Average Error: 0.7 → 0.7
Time: 1.0m
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\left(1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{1.0}{2.0}\]
\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)}
↓
\left(1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{1.0}{2.0}
double f(double alpha, double beta) {
        double r269149 = beta;
        double r269150 = alpha;
        double r269151 = r269149 - r269150;
        double r269152 = r269150 + r269149;
        double r269153 = 2.0;
        double r269154 = /* ERROR: no posit support in C */;
        double r269155 = r269152 + r269154;
        double r269156 = r269151 / r269155;
        double r269157 = 1.0;
        double r269158 = /* ERROR: no posit support in C */;
        double r269159 = r269156 + r269158;
        double r269160 = r269159 / r269154;
        return r269160;
}
↓
double f(double alpha, double beta) {
        double r269161 = 1.0;
        double r269162 = beta;
        double r269163 = alpha;
        double r269164 = r269162 - r269163;
        double r269165 = r269163 + r269162;
        double r269166 = 2.0;
        double r269167 = r269165 + r269166;
        double r269168 = r269164 / r269167;
        double r269169 = r269161 + r269168;
        double r269170 = r269161 / r269166;
        double r269171 = r269169 * r269170;
        return r269171;
}

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-lft-identity-expand0.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\color{blue}{\left(\frac{\left(0.0\right)}{\left(1.0\right)}\right)}}\right)}{\left(2.0\right)}\]

  4. Applied associate-+r+0.7

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(0.0\right)}\right)}{\left(1.0\right)}\right)}}{\left(2.0\right)}\]

  5. Simplified0.7

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(2.0\right)}\right)}\right)}\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  6. Using strategy rm

  7. Applied *p16-lft-identity-expand0.7

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

  8. Applied /p16-rgt-identity-expand0.7

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

  9. Applied associate-/r/0.7

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

  10. Applied distribute-lft1-in0.7

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

  11. Applied p16-times-frac0.7

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

  12. Simplified0.7

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

  13. Final simplification0.7

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

Reproduce

herbie shell --seed 2019152 
(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)))