Octave 3.8, jcobi/2

Average Error: 23.8 → 11.2

Time: 1.0m

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.042060258518315547008313851677255522197 \cdot 10^{134}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{i \cdot 2 + \left(\beta + \alpha\right)} + 1\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{i \cdot 2 + \left(\beta + \alpha\right)} + 1\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{i \cdot 2 + \left(\beta + \alpha\right)} + 1\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \le 2.709646863409569463522336468763265388785 \cdot 10^{149}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{elif}\;\alpha \le 1.371995439469866450102810596052968859341 \cdot 10^{190}:\\ \;\;\;\;\frac{1 + \left(\left(\beta + \alpha\right) \cdot \frac{\sqrt{\alpha} + \sqrt{\beta}}{\sqrt[3]{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}\right) \cdot \frac{\frac{\sqrt{\beta} - \sqrt{\alpha}}{\sqrt[3]{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{i \cdot 2 + \left(\beta + \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4605743 = alpha;
        double r4605744 = beta;
        double r4605745 = r4605743 + r4605744;
        double r4605746 = r4605744 - r4605743;
        double r4605747 = r4605745 * r4605746;
        double r4605748 = 2.0;
        double r4605749 = i;
        double r4605750 = r4605748 * r4605749;
        double r4605751 = r4605745 + r4605750;
        double r4605752 = r4605747 / r4605751;
        double r4605753 = r4605751 + r4605748;
        double r4605754 = r4605752 / r4605753;
        double r4605755 = 1.0;
        double r4605756 = r4605754 + r4605755;
        double r4605757 = r4605756 / r4605748;
        return r4605757;
}

↓

double f(double alpha, double beta, double i) {
        double r4605758 = alpha;
        double r4605759 = 2.0420602585183155e+134;
        bool r4605760 = r4605758 <= r4605759;
        double r4605761 = beta;
        double r4605762 = r4605761 + r4605758;
        double r4605763 = r4605761 - r4605758;
        double r4605764 = 2.0;
        double r4605765 = i;
        double r4605766 = r4605765 * r4605764;
        double r4605767 = r4605766 + r4605762;
        double r4605768 = r4605764 + r4605767;
        double r4605769 = r4605763 / r4605768;
        double r4605770 = r4605769 / r4605767;
        double r4605771 = r4605762 * r4605770;
        double r4605772 = 1.0;
        double r4605773 = r4605771 + r4605772;
        double r4605774 = r4605773 * r4605773;
        double r4605775 = r4605773 * r4605774;
        double r4605776 = cbrt(r4605775);
        double r4605777 = r4605776 / r4605764;
        double r4605778 = 2.7096468634095695e+149;
        bool r4605779 = r4605758 <= r4605778;
        double r4605780 = r4605764 / r4605758;
        double r4605781 = 8.0;
        double r4605782 = r4605758 * r4605758;
        double r4605783 = r4605758 * r4605782;
        double r4605784 = r4605781 / r4605783;
        double r4605785 = r4605780 + r4605784;
        double r4605786 = 4.0;
        double r4605787 = r4605786 / r4605782;
        double r4605788 = r4605785 - r4605787;
        double r4605789 = r4605788 / r4605764;
        double r4605790 = 1.3719954394698665e+190;
        bool r4605791 = r4605758 <= r4605790;
        double r4605792 = sqrt(r4605758);
        double r4605793 = sqrt(r4605761);
        double r4605794 = r4605792 + r4605793;
        double r4605795 = cbrt(r4605768);
        double r4605796 = r4605795 * r4605795;
        double r4605797 = r4605794 / r4605796;
        double r4605798 = r4605762 * r4605797;
        double r4605799 = r4605793 - r4605792;
        double r4605800 = r4605799 / r4605795;
        double r4605801 = r4605800 / r4605767;
        double r4605802 = r4605798 * r4605801;
        double r4605803 = r4605772 + r4605802;
        double r4605804 = r4605803 / r4605764;
        double r4605805 = r4605791 ? r4605804 : r4605789;
        double r4605806 = r4605779 ? r4605789 : r4605805;
        double r4605807 = r4605760 ? r4605777 : r4605806;
        return r4605807;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 2.0420602585183155e+134

   1. Initial program 15.0

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

   3. Applied *-un-lft-identity 15.0

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

   4. Applied *-un-lft-identity 15.0

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

   5. Applied times-frac 4.4

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

   6. Applied times-frac 4.3

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

   7. Simplified 4.3

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

   8. Simplified 4.3

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

   10. Applied add-cbrt-cube 4.4

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

   if 2.0420602585183155e+134 < alpha < 2.7096468634095695e+149 or 1.3719954394698665e+190 < alpha

   1. Initial program 62.3

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

   2. Taylor expanded around inf 39.8

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   3. Simplified 39.8

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{4}{\alpha \cdot \alpha}}}{2}\]

   if 2.7096468634095695e+149 < alpha < 1.3719954394698665e+190

   1. Initial program 61.9

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

   3. Applied *-un-lft-identity 61.9

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

   4. Applied *-un-lft-identity 61.9

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

   5. Applied times-frac 39.1

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

   6. Applied times-frac 39.0

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

   7. Simplified 39.0

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

   8. Simplified 39.0

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

   10. Applied *-un-lft-identity 39.0

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

   11. Applied add-cube-cbrt 39.2

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

   12. Applied add-sqr-sqrt 39.1

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

   13. Applied add-sqr-sqrt 46.5

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

   14. Applied difference-of-squares 46.5

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

   15. Applied times-frac 46.4

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

   16. Applied times-frac 46.4

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

   17. Applied associate-*r* 46.4

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

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.042060258518315547008313851677255522197 \cdot 10^{134}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{i \cdot 2 + \left(\beta + \alpha\right)} + 1\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{i \cdot 2 + \left(\beta + \alpha\right)} + 1\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{i \cdot 2 + \left(\beta + \alpha\right)} + 1\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \le 2.709646863409569463522336468763265388785 \cdot 10^{149}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{elif}\;\alpha \le 1.371995439469866450102810596052968859341 \cdot 10^{190}:\\ \;\;\;\;\frac{1 + \left(\left(\beta + \alpha\right) \cdot \frac{\sqrt{\alpha} + \sqrt{\beta}}{\sqrt[3]{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}\right) \cdot \frac{\frac{\sqrt{\beta} - \sqrt{\alpha}}{\sqrt[3]{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{i \cdot 2 + \left(\beta + \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))