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 r3172187 = i;
        double r3172188 = r3172187 * r3172187;
        double r3172189 = r3172188 * r3172188;
        double r3172190 = 2.0;
        double r3172191 = /* ERROR: no posit support in C */;
        double r3172192 = r3172191 * r3172187;
        double r3172193 = r3172192 * r3172192;
        double r3172194 = r3172189 / r3172193;
        double r3172195 = 1.0;
        double r3172196 = /* ERROR: no posit support in C */;
        double r3172197 = r3172193 - r3172196;
        double r3172198 = r3172194 / r3172197;
        return r3172198;
}

↓

double f(double i) {
        double r3172199 = i;
        double r3172200 = 2.0;
        double r3172201 = /* ERROR: no posit support in C */;
        double r3172202 = r3172199 / r3172201;
        double r3172203 = r3172199 * r3172201;
        double r3172204 = 1.0;
        double r3172205 = /* ERROR: no posit support in C */;
        double r3172206 = r3172203 + r3172205;
        double r3172207 = r3172202 / r3172206;
        double r3172208 = r3172203 - r3172205;
        double r3172209 = r3172202 / r3172208;
        double r3172210 = r3172207 * r3172209;
        return r3172210;
}

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 
(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))))