Octave 3.8, jcobi/4, as called

Average Error: 2.4 → 0.4

Time: 7.0m

Precision: 64

\[i \gt \left(0\right)\]

Math

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

↓

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

C

double f(double i) {
        double r3172059 = i;
        double r3172060 = r3172059 * r3172059;
        double r3172061 = r3172060 * r3172060;
        double r3172062 = 2.0;
        double r3172063 = /* ERROR: no posit support in C */;
        double r3172064 = r3172063 * r3172059;
        double r3172065 = r3172064 * r3172064;
        double r3172066 = r3172061 / r3172065;
        double r3172067 = 1.0;
        double r3172068 = /* ERROR: no posit support in C */;
        double r3172069 = r3172065 - r3172068;
        double r3172070 = r3172066 / r3172069;
        return r3172070;
}

↓

double f(double i) {
        double r3172071 = i;
        double r3172072 = 2.0;
        double r3172073 = /* ERROR: no posit support in C */;
        double r3172074 = r3172071 / r3172073;
        double r3172075 = r3172071 * r3172073;
        double r3172076 = 1.0;
        double r3172077 = /* ERROR: no posit support in C */;
        double r3172078 = r3172075 + r3172077;
        double r3172079 = r3172074 / r3172078;
        double r3172080 = r3172075 - r3172077;
        double r3172081 = r3172074 / r3172080;
        double r3172082 = r3172079 * r3172081;
        return r3172082;
}

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. Simplified 1.2

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

4. Applied associate-*r/ 1.1

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

6. Applied associate-/r* 0.9

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

8. Applied associate-/r* 0.8

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

9. Simplified 0.8

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

11. Applied p16-*-un-lft-identity 0.8

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

12. Applied difference-of-squares 0.8

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

13. Applied p16-times-frac 0.8

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

14. Applied p16-times-frac 0.4

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

15. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019168 +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))))