Area of a triangle

Average Error: 0.2 → 0.1

Time: 1.0m

Precision: 64

\[\left(0\right) \lt a \land \left(0\right) \lt b \land \left(0\right) \lt c\]

Math

\[\sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

↓

\[\sqrt{\left(\left(\frac{\left(a + b\right) + c}{2} \cdot \left(\frac{a + \left(c + b\right)}{2} - a\right)\right) \cdot \left(\left(\frac{b + \left(a + c\right)}{2} - b\right)\right)\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - c\right)}\]

C

double f(double a, double b, double c) {
        double r4070828 = a;
        double r4070829 = b;
        double r4070830 = r4070828 + r4070829;
        double r4070831 = c;
        double r4070832 = r4070830 + r4070831;
        double r4070833 = 2.0;
        double r4070834 = /* ERROR: no posit support in C */;
        double r4070835 = r4070832 / r4070834;
        double r4070836 = r4070835 - r4070828;
        double r4070837 = r4070835 * r4070836;
        double r4070838 = r4070835 - r4070829;
        double r4070839 = r4070837 * r4070838;
        double r4070840 = r4070835 - r4070831;
        double r4070841 = r4070839 * r4070840;
        double r4070842 = sqrt(r4070841);
        return r4070842;
}

↓

double f(double a, double b, double c) {
        double r4070843 = a;
        double r4070844 = b;
        double r4070845 = r4070843 + r4070844;
        double r4070846 = c;
        double r4070847 = r4070845 + r4070846;
        double r4070848 = 2.0;
        double r4070849 = r4070847 / r4070848;
        double r4070850 = r4070846 + r4070844;
        double r4070851 = r4070843 + r4070850;
        double r4070852 = r4070851 / r4070848;
        double r4070853 = r4070852 - r4070843;
        double r4070854 = r4070849 * r4070853;
        double r4070855 = r4070843 + r4070846;
        double r4070856 = r4070844 + r4070855;
        double r4070857 = r4070856 / r4070848;
        double r4070858 = r4070857 - r4070844;
        double r4070859 = /*Error: no posit support in C */;
        double r4070860 = /*Error: no posit support in C */;
        double r4070861 = r4070854 * r4070860;
        double r4070862 = r4070849 - r4070846;
        double r4070863 = r4070861 * r4070862;
        double r4070864 = sqrt(r4070863);
        return r4070864;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 0.2

   \[\sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]
2. Using strategy rm

3. Applied *p16-rgt-identity-expand 0.2

   \[\leadsto \sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\color{blue}{\left(\left(2\right) \cdot \left(1.0\right)\right)}}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

4. Applied *p16-rgt-identity-expand 0.2

   \[\leadsto \sqrt{\left(\left(\left(\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{a}{b}\right)}{c}\right) \cdot \left(1.0\right)\right)}}{\left(\left(2\right) \cdot \left(1.0\right)\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

5. Applied p16-times-frac 0.2

   \[\leadsto \sqrt{\left(\left(\left(\color{blue}{\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(1.0\right)}\right)\right)} \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

6. Applied associate-*l* 0.2

   \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(1.0\right)}{\left(1.0\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right)\right)} \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

7. Simplified 0.1

   \[\leadsto \sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \color{blue}{\left(\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right) - a\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]
8. Using strategy rm

9. Applied p16-flip-- 0.2

   \[\leadsto \sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \color{blue}{\left(\frac{\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right)\right) - \left(b \cdot b\right)\right)}{\left(\frac{\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right)}{b}\right)}\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

10. Simplified 0.1

    \[\leadsto \sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\frac{\color{blue}{\left(\left(\frac{b}{\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right)}\right) \cdot \left(\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right) - b\right)\right)}}{\left(\frac{\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right)}{b}\right)}\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

11. Simplified 0.1

    \[\leadsto \sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\frac{\left(\left(\frac{b}{\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right)}\right) \cdot \left(\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right) - b\right)\right)}{\color{blue}{\left(\frac{b}{\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right)}\right)}}\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]
12. Using strategy rm

13. Applied introduce-quire 0.1

    \[\leadsto \sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \color{blue}{\left(\left(\left(\frac{\left(\left(\frac{b}{\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right)}\right) \cdot \left(\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right) - b\right)\right)}{\left(\frac{b}{\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right)}\right)}\right)\right)\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

14. Simplified 0.1

    \[\leadsto \sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{a}{\left(\frac{c}{b}\right)}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\color{blue}{\left(\left(\left(\frac{\left(\frac{b}{\left(\frac{a}{c}\right)}\right)}{\left(2\right)}\right) - b\right)\right)}\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

15. Final simplification 0.1

    \[\leadsto \sqrt{\left(\left(\frac{\left(a + b\right) + c}{2} \cdot \left(\frac{a + \left(c + b\right)}{2} - a\right)\right) \cdot \left(\left(\frac{b + \left(a + c\right)}{2} - b\right)\right)\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - c\right)}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a b c)
  :name "Area of a triangle"
  :pre (and (<.p16 (real->posit16 0) a) (<.p16 (real->posit16 0) b) (<.p16 (real->posit16 0) c))
  (sqrt.p16 (*.p16 (*.p16 (*.p16 (/.p16 (+.p16 (+.p16 a b) c) (real->posit16 2)) (-.p16 (/.p16 (+.p16 (+.p16 a b) c) (real->posit16 2)) a)) (-.p16 (/.p16 (+.p16 (+.p16 a b) c) (real->posit16 2)) b)) (-.p16 (/.p16 (+.p16 (+.p16 a b) c) (real->posit16 2)) c))))