Average Error: 0.4 → 0.4
Time: 17.7s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
↓
\[\frac{\frac{\frac{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}
↓
\frac{\frac{\frac{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
double f(double alpha, double beta) {
        double r1545644 = alpha;
        double r1545645 = beta;
        double r1545646 = r1545644 + r1545645;
        double r1545647 = r1545645 * r1545644;
        double r1545648 = r1545646 + r1545647;
        double r1545649 = 1.0;
        double r1545650 = /* ERROR: no posit support in C */;
        double r1545651 = r1545648 + r1545650;
        double r1545652 = 2.0;
        double r1545653 = /* ERROR: no posit support in C */;
        double r1545654 = 1.0;
        double r1545655 = /* ERROR: no posit support in C */;
        double r1545656 = r1545653 * r1545655;
        double r1545657 = r1545646 + r1545656;
        double r1545658 = r1545651 / r1545657;
        double r1545659 = r1545658 / r1545657;
        double r1545660 = r1545657 + r1545650;
        double r1545661 = r1545659 / r1545660;
        return r1545661;
}
↓
double f(double alpha, double beta) {
        double r1545662 = alpha;
        double r1545663 = beta;
        double r1545664 = r1545663 * r1545662;
        double r1545665 = r1545663 + r1545664;
        double r1545666 = r1545662 + r1545665;
        double r1545667 = 1.0;
        double r1545668 = r1545666 + r1545667;
        double r1545669 = r1545662 + r1545663;
        double r1545670 = 2.0;
        double r1545671 = 1.0;
        double r1545672 = r1545670 * r1545671;
        double r1545673 = r1545669 + r1545672;
        double r1545674 = r1545668 / r1545673;
        double r1545675 = r1545674 / r1545673;
        double r1545676 = r1545673 + r1545667;
        double r1545677 = r1545675 / r1545676;
        return r1545677;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.4

    \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
  2. Using strategy rm

  3. Applied associate-+l+0.4

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(\beta \cdot \alpha\right)}\right)}\right)}}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  4. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (/.p16 (/.p16 (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 beta alpha)) (real->posit16 1.0)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1))) (real->posit16 1.0))))