Octave 3.8, jcobi/2

Average Error: 23.6 → 11.1

Time: 24.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.3451371616542299 \cdot 10^{+213}:\\ \;\;\;\;\frac{\log \left(e^{1.0 + \left(\left(\beta + \alpha\right) \cdot \frac{1}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r2106768 = alpha;
        double r2106769 = beta;
        double r2106770 = r2106768 + r2106769;
        double r2106771 = r2106769 - r2106768;
        double r2106772 = r2106770 * r2106771;
        double r2106773 = 2.0;
        double r2106774 = i;
        double r2106775 = r2106773 * r2106774;
        double r2106776 = r2106770 + r2106775;
        double r2106777 = r2106772 / r2106776;
        double r2106778 = 2.0;
        double r2106779 = r2106776 + r2106778;
        double r2106780 = r2106777 / r2106779;
        double r2106781 = 1.0;
        double r2106782 = r2106780 + r2106781;
        double r2106783 = r2106782 / r2106778;
        return r2106783;
}

↓

double f(double alpha, double beta, double i) {
        double r2106784 = alpha;
        double r2106785 = 1.3451371616542299e+213;
        bool r2106786 = r2106784 <= r2106785;
        double r2106787 = 1.0;
        double r2106788 = beta;
        double r2106789 = r2106788 + r2106784;
        double r2106790 = 1.0;
        double r2106791 = 2.0;
        double r2106792 = i;
        double r2106793 = 2.0;
        double r2106794 = r2106792 * r2106793;
        double r2106795 = r2106794 + r2106789;
        double r2106796 = r2106791 + r2106795;
        double r2106797 = sqrt(r2106796);
        double r2106798 = r2106790 / r2106797;
        double r2106799 = r2106789 * r2106798;
        double r2106800 = r2106788 - r2106784;
        double r2106801 = r2106800 / r2106795;
        double r2106802 = r2106801 / r2106797;
        double r2106803 = r2106799 * r2106802;
        double r2106804 = r2106787 + r2106803;
        double r2106805 = exp(r2106804);
        double r2106806 = log(r2106805);
        double r2106807 = r2106806 / r2106791;
        double r2106808 = r2106791 / r2106784;
        double r2106809 = 8.0;
        double r2106810 = r2106784 * r2106784;
        double r2106811 = r2106784 * r2106810;
        double r2106812 = r2106809 / r2106811;
        double r2106813 = 4.0;
        double r2106814 = r2106813 / r2106810;
        double r2106815 = r2106812 - r2106814;
        double r2106816 = r2106808 + r2106815;
        double r2106817 = r2106816 / r2106791;
        double r2106818 = r2106786 ? r2106807 : r2106817;
        return r2106818;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.3451371616542299e+213

   1. Initial program 19.2

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

   3. Applied *-un-lft-identity 19.2

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

   4. Applied *-un-lft-identity 19.2

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

   5. Applied times-frac 7.6

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

   6. Applied times-frac 7.6

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

   7. Simplified 7.6

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

   9. Applied add-log-exp 7.6

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

   11. Applied add-sqr-sqrt 7.6

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

   12. Applied *-un-lft-identity 7.6

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

   13. Applied times-frac 7.6

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

   14. Applied associate-*r* 7.6

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

   if 1.3451371616542299e+213 < alpha

   1. Initial program 63.2

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

   3. Applied *-un-lft-identity 63.2

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

   4. Applied *-un-lft-identity 63.2

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

   5. Applied times-frac 51.8

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

   6. Applied times-frac 51.9

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

   7. Simplified 51.9

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

   9. Applied add-log-exp 51.9

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

   10. Taylor expanded around inf 41.6

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

   11. Simplified 41.6

       \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.3451371616542299 \cdot 10^{+213}:\\ \;\;\;\;\frac{\log \left(e^{1.0 + \left(\left(\beta + \alpha\right) \cdot \frac{1}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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))