Octave 3.8, jcobi/2

Average Error: 24.2 → 12.3

Time: 14.1s

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r103759 = alpha;
        double r103760 = beta;
        double r103761 = r103759 + r103760;
        double r103762 = r103760 - r103759;
        double r103763 = r103761 * r103762;
        double r103764 = 2.0;
        double r103765 = i;
        double r103766 = r103764 * r103765;
        double r103767 = r103761 + r103766;
        double r103768 = r103763 / r103767;
        double r103769 = r103767 + r103764;
        double r103770 = r103768 / r103769;
        double r103771 = 1.0;
        double r103772 = r103770 + r103771;
        double r103773 = r103772 / r103764;
        return r103773;
}

↓

double f(double alpha, double beta, double i) {
        double r103774 = beta;
        double r103775 = alpha;
        double r103776 = r103775 + r103774;
        double r103777 = 2.0;
        double r103778 = i;
        double r103779 = r103777 * r103778;
        double r103780 = r103776 + r103779;
        double r103781 = r103774 / r103780;
        double r103782 = cbrt(r103781);
        double r103783 = r103782 * r103782;
        double r103784 = r103783 * r103776;
        double r103785 = r103780 + r103777;
        double r103786 = r103782 / r103785;
        double r103787 = r103784 * r103786;
        double r103788 = r103775 / r103780;
        double r103789 = r103788 / r103785;
        double r103790 = -r103789;
        double r103791 = r103790 * r103776;
        double r103792 = 1.0;
        double r103793 = r103791 + r103792;
        double r103794 = r103787 + r103793;
        double r103795 = r103794 / r103777;
        return r103795;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Initial program 24.2

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

3. Applied *-un-lft-identity 24.2

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

4. Applied *-un-lft-identity 24.2

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

5. Applied times-frac 12.5

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

6. Applied times-frac 12.5

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

7. Simplified 12.5

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

9. Applied div-sub 12.5

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

10. Applied div-sub 12.5

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

12. Applied sub-neg 12.5

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

13. Applied distribute-lft-in 12.5

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

14. Applied associate-+l+ 12.3

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

15. Simplified 12.3

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

17. Applied *-un-lft-identity 12.3

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

18. Applied add-cube-cbrt 12.3

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

19. Applied times-frac 12.3

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

20. Applied associate-*r* 12.3

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

21. Simplified 12.3

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

22. Final simplification 12.3

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

Reproduce

herbie shell --seed 2020003 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))