Average Error: 1.0 → 1.0
Time: 1.2m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\mathsf{qms}\left(\left(\left(\frac{1}{x + 1}\right)\right), \left(\frac{2}{x}\right), 1.0\right)\right) + \frac{1}{x - 1}\]
\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}
↓
\left(\mathsf{qms}\left(\left(\left(\frac{1}{x + 1}\right)\right), \left(\frac{2}{x}\right), 1.0\right)\right) + \frac{1}{x - 1}
double f(double x) {
        double r2177011 = 1.0;
        double r2177012 = /* ERROR: no posit support in C */;
        double r2177013 = x;
        double r2177014 = r2177013 + r2177012;
        double r2177015 = r2177012 / r2177014;
        double r2177016 = 2.0;
        double r2177017 = /* ERROR: no posit support in C */;
        double r2177018 = r2177017 / r2177013;
        double r2177019 = r2177015 - r2177018;
        double r2177020 = r2177013 - r2177012;
        double r2177021 = r2177012 / r2177020;
        double r2177022 = r2177019 + r2177021;
        return r2177022;
}
↓
double f(double x) {
        double r2177023 = 1.0;
        double r2177024 = x;
        double r2177025 = r2177024 + r2177023;
        double r2177026 = r2177023 / r2177025;
        double r2177027 = /*Error: no posit support in C */;
        double r2177028 = 2.0;
        double r2177029 = r2177028 / r2177024;
        double r2177030 = 1.0;
        double r2177031 = /*Error: no posit support in C */;
        double r2177032 = /*Error: no posit support in C */;
        double r2177033 = r2177024 - r2177023;
        double r2177034 = r2177023 / r2177033;
        double r2177035 = r2177032 + r2177034;
        return r2177035;
}

Error

Bits error versus x

Derivation

  1. Initial program 1.0

    \[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
  2. Using strategy rm

  3. Applied introduce-quire1.0

    \[\leadsto \frac{\left(\color{blue}{\left(\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)\right)\right)} - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]

  4. Applied insert-quire-sub1.0

    \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{qms}\left(\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)\right), \left(\frac{\left(2\right)}{x}\right), \left(1.0\right)\right)\right)\right)}}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]

  5. Final simplification1.0

    \[\leadsto \left(\mathsf{qms}\left(\left(\left(\frac{1}{x + 1}\right)\right), \left(\frac{2}{x}\right), 1.0\right)\right) + \frac{1}{x - 1}\]

Reproduce

herbie shell --seed 2019155 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  (+.p16 (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 2) x)) (/.p16 (real->posit16 1) (-.p16 x (real->posit16 1)))))