Average Error: 2.4 → 0.4
Time: 25.6s
Precision: 64
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{\frac{\frac{\frac{i}{2}}{2 \cdot i - 1.0} \cdot i}{2}}{2 \cdot i + 1.0}\]
double f(double i) {
        double r2300573 = i;
        double r2300574 = r2300573 * r2300573;
        double r2300575 = r2300574 * r2300574;
        double r2300576 = 2.0;
        double r2300577 = r2300576 * r2300573;
        double r2300578 = r2300577 * r2300577;
        double r2300579 = r2300575 / r2300578;
        double r2300580 = 1.0;
        double r2300581 = r2300578 - r2300580;
        double r2300582 = r2300579 / r2300581;
        return r2300582;
}
↓
double f(double i) {
        double r2300583 = i;
        double r2300584 = 2.0;
        double r2300585 = r2300583 / r2300584;
        double r2300586 = r2300584 * r2300583;
        double r2300587 = 1.0;
        double r2300588 = r2300586 - r2300587;
        double r2300589 = r2300585 / r2300588;
        double r2300590 = r2300589 * r2300583;
        double r2300591 = r2300590 / r2300584;
        double r2300592 = r2300586 + r2300587;
        double r2300593 = r2300591 / r2300592;
        return r2300593;
}


\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}
↓
\frac{\frac{\frac{\frac{i}{2}}{2 \cdot i - 1.0} \cdot i}{2}}{2 \cdot i + 1.0}

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-12.3

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

  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 p16-times-frac0.7

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

  8. Simplified0.7

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

  10. Applied associate-*l/0.7

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

  11. Simplified0.4

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

  13. Applied associate-*r/0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019102 +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))))