Average Error: 0.9 → 0.6
Time: 34.6s
Precision: 64
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{\frac{\frac{\alpha + \beta}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\beta - \alpha}}}{\left(\alpha + \beta\right) + 2 \cdot i} + 1.0}{2.0}\]
double f(double alpha, double beta, double i) {
        double r968194 = alpha;
        double r968195 = beta;
        double r968196 = r968194 + r968195;
        double r968197 = r968195 - r968194;
        double r968198 = r968196 * r968197;
        double r968199 = 2.0;
        double r968200 = i;
        double r968201 = r968199 * r968200;
        double r968202 = r968196 + r968201;
        double r968203 = r968198 / r968202;
        double r968204 = 2.0;
        double r968205 = r968202 + r968204;
        double r968206 = r968203 / r968205;
        double r968207 = 1.0;
        double r968208 = r968206 + r968207;
        double r968209 = r968208 / r968204;
        return r968209;
}
↓
double f(double alpha, double beta, double i) {
        double r968210 = alpha;
        double r968211 = beta;
        double r968212 = r968210 + r968211;
        double r968213 = 2.0;
        double r968214 = i;
        double r968215 = r968213 * r968214;
        double r968216 = r968212 + r968215;
        double r968217 = 2.0;
        double r968218 = r968216 + r968217;
        double r968219 = r968211 - r968210;
        double r968220 = r968218 / r968219;
        double r968221 = r968212 / r968220;
        double r968222 = r968221 / r968216;
        double r968223 = 1.0;
        double r968224 = r968222 + r968223;
        double r968225 = r968224 / r968217;
        return r968225;
}


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

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 0.9

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

  8. Applied associate-/l/0.6

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

  10. Applied associate-/r*0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2019102 +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)))