Average Error: 2.4 → 0.4
Time: 54.3s
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 r4225126 = i;
        double r4225127 = r4225126 * r4225126;
        double r4225128 = r4225127 * r4225127;
        double r4225129 = 2.0;
        double r4225130 = /* ERROR: no posit support in C */;
        double r4225131 = r4225130 * r4225126;
        double r4225132 = r4225131 * r4225131;
        double r4225133 = r4225128 / r4225132;
        double r4225134 = 1.0;
        double r4225135 = /* ERROR: no posit support in C */;
        double r4225136 = r4225132 - r4225135;
        double r4225137 = r4225133 / r4225136;
        return r4225137;
}
↓
double f(double i) {
        double r4225138 = i;
        double r4225139 = 2.0;
        double r4225140 = r4225138 / r4225139;
        double r4225141 = r4225138 * r4225139;
        double r4225142 = 1.0;
        double r4225143 = r4225141 - r4225142;
        double r4225144 = r4225143 / r4225140;
        double r4225145 = r4225140 / r4225144;
        double r4225146 = r4225141 + r4225142;
        double r4225147 = r4225145 / r4225146;
        return r4225147;
}

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 2019134 +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))))