Average Error: 0.7 → 0.8
Time: 18.5s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2.0\right)} + 1.0}{2.0}\]
\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
↓
\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2.0\right)} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r3280120 = beta;
        double r3280121 = alpha;
        double r3280122 = r3280120 - r3280121;
        double r3280123 = r3280121 + r3280120;
        double r3280124 = 2.0;
        double r3280125 = /* ERROR: no posit support in C */;
        double r3280126 = r3280123 + r3280125;
        double r3280127 = r3280122 / r3280126;
        double r3280128 = 1.0;
        double r3280129 = /* ERROR: no posit support in C */;
        double r3280130 = r3280127 + r3280129;
        double r3280131 = r3280130 / r3280125;
        return r3280131;
}
↓
double f(double alpha, double beta) {
        double r3280132 = beta;
        double r3280133 = alpha;
        double r3280134 = r3280132 - r3280133;
        double r3280135 = 2.0;
        double r3280136 = r3280133 + r3280135;
        double r3280137 = r3280132 + r3280136;
        double r3280138 = r3280134 / r3280137;
        double r3280139 = 1.0;
        double r3280140 = r3280138 + r3280139;
        double r3280141 = r3280140 / r3280135;
        return r3280141;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.7

    \[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  2. Using strategy rm

  3. Applied p16-flip--1.3

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\left(\left(\beta \cdot \beta\right) - \left(\alpha \cdot \alpha\right)\right)}{\left(\frac{\beta}{\alpha}\right)}\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  4. Applied associate-/l/1.3

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(\left(\beta \cdot \beta\right) - \left(\alpha \cdot \alpha\right)\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right) \cdot \left(\frac{\beta}{\alpha}\right)\right)}\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  5. Simplified1.2

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\beta - \alpha\right) \cdot \left(\frac{\alpha}{\beta}\right)\right)}}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right) \cdot \left(\frac{\beta}{\alpha}\right)\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  6. Simplified0.8

    \[\leadsto \color{blue}{\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\beta}{\left(\frac{\alpha}{\left(2.0\right)}\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}}\]

  7. Final simplification0.8

    \[\leadsto \frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2.0\right)} + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (+.p16 (/.p16 (-.p16 beta alpha) (+.p16 (+.p16 alpha beta) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))