
        double r1599259 = r1599258 / r1599255;
        double r1599260 = r1599257 - r1599259;
        double r1599261 = r1599242 / r1599231;
        double r1599262 = r1599260 + r1599261;
        double r1599263 = r1599262 / r1599242;
        double r1599264 = 8.621431291668078e+174;
        bool r1599265 = r1599231 <= r1599264;
        double r1599266 = sqrt(r1599231);
        double r1599267 = -r1599266;
        double r1599268 = r1599267 * r1599266;
        double r1599269 = fma(r1599236, r1599234, r1599268);
        double r1599270 = r1599269 * r1599245;
        double r1599271 = r1599266 * r1599266;
        double r1599272 = fma(r1599267, r1599266, r1599271);
        double r1599273 = r1599272 * r1599245;
        double r1599274 = r1599273 + r1599247;
        double r1599275 = r1599270 + r1599274;
        double r1599276 = log(r1599275);
        double r1599277 = exp(r1599276);
        double r1599278 = r1599277 / r1599242;
        double r1599279 = r1599265 ? r1599278 : r1599263;
        double r1599280 = r1599253 ? r1599263 : r1599279;
        double r1599281 = r1599233 ? r1599251 : r1599280;
        return r1599281;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 5.841782349532883e+51

   1. Initial program 11.6

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Simplified 11.6

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)} + 1.0}{2.0}}\]
   3. Using strategy rm

   4. Applied add-exp-log 11.6

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)} + 1.0\right)}}}{2.0}\]

   5. Simplified 8.8

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0\right)}, \beta - \alpha, 1.0\right)\right)}}}{2.0}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 8.8

      \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\frac{\color{blue}{1 \cdot \left(\alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0\right)}, \beta - \alpha, 1.0\right)\right)}}{2.0}\]

   8. Applied times-frac 1.2

      \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\color{blue}{\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}}, \beta - \alpha, 1.0\right)\right)}}{2.0}\]
   9. Using strategy rm

   10. Applied fma-udef 1.2

       \[\leadsto \frac{e^{\log \color{blue}{\left(\left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}\right) \cdot \left(\beta - \alpha\right) + 1.0\right)}}}{2.0}\]

   if 5.841782349532883e+51 < alpha < 1.8551287992761453e+84 or 8.621431291668078e+174 < alpha

   1. Initial program 57.7

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Simplified 57.1

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)} + 1.0}{2.0}}\]
   3. Using strategy rm

   4. Applied add-exp-log 57.1

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)} + 1.0\right)}}}{2.0}\]

   5. Simplified 49.2

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0\right)}, \beta - \alpha, 1.0\right)\right)}}}{2.0}\]

   6. Taylor expanded around inf 41.3

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

   7. Simplified 41.3

      \[\leadsto \frac{\color{blue}{\left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}}{2.0}\]

   if 1.8551287992761453e+84 < alpha < 8.621431291668078e+174

   1. Initial program 48.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Simplified 47.8

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)} + 1.0}{2.0}}\]
   3. Using strategy rm

   4. Applied add-exp-log 47.8

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot 2.0\right)} + 1.0\right)}}}{2.0}\]

   5. Simplified 42.0

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0\right)}, \beta - \alpha, 1.0\right)\right)}}}{2.0}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 42.0

      \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\frac{\color{blue}{1 \cdot \left(\alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0\right)}, \beta - \alpha, 1.0\right)\right)}}{2.0}\]

   8. Applied times-frac 35.1

      \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\color{blue}{\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}}, \beta - \alpha, 1.0\right)\right)}}{2.0}\]
   9. Using strategy rm

   10. Applied fma-udef 35.1

       \[\leadsto \frac{e^{\log \color{blue}{\left(\left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}\right) \cdot \left(\beta - \alpha\right) + 1.0\right)}}}{2.0}\]
   11. Using strategy rm

   12. Applied add-sqr-sqrt 35.1

       \[\leadsto \frac{e^{\log \left(\left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}\right) \cdot \left(\beta - \color{blue}{\sqrt{\alpha} \cdot \sqrt{\alpha}}\right) + 1.0\right)}}{2.0}\]

   13. Applied *-un-lft-identity 35.1

       \[\leadsto \frac{e^{\log \left(\left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}\right) \cdot \left(\color{blue}{1 \cdot \beta} - \sqrt{\alpha} \cdot \sqrt{\alpha}\right) + 1.0\right)}}{2.0}\]

   14. Applied prod-diff 35.1

       \[\leadsto \frac{e^{\log \left(\left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(1, \beta, -\sqrt{\alpha} \cdot \sqrt{\alpha}\right) + \mathsf{fma}\left(-\sqrt{\alpha}, \sqrt{\alpha}, \sqrt{\alpha} \cdot \sqrt{\alpha}\right)\right)} + 1.0\right)}}{2.0}\]

   15. Applied distribute-lft-in 35.0

       \[\leadsto \frac{e^{\log \left(\color{blue}{\left(\left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}\right) \cdot \mathsf{fma}\left(1, \beta, -\sqrt{\alpha} \cdot \sqrt{\alpha}\right) + \left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}\right) \cdot \mathsf{fma}\left(-\sqrt{\alpha}, \sqrt{\alpha}, \sqrt{\alpha} \cdot \sqrt{\alpha}\right)\right)} + 1.0\right)}}{2.0}\]

   16. Applied associate-+l+ 35.2

       \[\leadsto \frac{e^{\log \color{blue}{\left(\left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}\right) \cdot \mathsf{fma}\left(1, \beta, -\sqrt{\alpha} \cdot \sqrt{\alpha}\right) + \left(\left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 2.0}\right) \cdot \mathsf{fma}\left(-\sqrt{\alpha}, \sqrt{\alpha}, \sqrt{\alpha} \cdot \sqrt{\alpha}\right) + 1.0\right)\right)}}}{2.0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.841782349532883 \cdot 10^{+51}:\\ \;\;\;\;\frac{e^{\log \left(\left(\beta - \alpha\right) \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) + 1.0\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.8551287992761453 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 8.621431291668078 \cdot 10^{+174}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{fma}\left(1, \beta, \left(-\sqrt{\alpha}\right) \cdot \sqrt{\alpha}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) + \left(\mathsf{fma}\left(-\sqrt{\alpha}, \sqrt{\alpha}, \sqrt{\alpha} \cdot \sqrt{\alpha}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) + 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))