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 r4797845 = a;
        double r4797846 = b;
        double r4797847 = r4797845 + r4797846;
        double r4797848 = c;
        double r4797849 = r4797847 + r4797848;
        double r4797850 = 2.0;
        double r4797851 = /* ERROR: no posit support in C */;
        double r4797852 = r4797849 / r4797851;
        double r4797853 = r4797852 - r4797845;
        double r4797854 = r4797852 * r4797853;
        double r4797855 = r4797852 - r4797846;
        double r4797856 = r4797854 * r4797855;
        double r4797857 = r4797852 - r4797848;
        double r4797858 = r4797856 * r4797857;
        double r4797859 = sqrt(r4797858);
        return r4797859;
}

↓

double f(double a, double b, double c) {
        double r4797860 = a;
        double r4797861 = b;
        double r4797862 = r4797860 + r4797861;
        double r4797863 = c;
        double r4797864 = r4797862 + r4797863;
        double r4797865 = 2.0;
        double r4797866 = r4797864 / r4797865;
        double r4797867 = r4797863 + r4797861;
        double r4797868 = r4797860 + r4797867;
        double r4797869 = r4797868 / r4797865;
        double r4797870 = r4797869 - r4797860;
        double r4797871 = r4797866 * r4797870;
        double r4797872 = r4797860 + r4797863;
        double r4797873 = r4797861 + r4797872;
        double r4797874 = r4797873 / r4797865;
        double r4797875 = r4797874 - r4797861;
        double r4797876 = /*Error: no posit support in C */;
        double r4797877 = /*Error: no posit support in C */;
        double r4797878 = r4797871 * r4797877;
        double r4797879 = r4797866 - r4797863;
        double r4797880 = r4797878 * r4797879;
        double r4797881 = sqrt(r4797880);
        return r4797881;
}

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 
(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))))