Average Error: 2.4 → 0.4
Time: 28.4s
Precision: 64
\[i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\left(i \cdot i\right) \cdot \left(i \cdot i\right)\right)}{\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right)}\right)}{\left(\left(\left(\left(2\right) \cdot i\right) \cdot \left(\left(2\right) \cdot i\right)\right) - \left(1.0\right)\right)}\]
↓
\[\frac{\frac{i}{\frac{2}{\frac{i}{i}}}}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{\frac{2}{\frac{i}{i}}}}{2 \cdot i - 1.0}\]
\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}{\frac{i}{i}}}}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{\frac{2}{\frac{i}{i}}}}{2 \cdot i - 1.0}
double f(double i) {
        double r1390089 = i;
        double r1390090 = r1390089 * r1390089;
        double r1390091 = r1390090 * r1390090;
        double r1390092 = 2.0;
        double r1390093 = /* ERROR: no posit support in C */;
        double r1390094 = r1390093 * r1390089;
        double r1390095 = r1390094 * r1390094;
        double r1390096 = r1390091 / r1390095;
        double r1390097 = 1.0;
        double r1390098 = /* ERROR: no posit support in C */;
        double r1390099 = r1390095 - r1390098;
        double r1390100 = r1390096 / r1390099;
        return r1390100;
}
↓
double f(double i) {
        double r1390101 = i;
        double r1390102 = 2.0;
        double r1390103 = r1390101 / r1390101;
        double r1390104 = r1390102 / r1390103;
        double r1390105 = r1390101 / r1390104;
        double r1390106 = r1390102 * r1390101;
        double r1390107 = 1.0;
        double r1390108 = r1390106 + r1390107;
        double r1390109 = r1390105 / r1390108;
        double r1390110 = r1390106 - r1390107;
        double r1390111 = r1390105 / r1390110;
        double r1390112 = r1390109 * r1390111;
        return r1390112;
}

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 associate-/l*1.0

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

  5. Applied difference-of-sqr-11.0

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

  6. Applied p16-times-frac0.8

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

  7. Applied p16-times-frac0.9

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

  8. Applied p16-times-frac0.5

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

  10. Applied associate-/l*0.4

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

  12. Applied associate-/l*0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019112 +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))))