Octave 3.8, jcobi/2

Average Error: 23.8 → 12.4

Time: 3.6m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.39654162857562 \cdot 10^{+62}:\\ \;\;\;\;\frac{(\left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{2.0 + (i \cdot 2 + \left(\beta + \alpha\right))_*} \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\beta + \alpha\right))_*}}\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha}}{(i \cdot 2 + \left(\beta + \alpha\right))_*}}{\sqrt[3]{2.0 + (i \cdot 2 + \left(\beta + \alpha\right))_*}}\right) + 1.0)_*}{2.0}\\ \mathbf{elif}\;\alpha \le 2.703385935698668 \cdot 10^{+135}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{elif}\;\alpha \le 3.857914563165407 \cdot 10^{+217}:\\ \;\;\;\;\frac{e^{\log \left((\left(\beta + \alpha\right) \cdot \left(\frac{\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{(2 \cdot i + \left(\beta + \alpha\right))_* + 2.0}\right) + 1.0)_*\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r28072150 = alpha;
        double r28072151 = beta;
        double r28072152 = r28072150 + r28072151;
        double r28072153 = r28072151 - r28072150;
        double r28072154 = r28072152 * r28072153;
        double r28072155 = 2.0;
        double r28072156 = i;
        double r28072157 = r28072155 * r28072156;
        double r28072158 = r28072152 + r28072157;
        double r28072159 = r28072154 / r28072158;
        double r28072160 = 2.0;
        double r28072161 = r28072158 + r28072160;
        double r28072162 = r28072159 / r28072161;
        double r28072163 = 1.0;
        double r28072164 = r28072162 + r28072163;
        double r28072165 = r28072164 / r28072160;
        return r28072165;
}

↓

double f(double alpha, double beta, double i) {
        double r28072166 = alpha;
        double r28072167 = 1.39654162857562e+62;
        bool r28072168 = r28072166 <= r28072167;
        double r28072169 = beta;
        double r28072170 = r28072169 + r28072166;
        double r28072171 = r28072169 - r28072166;
        double r28072172 = cbrt(r28072171);
        double r28072173 = r28072172 * r28072172;
        double r28072174 = 2.0;
        double r28072175 = i;
        double r28072176 = 2.0;
        double r28072177 = fma(r28072175, r28072176, r28072170);
        double r28072178 = r28072174 + r28072177;
        double r28072179 = cbrt(r28072178);
        double r28072180 = r28072179 * r28072179;
        double r28072181 = r28072173 / r28072180;
        double r28072182 = r28072170 * r28072181;
        double r28072183 = r28072172 / r28072177;
        double r28072184 = r28072183 / r28072179;
        double r28072185 = 1.0;
        double r28072186 = fma(r28072182, r28072184, r28072185);
        double r28072187 = r28072186 / r28072174;
        double r28072188 = 2.703385935698668e+135;
        bool r28072189 = r28072166 <= r28072188;
        double r28072190 = 1.0;
        double r28072191 = r28072166 * r28072166;
        double r28072192 = r28072190 / r28072191;
        double r28072193 = 8.0;
        double r28072194 = r28072193 / r28072166;
        double r28072195 = 4.0;
        double r28072196 = r28072194 - r28072195;
        double r28072197 = r28072174 / r28072166;
        double r28072198 = fma(r28072192, r28072196, r28072197);
        double r28072199 = r28072198 / r28072174;
        double r28072200 = 3.857914563165407e+217;
        bool r28072201 = r28072166 <= r28072200;
        double r28072202 = fma(r28072176, r28072175, r28072170);
        double r28072203 = r28072171 / r28072202;
        double r28072204 = r28072202 + r28072174;
        double r28072205 = r28072203 / r28072204;
        double r28072206 = fma(r28072170, r28072205, r28072185);
        double r28072207 = log(r28072206);
        double r28072208 = exp(r28072207);
        double r28072209 = r28072208 / r28072174;
        double r28072210 = r28072201 ? r28072209 : r28072199;
        double r28072211 = r28072189 ? r28072199 : r28072210;
        double r28072212 = r28072168 ? r28072187 : r28072211;
        return r28072212;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 1.39654162857562e+62

   1. Initial program 12.1

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

   3. Applied *-un-lft-identity 12.1

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

   4. Applied *-un-lft-identity 12.1

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   5. Applied times-frac 1.9

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   6. Applied times-frac 1.9

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   7. Simplified 1.9

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

   8. Simplified 1.9

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\frac{\frac{\beta - \alpha}{(i \cdot 2 + \left(\alpha + \beta\right))_*}}{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}} + 1.0}{2.0}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 2.0

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{(i \cdot 2 + \left(\alpha + \beta\right))_*}}{\color{blue}{\left(\sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*} \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}\right) \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}}} + 1.0}{2.0}\]

   11. Applied *-un-lft-identity 2.0

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\color{blue}{1 \cdot (i \cdot 2 + \left(\alpha + \beta\right))_*}}}{\left(\sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*} \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}\right) \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}} + 1.0}{2.0}\]

   12. Applied add-cube-cbrt 1.9

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \sqrt[3]{\beta - \alpha}}}{1 \cdot (i \cdot 2 + \left(\alpha + \beta\right))_*}}{\left(\sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*} \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}\right) \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}} + 1.0}{2.0}\]

   13. Applied times-frac 1.9

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1} \cdot \frac{\sqrt[3]{\beta - \alpha}}{(i \cdot 2 + \left(\alpha + \beta\right))_*}}}{\left(\sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*} \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}\right) \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}} + 1.0}{2.0}\]

   14. Applied times-frac 1.9

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*} \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{(i \cdot 2 + \left(\alpha + \beta\right))_*}}{\sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}}\right)} + 1.0}{2.0}\]

   15. Applied associate-*r* 1.9

       \[\leadsto \frac{\color{blue}{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*} \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{(i \cdot 2 + \left(\alpha + \beta\right))_*}}{\sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}}} + 1.0}{2.0}\]
   16. Using strategy rm

   17. Applied fma-def 1.9

       \[\leadsto \frac{\color{blue}{(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*} \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}}\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha}}{(i \cdot 2 + \left(\alpha + \beta\right))_*}}{\sqrt[3]{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}}\right) + 1.0)_*}}{2.0}\]

   if 1.39654162857562e+62 < alpha < 2.703385935698668e+135 or 3.857914563165407e+217 < alpha

   1. Initial program 52.8

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

   2. Taylor expanded around inf 40.7

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

   3. Simplified 40.7

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

   if 2.703385935698668e+135 < alpha < 3.857914563165407e+217

   1. Initial program 58.8

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

   3. Applied *-un-lft-identity 58.8

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

   4. Applied *-un-lft-identity 58.8

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   5. Applied times-frac 39.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   6. Applied times-frac 39.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   7. Simplified 39.3

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

   8. Simplified 39.3

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\frac{\frac{\beta - \alpha}{(i \cdot 2 + \left(\alpha + \beta\right))_*}}{2.0 + (i \cdot 2 + \left(\alpha + \beta\right))_*}} + 1.0}{2.0}\]
   9. Using strategy rm

   10. Applied add-exp-log 39.3

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

   12. Applied add-exp-log 39.3

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

   13. Applied rem-log-exp 39.3

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

   14. Simplified 39.6

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

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.39654162857562 \cdot 10^{+62}:\\ \;\;\;\;\frac{(\left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{2.0 + (i \cdot 2 + \left(\beta + \alpha\right))_*} \cdot \sqrt[3]{2.0 + (i \cdot 2 + \left(\beta + \alpha\right))_*}}\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha}}{(i \cdot 2 + \left(\beta + \alpha\right))_*}}{\sqrt[3]{2.0 + (i \cdot 2 + \left(\beta + \alpha\right))_*}}\right) + 1.0)_*}{2.0}\\ \mathbf{elif}\;\alpha \le 2.703385935698668 \cdot 10^{+135}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{elif}\;\alpha \le 3.857914563165407 \cdot 10^{+217}:\\ \;\;\;\;\frac{e^{\log \left((\left(\beta + \alpha\right) \cdot \left(\frac{\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{(2 \cdot i + \left(\beta + \alpha\right))_* + 2.0}\right) + 1.0)_*\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +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))