Average Error: 0.9 → 0.6
Time: 29.0s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}\]
\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(\alpha + \left(\beta + 2 \cdot i\right)\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r1416702 = alpha;
        double r1416703 = beta;
        double r1416704 = r1416702 + r1416703;
        double r1416705 = r1416703 - r1416702;
        double r1416706 = r1416704 * r1416705;
        double r1416707 = 2.0;
        double r1416708 = /* ERROR: no posit support in C */;
        double r1416709 = i;
        double r1416710 = r1416708 * r1416709;
        double r1416711 = r1416704 + r1416710;
        double r1416712 = r1416706 / r1416711;
        double r1416713 = 2.0;
        double r1416714 = /* ERROR: no posit support in C */;
        double r1416715 = r1416711 + r1416714;
        double r1416716 = r1416712 / r1416715;
        double r1416717 = 1.0;
        double r1416718 = /* ERROR: no posit support in C */;
        double r1416719 = r1416716 + r1416718;
        double r1416720 = r1416719 / r1416714;
        return r1416720;
}
↓
double f(double alpha, double beta, double i) {
        double r1416721 = alpha;
        double r1416722 = beta;
        double r1416723 = r1416721 + r1416722;
        double r1416724 = 2.0;
        double r1416725 = i;
        double r1416726 = r1416724 * r1416725;
        double r1416727 = r1416723 + r1416726;
        double r1416728 = r1416723 / r1416727;
        double r1416729 = r1416722 + r1416726;
        double r1416730 = r1416721 + r1416729;
        double r1416731 = 2.0;
        double r1416732 = r1416730 + r1416731;
        double r1416733 = r1416722 - r1416721;
        double r1416734 = r1416732 / r1416733;
        double r1416735 = r1416728 / r1416734;
        double r1416736 = 1.0;
        double r1416737 = r1416735 + r1416736;
        double r1416738 = r1416737 / r1416731;
        return r1416738;
}

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-+l+0.6

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

  7. Applied associate-/r/0.6

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

  8. Applied associate-/l*0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019124 +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)))