Average Error: 2.3 → 0.4
Time: 22.4s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i}{2} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}}{i \cdot 2 + 1.0}\]
\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}
↓
\frac{\frac{i}{2} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}}{i \cdot 2 + 1.0}
double f(double i) {
        double r2029764 = i;
        double r2029765 = r2029764 * r2029764;
        double r2029766 = r2029765 * r2029765;
        double r2029767 = 2.0;
        double r2029768 = /* ERROR: no posit support in C */;
        double r2029769 = r2029768 * r2029764;
        double r2029770 = r2029769 * r2029769;
        double r2029771 = r2029766 / r2029770;
        double r2029772 = 1.0;
        double r2029773 = /* ERROR: no posit support in C */;
        double r2029774 = r2029770 - r2029773;
        double r2029775 = r2029771 / r2029774;
        return r2029775;
}
↓
double f(double i) {
        double r2029776 = i;
        double r2029777 = 2.0;
        double r2029778 = r2029776 / r2029777;
        double r2029779 = r2029776 * r2029777;
        double r2029780 = 1.0;
        double r2029781 = r2029779 - r2029780;
        double r2029782 = r2029778 / r2029781;
        double r2029783 = r2029778 * r2029782;
        double r2029784 = r2029779 + r2029780;
        double r2029785 = r2029783 / r2029784;
        return r2029785;
}

Error

Bits error versus i

Derivation

  1. Initial program 2.3

    \[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]

  2. Simplified2.3

    \[\leadsto \color{blue}{\left(\frac{\left(i \cdot i\right)}{\left(\left(\left(i \cdot \left(2\right)\right) \cdot \left(i \cdot \left(2\right)\right)\right) \cdot \left(\left(\left(i \cdot \left(2\right)\right) \cdot \left(i \cdot \left(2\right)\right)\right) - \left(1.0\right)\right)\right)}\right) \cdot \left(i \cdot i\right)}\]
  3. Using strategy rm

  4. Applied p16-times-frac1.2

    \[\leadsto \color{blue}{\left(\left(\frac{i}{\left(\left(i \cdot \left(2\right)\right) \cdot \left(i \cdot \left(2\right)\right)\right)}\right) \cdot \left(\frac{i}{\left(\left(\left(i \cdot \left(2\right)\right) \cdot \left(i \cdot \left(2\right)\right)\right) - \left(1.0\right)\right)}\right)\right)} \cdot \left(i \cdot i\right)\]
  5. Using strategy rm

  6. Applied associate-*r/1.2

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

  7. Applied associate-*l/1.1

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

  8. Simplified0.9

    \[\leadsto \frac{\color{blue}{\left(\left(\frac{i}{\left(2\right)}\right) \cdot \left(\frac{i}{\left(2\right)}\right)\right)}}{\left(\left(\left(i \cdot \left(2\right)\right) \cdot \left(i \cdot \left(2\right)\right)\right) - \left(1.0\right)\right)}\]
  9. Using strategy rm

  10. Applied difference-of-sqr-10.8

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

  11. Applied p16-times-frac0.4

    \[\leadsto \color{blue}{\left(\frac{\left(\frac{i}{\left(2\right)}\right)}{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(1.0\right)}\right)}\right) \cdot \left(\frac{\left(\frac{i}{\left(2\right)}\right)}{\left(\left(i \cdot \left(2\right)\right) - \left(1.0\right)\right)}\right)}\]
  12. Using strategy rm

  13. Applied associate-*l/0.4

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

  14. Final simplification0.4

    \[\leadsto \frac{\frac{i}{2} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}}{i \cdot 2 + 1.0}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (>.p16 i (real->posit16 0)))
  (/.p16 (/.p16 (*.p16 (*.p16 i i) (*.p16 i i)) (*.p16 (*.p16 (real->posit16 2) i) (*.p16 (real->posit16 2) i))) (-.p16 (*.p16 (*.p16 (real->posit16 2) i) (*.p16 (real->posit16 2) i)) (real->posit16 1.0))))