Octave 3.8, jcobi/3

Average Error: 4.2 → 1.3

Time: 2.9m

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta) {
        double r610161 = alpha;
        double r610162 = beta;
        double r610163 = r610161 + r610162;
        double r610164 = r610162 * r610161;
        double r610165 = r610163 + r610164;
        double r610166 = 1.0;
        double r610167 = r610165 + r610166;
        double r610168 = 2.0;
        double r610169 = r610168 * r610166;
        double r610170 = r610163 + r610169;
        double r610171 = r610167 / r610170;
        double r610172 = r610171 / r610170;
        double r610173 = r610170 + r610166;
        double r610174 = r610172 / r610173;
        return r610174;
}

↓

double f(double alpha, double beta) {
        double r610175 = alpha;
        double r610176 = 1.8040289506123877e+158;
        bool r610177 = r610175 <= r610176;
        double r610178 = 1.0;
        double r610179 = beta;
        double r610180 = 1.0;
        double r610181 = 2.0;
        double r610182 = r610180 * r610181;
        double r610183 = r610179 + r610182;
        double r610184 = r610175 + r610183;
        double r610185 = r610179 + r610175;
        double r610186 = r610180 + r610185;
        double r610187 = r610179 * r610175;
        double r610188 = r610186 + r610187;
        double r610189 = r610184 / r610188;
        double r610190 = r610178 / r610184;
        double r610191 = r610189 / r610190;
        double r610192 = r610178 / r610191;
        double r610193 = sqrt(r610192);
        double r610194 = r610185 + r610182;
        double r610195 = r610194 + r610180;
        double r610196 = r610193 / r610195;
        double r610197 = r610196 * r610193;
        double r610198 = 2.0;
        double r610199 = r610175 / r610179;
        double r610200 = r610198 + r610199;
        double r610201 = r610179 / r610175;
        double r610202 = r610200 + r610201;
        double r610203 = r610178 / r610202;
        double r610204 = r610203 / r610195;
        double r610205 = r610177 ? r610197 : r610204;
        return r610205;
}

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

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

   4. Applied *-un-lft-identity 1.4

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

   5. Applied times-frac 1.4

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

   6. Applied associate-/l* 1.4

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

   7. Simplified 1.4

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

   9. Applied div-inv 1.4

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

   10. Applied associate-/r* 1.4

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

   12. Applied *-un-lft-identity 1.4

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

   13. Applied add-sqr-sqrt 1.5

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

   14. Applied times-frac 1.5

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

   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

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

   4. Applied *-un-lft-identity 18.0

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

   5. Applied times-frac 18.0

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

   6. Applied associate-/l* 18.0

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

   7. Simplified 18.0

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

   8. Taylor expanded around inf 0.1

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

4. Final simplification 1.3

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

Reproduce

herbie shell --seed 2019194 
(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)))