Average Error: 0.4 → 0.4
Time: 28.4s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
↓
\[\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
\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{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
double f(double alpha, double beta) {
        double r4324558 = alpha;
        double r4324559 = beta;
        double r4324560 = r4324558 + r4324559;
        double r4324561 = r4324559 * r4324558;
        double r4324562 = r4324560 + r4324561;
        double r4324563 = 1.0;
        double r4324564 = /* ERROR: no posit support in C */;
        double r4324565 = r4324562 + r4324564;
        double r4324566 = 2.0;
        double r4324567 = /* ERROR: no posit support in C */;
        double r4324568 = 1.0;
        double r4324569 = /* ERROR: no posit support in C */;
        double r4324570 = r4324567 * r4324569;
        double r4324571 = r4324560 + r4324570;
        double r4324572 = r4324565 / r4324571;
        double r4324573 = r4324572 / r4324571;
        double r4324574 = r4324571 + r4324564;
        double r4324575 = r4324573 / r4324574;
        return r4324575;
}
↓
double f(double alpha, double beta) {
        double r4324576 = alpha;
        double r4324577 = beta;
        double r4324578 = r4324576 + r4324577;
        double r4324579 = r4324577 * r4324576;
        double r4324580 = r4324578 + r4324579;
        double r4324581 = 1.0;
        double r4324582 = r4324580 + r4324581;
        double r4324583 = 2.0;
        double r4324584 = 1.0;
        double r4324585 = r4324583 * r4324584;
        double r4324586 = r4324578 + r4324585;
        double r4324587 = r4324586 * r4324578;
        double r4324588 = r4324586 * r4324585;
        double r4324589 = r4324587 + r4324588;
        double r4324590 = r4324582 / r4324589;
        double r4324591 = r4324586 + r4324581;
        double r4324592 = r4324590 / r4324591;
        return r4324592;
}

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

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

  5. Applied distribute-lft-in0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019135 
(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))))