Average Error: 2.4 → 0.4
Time: 58.1s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{\frac{i}{2}}{\frac{i \cdot 2 - 1.0}{\frac{i}{2}}}}{i \cdot 2 + 1.0}\]
\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{\frac{i}{2}}{\frac{i \cdot 2 - 1.0}{\frac{i}{2}}}}{i \cdot 2 + 1.0}
double f(double i) {
        double r5113699 = i;
        double r5113700 = r5113699 * r5113699;
        double r5113701 = r5113700 * r5113700;
        double r5113702 = 2.0;
        double r5113703 = /* ERROR: no posit support in C */;
        double r5113704 = r5113703 * r5113699;
        double r5113705 = r5113704 * r5113704;
        double r5113706 = r5113701 / r5113705;
        double r5113707 = 1.0;
        double r5113708 = /* ERROR: no posit support in C */;
        double r5113709 = r5113705 - r5113708;
        double r5113710 = r5113706 / r5113709;
        return r5113710;
}
↓
double f(double i) {
        double r5113711 = i;
        double r5113712 = 2.0;
        double r5113713 = r5113711 / r5113712;
        double r5113714 = r5113711 * r5113712;
        double r5113715 = 1.0;
        double r5113716 = r5113714 - r5113715;
        double r5113717 = r5113716 / r5113713;
        double r5113718 = r5113713 / r5113717;
        double r5113719 = r5113714 + r5113715;
        double r5113720 = r5113718 / r5113719;
        return r5113720;
}

Error

Bits error versus i

Derivation

  1. Initial program 2.4

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

  2. Simplified0.9

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

  4. Applied difference-of-sqr-10.9

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

  5. Applied associate-/r*0.6

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

  7. Applied associate-*r/0.5

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

  9. Applied associate-/l*0.5

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

  11. Applied associate-/r/0.5

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

  12. Applied associate-/r*0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 +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))))