Octave 3.8, jcobi/3

Average Error: 4.2 → 1.2

Time: 26.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.80402895061238771139523421196263224057 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\beta - 2 \cdot 1\right) + \alpha}{\mathsf{fma}\left(1, 2, 1\right) + \left(\beta + \alpha\right)} \cdot \frac{\frac{1}{\mathsf{fma}\left(2, 1, \beta + \alpha\right) \cdot \frac{\beta + \left(\alpha - 2 \cdot 1\right)}{\left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right) + \alpha}}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - 2 \cdot 1\right) + \alpha}{\mathsf{fma}\left(1, 2, 1\right) + \left(\beta + \alpha\right)} \cdot \frac{\frac{1}{\frac{\alpha}{\beta} + \left(\frac{\beta}{\alpha} + 2\right)}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r78295 = alpha;
        double r78296 = beta;
        double r78297 = r78295 + r78296;
        double r78298 = r78296 * r78295;
        double r78299 = r78297 + r78298;
        double r78300 = 1.0;
        double r78301 = r78299 + r78300;
        double r78302 = 2.0;
        double r78303 = r78302 * r78300;
        double r78304 = r78297 + r78303;
        double r78305 = r78301 / r78304;
        double r78306 = r78305 / r78304;
        double r78307 = r78304 + r78300;
        double r78308 = r78306 / r78307;
        return r78308;
}

↓

double f(double alpha, double beta) {
        double r78309 = alpha;
        double r78310 = 1.8040289506123877e+158;
        bool r78311 = r78309 <= r78310;
        double r78312 = beta;
        double r78313 = 2.0;
        double r78314 = 1.0;
        double r78315 = r78313 * r78314;
        double r78316 = r78312 - r78315;
        double r78317 = r78316 + r78309;
        double r78318 = fma(r78314, r78313, r78314);
        double r78319 = r78312 + r78309;
        double r78320 = r78318 + r78319;
        double r78321 = r78317 / r78320;
        double r78322 = 1.0;
        double r78323 = fma(r78313, r78314, r78319);
        double r78324 = r78309 - r78315;
        double r78325 = r78312 + r78324;
        double r78326 = fma(r78309, r78312, r78314);
        double r78327 = r78312 + r78326;
        double r78328 = r78327 + r78309;
        double r78329 = r78325 / r78328;
        double r78330 = r78323 * r78329;
        double r78331 = r78322 / r78330;
        double r78332 = r78331 / r78323;
        double r78333 = r78321 * r78332;
        double r78334 = r78309 / r78312;
        double r78335 = r78312 / r78309;
        double r78336 = 2.0;
        double r78337 = r78335 + r78336;
        double r78338 = r78334 + r78337;
        double r78339 = r78322 / r78338;
        double r78340 = r78339 / r78323;
        double r78341 = r78321 * r78340;
        double r78342 = r78311 ? r78333 : r78341;
        return r78342;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 1.8040289506123877e+158

   1. Initial program 1.4

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

   3. Applied *-un-lft-identity 1.4

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

   4. Applied flip-+ 2.1

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

   5. Applied associate-/r/ 2.1

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

   6. Applied times-frac 2.1

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

   7. Simplified 1.4

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

   8. Simplified 1.4

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

   10. Applied *-un-lft-identity 1.4

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

   11. Applied *-un-lft-identity 1.4

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

   12. Applied times-frac 1.4

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

   13. Applied associate-/l* 1.4

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

   14. Simplified 1.4

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

   if 1.8040289506123877e+158 < alpha

   1. Initial program 18.0

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

   3. Applied *-un-lft-identity 18.0

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

   4. Applied flip-+ 19.4

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

   5. Applied associate-/r/ 19.4

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

   6. Applied times-frac 19.4

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

   7. Simplified 18.0

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

   8. Simplified 18.0

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

   10. Applied *-un-lft-identity 18.0

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

   11. Applied *-un-lft-identity 18.0

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

   12. Applied times-frac 18.0

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

   13. Applied associate-/l* 18.0

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

   14. Simplified 18.0

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

   15. Taylor expanded around inf 0.1

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

   16. Simplified 0.1

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.80402895061238771139523421196263224057 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\beta - 2 \cdot 1\right) + \alpha}{\mathsf{fma}\left(1, 2, 1\right) + \left(\beta + \alpha\right)} \cdot \frac{\frac{1}{\mathsf{fma}\left(2, 1, \beta + \alpha\right) \cdot \frac{\beta + \left(\alpha - 2 \cdot 1\right)}{\left(\beta + \mathsf{fma}\left(\alpha, \beta, 1\right)\right) + \alpha}}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - 2 \cdot 1\right) + \alpha}{\mathsf{fma}\left(1, 2, 1\right) + \left(\beta + \alpha\right)} \cdot \frac{\frac{1}{\frac{\alpha}{\beta} + \left(\frac{\beta}{\alpha} + 2\right)}}{\mathsf{fma}\left(2, 1, \beta + \alpha\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))