Average Error: 2.4 → 0.4
Time: 20.2s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i}{\frac{2 \cdot i - 1.0}{\frac{i}{2}} \cdot 2}}{2 \cdot i + 1.0}\]
\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}{\frac{2 \cdot i - 1.0}{\frac{i}{2}} \cdot 2}}{2 \cdot i + 1.0}
double f(double i) {
        double r1868128 = i;
        double r1868129 = r1868128 * r1868128;
        double r1868130 = r1868129 * r1868129;
        double r1868131 = 2.0;
        double r1868132 = /* ERROR: no posit support in C */;
        double r1868133 = r1868132 * r1868128;
        double r1868134 = r1868133 * r1868133;
        double r1868135 = r1868130 / r1868134;
        double r1868136 = 1.0;
        double r1868137 = /* ERROR: no posit support in C */;
        double r1868138 = r1868134 - r1868137;
        double r1868139 = r1868135 / r1868138;
        return r1868139;
}
↓
double f(double i) {
        double r1868140 = i;
        double r1868141 = 2.0;
        double r1868142 = r1868141 * r1868140;
        double r1868143 = 1.0;
        double r1868144 = r1868142 - r1868143;
        double r1868145 = r1868140 / r1868141;
        double r1868146 = r1868144 / r1868145;
        double r1868147 = r1868146 * r1868141;
        double r1868148 = r1868140 / r1868147;
        double r1868149 = r1868142 + r1868143;
        double r1868150 = r1868148 / r1868149;
        return r1868150;
}

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. Using strategy rm

  3. Applied difference-of-sqr-12.4

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

  4. Applied p16-times-frac1.0

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

  5. Applied p16-times-frac1.0

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

  7. Applied associate-*l/1.0

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

  8. Simplified0.6

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

  10. Applied associate-/l*0.4

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

  12. Applied associate-/l/0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019107 +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))))