Octave 3.8, jcobi/4

Average Error: 3.3 → 1.4

Time: 1.2m

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r1763099 = i;
        double r1763100 = alpha;
        double r1763101 = beta;
        double r1763102 = r1763100 + r1763101;
        double r1763103 = r1763102 + r1763099;
        double r1763104 = r1763099 * r1763103;
        double r1763105 = r1763101 * r1763100;
        double r1763106 = r1763105 + r1763104;
        double r1763107 = r1763104 * r1763106;
        double r1763108 = 2.0;
        double r1763109 = /* ERROR: no posit support in C */;
        double r1763110 = r1763109 * r1763099;
        double r1763111 = r1763102 + r1763110;
        double r1763112 = r1763111 * r1763111;
        double r1763113 = r1763107 / r1763112;
        double r1763114 = 1.0;
        double r1763115 = /* ERROR: no posit support in C */;
        double r1763116 = r1763112 - r1763115;
        double r1763117 = r1763113 / r1763116;
        return r1763117;
}

↓

double f(double alpha, double beta, double i) {
        double r1763118 = i;
        double r1763119 = 1.0;
        double r1763120 = r1763118 / r1763119;
        double r1763121 = alpha;
        double r1763122 = beta;
        double r1763123 = r1763121 + r1763122;
        double r1763124 = 2.0;
        double r1763125 = r1763124 * r1763118;
        double r1763126 = r1763123 + r1763125;
        double r1763127 = r1763126 + r1763119;
        double r1763128 = r1763123 + r1763118;
        double r1763129 = r1763128 / r1763126;
        double r1763130 = r1763127 / r1763129;
        double r1763131 = r1763120 / r1763130;
        double r1763132 = r1763123 - r1763119;
        double r1763133 = r1763125 + r1763132;
        double r1763134 = r1763121 * r1763122;
        double r1763135 = r1763118 * r1763128;
        double r1763136 = r1763134 + r1763135;
        double r1763137 = r1763133 / r1763136;
        double r1763138 = r1763137 * r1763125;
        double r1763139 = r1763137 * r1763123;
        double r1763140 = r1763138 + r1763139;
        double r1763141 = r1763119 / r1763140;
        double r1763142 = r1763131 * r1763141;
        return r1763142;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Initial program 3.3

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

3. Applied p16-*-un-lft-identity 3.3

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

4. Applied difference-of-squares 3.3

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

5. Applied p16-times-frac 1.8

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

6. Applied p16-times-frac 1.7

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

8. Applied p16-*-un-lft-identity 1.7

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

9. Applied p16-times-frac 1.5

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

10. Applied associate-/l* 1.5

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

12. Applied p16-*-un-lft-identity 1.5

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

13. Applied associate-/l* 1.5

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

14. Simplified 1.5

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

16. Applied distribute-lft-in 1.4

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

17. Final simplification 1.4

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

Reproduce

herbie shell --seed 2019156 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 1)))
  (/.p16 (/.p16 (*.p16 (*.p16 i (+.p16 (+.p16 alpha beta) i)) (+.p16 (*.p16 beta alpha) (*.p16 i (+.p16 (+.p16 alpha beta) i)))) (*.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)))) (-.p16 (*.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (real->posit16 1.0))))