Octave 3.8, jcobi/3

Average Error: 3.9 → 2.9

Time: 35.8s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 8.099889163284841219645855361269251574276 \cdot 10^{151}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{\left(2 \cdot 1 + \left(\beta + \alpha\right)\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\alpha + \left(2 \cdot 1 + \beta\right)}}{\sqrt{\alpha + \left(2 \cdot 1 + \beta\right)}}}{\sqrt{1 + \left(\alpha + \left(2 \cdot 1 + \beta\right)\right)}} \cdot \left(\left(\left(\beta \cdot \sqrt{0.1666666666666666574148081281236954964697} + \left(\alpha \cdot \sqrt{0.1666666666666666574148081281236954964697} + \sqrt{0.1666666666666666574148081281236954964697} \cdot 1\right)\right) - \left(\sqrt{0.004629629629629629372633559114547097124159} \cdot \alpha\right) \cdot 2.5\right) - \frac{\beta \cdot 0.06944444444444444752839729062543483451009}{\sqrt{0.1666666666666666574148081281236954964697}}\right)\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r92208 = alpha;
        double r92209 = beta;
        double r92210 = r92208 + r92209;
        double r92211 = r92209 * r92208;
        double r92212 = r92210 + r92211;
        double r92213 = 1.0;
        double r92214 = r92212 + r92213;
        double r92215 = 2.0;
        double r92216 = r92215 * r92213;
        double r92217 = r92210 + r92216;
        double r92218 = r92214 / r92217;
        double r92219 = r92218 / r92217;
        double r92220 = r92217 + r92213;
        double r92221 = r92219 / r92220;
        return r92221;
}

↓

double f(double alpha, double beta) {
        double r92222 = alpha;
        double r92223 = 8.099889163284841e+151;
        bool r92224 = r92222 <= r92223;
        double r92225 = 1.0;
        double r92226 = beta;
        double r92227 = r92226 * r92222;
        double r92228 = r92226 + r92222;
        double r92229 = r92227 + r92228;
        double r92230 = r92225 + r92229;
        double r92231 = 2.0;
        double r92232 = r92231 * r92225;
        double r92233 = r92232 + r92228;
        double r92234 = r92230 / r92233;
        double r92235 = r92234 / r92233;
        double r92236 = r92233 + r92225;
        double r92237 = r92235 / r92236;
        double r92238 = 1.0;
        double r92239 = r92232 + r92226;
        double r92240 = r92222 + r92239;
        double r92241 = r92238 / r92240;
        double r92242 = sqrt(r92240);
        double r92243 = r92241 / r92242;
        double r92244 = r92225 + r92240;
        double r92245 = sqrt(r92244);
        double r92246 = r92243 / r92245;
        double r92247 = 0.16666666666666666;
        double r92248 = sqrt(r92247);
        double r92249 = r92226 * r92248;
        double r92250 = r92222 * r92248;
        double r92251 = r92248 * r92225;
        double r92252 = r92250 + r92251;
        double r92253 = r92249 + r92252;
        double r92254 = 0.004629629629629629;
        double r92255 = sqrt(r92254);
        double r92256 = r92255 * r92222;
        double r92257 = 2.5;
        double r92258 = r92256 * r92257;
        double r92259 = r92253 - r92258;
        double r92260 = 0.06944444444444445;
        double r92261 = r92226 * r92260;
        double r92262 = r92261 / r92248;
        double r92263 = r92259 - r92262;
        double r92264 = r92246 * r92263;
        double r92265 = r92224 ? r92237 : r92264;
        return r92265;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 8.099889163284841e+151

   1. Initial program 1.3

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

   3. Applied pow1 1.3

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

   if 8.099889163284841e+151 < alpha

   1. Initial program 16.1

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

   3. Applied pow1 16.1

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

   5. Applied add-sqr-sqrt 16.2

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

   6. Applied add-sqr-sqrt 16.2

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

   7. Applied div-inv 16.2

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

   8. Applied times-frac 16.2

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

   9. Applied unpow-prod-down 16.2

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

   10. Applied times-frac 17.5

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

   11. Simplified 17.5

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

   12. Simplified 17.5

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

   13. Taylor expanded around 0 10.8

       \[\leadsto \color{blue}{\left(\left(\sqrt{0.1666666666666666574148081281236954964697} \cdot \alpha + \left(1 \cdot \sqrt{0.1666666666666666574148081281236954964697} + \sqrt{0.1666666666666666574148081281236954964697} \cdot \beta\right)\right) - \left(2.5 \cdot \left(\sqrt{0.004629629629629629372633559114547097124159} \cdot \alpha\right) + 0.06944444444444444752839729062543483451009 \cdot \frac{\beta}{\sqrt{0.1666666666666666574148081281236954964697}}\right)\right)} \cdot \frac{\frac{\frac{1}{\alpha + \left(\beta + 2 \cdot 1\right)}}{\sqrt{\alpha + \left(\beta + 2 \cdot 1\right)}}}{\sqrt{\left(\alpha + \left(\beta + 2 \cdot 1\right)\right) + 1}}\]

   14. Simplified 10.8

       \[\leadsto \color{blue}{\left(\left(\left(\left(\sqrt{0.1666666666666666574148081281236954964697} \cdot \alpha + 1 \cdot \sqrt{0.1666666666666666574148081281236954964697}\right) + \sqrt{0.1666666666666666574148081281236954964697} \cdot \beta\right) - 2.5 \cdot \left(\alpha \cdot \sqrt{0.004629629629629629372633559114547097124159}\right)\right) - \frac{0.06944444444444444752839729062543483451009 \cdot \beta}{\sqrt{0.1666666666666666574148081281236954964697}}\right)} \cdot \frac{\frac{\frac{1}{\alpha + \left(\beta + 2 \cdot 1\right)}}{\sqrt{\alpha + \left(\beta + 2 \cdot 1\right)}}}{\sqrt{\left(\alpha + \left(\beta + 2 \cdot 1\right)\right) + 1}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8.099889163284841219645855361269251574276 \cdot 10^{151}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{2 \cdot 1 + \left(\beta + \alpha\right)}}{2 \cdot 1 + \left(\beta + \alpha\right)}}{\left(2 \cdot 1 + \left(\beta + \alpha\right)\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\alpha + \left(2 \cdot 1 + \beta\right)}}{\sqrt{\alpha + \left(2 \cdot 1 + \beta\right)}}}{\sqrt{1 + \left(\alpha + \left(2 \cdot 1 + \beta\right)\right)}} \cdot \left(\left(\left(\beta \cdot \sqrt{0.1666666666666666574148081281236954964697} + \left(\alpha \cdot \sqrt{0.1666666666666666574148081281236954964697} + \sqrt{0.1666666666666666574148081281236954964697} \cdot 1\right)\right) - \left(\sqrt{0.004629629629629629372633559114547097124159} \cdot \alpha\right) \cdot 2.5\right) - \frac{\beta \cdot 0.06944444444444444752839729062543483451009}{\sqrt{0.1666666666666666574148081281236954964697}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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)))