Average Error: 2.3 → 0.4
Time: 21.8s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i}{2}}{i \cdot 2 + 1.0} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}\]
\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{i}{2}}{i \cdot 2 + 1.0} \cdot \frac{\frac{i}{2}}{i \cdot 2 - 1.0}
double f(double i) {
        double r1929202 = i;
        double r1929203 = r1929202 * r1929202;
        double r1929204 = r1929203 * r1929203;
        double r1929205 = 2.0;
        double r1929206 = /* ERROR: no posit support in C */;
        double r1929207 = r1929206 * r1929202;
        double r1929208 = r1929207 * r1929207;
        double r1929209 = r1929204 / r1929208;
        double r1929210 = 1.0;
        double r1929211 = /* ERROR: no posit support in C */;
        double r1929212 = r1929208 - r1929211;
        double r1929213 = r1929209 / r1929212;
        return r1929213;
}
↓
double f(double i) {
        double r1929214 = i;
        double r1929215 = 2.0;
        double r1929216 = r1929214 / r1929215;
        double r1929217 = r1929214 * r1929215;
        double r1929218 = 1.0;
        double r1929219 = r1929217 + r1929218;
        double r1929220 = r1929216 / r1929219;
        double r1929221 = r1929217 - r1929218;
        double r1929222 = r1929216 / r1929221;
        double r1929223 = r1929220 * r1929222;
        return r1929223;
}

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. Simplified2.4

    \[\leadsto \color{blue}{\left(\frac{\left(i \cdot i\right)}{\left(\left(\left(i \cdot \left(2\right)\right) \cdot \left(i \cdot \left(2\right)\right)\right) \cdot \left(\left(\left(i \cdot \left(2\right)\right) \cdot \left(i \cdot \left(2\right)\right)\right) - \left(1.0\right)\right)\right)}\right) \cdot \left(i \cdot i\right)}\]
  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019104 
(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))))