Octave 3.8, jcobi/3

Average Error: 0.4 → 0.3

Time: 2.3m

Precision: 64

\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]

Math

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

↓

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

C

double f(double alpha, double beta) {
        double r1545071 = alpha;
        double r1545072 = beta;
        double r1545073 = r1545071 + r1545072;
        double r1545074 = r1545072 * r1545071;
        double r1545075 = r1545073 + r1545074;
        double r1545076 = 1.0;
        double r1545077 = /* ERROR: no posit support in C */;
        double r1545078 = r1545075 + r1545077;
        double r1545079 = 2.0;
        double r1545080 = /* ERROR: no posit support in C */;
        double r1545081 = 1.0;
        double r1545082 = /* ERROR: no posit support in C */;
        double r1545083 = r1545080 * r1545082;
        double r1545084 = r1545073 + r1545083;
        double r1545085 = r1545078 / r1545084;
        double r1545086 = r1545085 / r1545084;
        double r1545087 = r1545084 + r1545077;
        double r1545088 = r1545086 / r1545087;
        return r1545088;
}

↓

double f(double alpha, double beta) {
        double r1545089 = 1.0;
        double r1545090 = alpha;
        double r1545091 = beta;
        double r1545092 = r1545090 + r1545091;
        double r1545093 = 2.0;
        double r1545094 = 1.0;
        double r1545095 = r1545093 * r1545094;
        double r1545096 = r1545092 + r1545095;
        double r1545097 = r1545096 / r1545089;
        double r1545098 = r1545089 / r1545097;
        double r1545099 = r1545090 + r1545095;
        double r1545100 = r1545099 + r1545091;
        double r1545101 = r1545091 * r1545090;
        double r1545102 = r1545091 + r1545090;
        double r1545103 = r1545089 + r1545102;
        double r1545104 = r1545101 + r1545103;
        double r1545105 = r1545100 / r1545104;
        double r1545106 = r1545098 / r1545105;
        double r1545107 = r1545096 + r1545089;
        double r1545108 = r1545106 / r1545107;
        return r1545108;
}

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 *p16-lft-identity-expand 0.4

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

4. Applied *p16-lft-identity-expand 0.4

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

5. Applied distribute-lft-out 0.4

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

6. Applied associate-/l* 0.4

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

8. Applied *p16-rgt-identity-expand 0.4

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

9. Applied *p16-lft-identity-expand 0.4

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

10. Applied p16-distribute-lft-out 0.4

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

11. Applied *p16-rgt-identity-expand 0.4

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

12. Applied p16-times-frac 0.4

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

13. Applied *p16-rgt-identity-expand 0.4

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

14. Applied p16-times-frac 0.4

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

15. Applied associate-/l* 0.4

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

16. Simplified 0.3

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

17. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019152 
(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))))