Average Error: 1.0 → 0.6
Time: 34.8s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}\]
\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
↓
\frac{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r2476217 = alpha;
        double r2476218 = beta;
        double r2476219 = r2476217 + r2476218;
        double r2476220 = r2476218 - r2476217;
        double r2476221 = r2476219 * r2476220;
        double r2476222 = 2.0;
        double r2476223 = /* ERROR: no posit support in C */;
        double r2476224 = i;
        double r2476225 = r2476223 * r2476224;
        double r2476226 = r2476219 + r2476225;
        double r2476227 = r2476221 / r2476226;
        double r2476228 = 2.0;
        double r2476229 = /* ERROR: no posit support in C */;
        double r2476230 = r2476226 + r2476229;
        double r2476231 = r2476227 / r2476230;
        double r2476232 = 1.0;
        double r2476233 = /* ERROR: no posit support in C */;
        double r2476234 = r2476231 + r2476233;
        double r2476235 = r2476234 / r2476229;
        return r2476235;
}
↓
double f(double alpha, double beta, double i) {
        double r2476236 = alpha;
        double r2476237 = beta;
        double r2476238 = r2476236 + r2476237;
        double r2476239 = 2.0;
        double r2476240 = i;
        double r2476241 = r2476239 * r2476240;
        double r2476242 = r2476238 + r2476241;
        double r2476243 = r2476238 / r2476242;
        double r2476244 = 2.0;
        double r2476245 = r2476242 + r2476244;
        double r2476246 = r2476237 - r2476236;
        double r2476247 = r2476245 / r2476246;
        double r2476248 = r2476243 / r2476247;
        double r2476249 = 1.0;
        double r2476250 = r2476248 + r2476249;
        double r2476251 = r2476250 / r2476244;
        return r2476251;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 1.0

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

  3. Applied associate-/l*0.6

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  4. Using strategy rm

  5. Applied associate-/r/0.6

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(\beta - \alpha\right)\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  6. Applied associate-/l*0.6

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  7. Final simplification0.6

    \[\leadsto \frac{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 0)))
  (/.p16 (+.p16 (/.p16 (/.p16 (*.p16 (+.p16 alpha beta) (-.p16 beta alpha)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))