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 r4025816 = a;
        double r4025817 = b;
        double r4025818 = r4025816 + r4025817;
        double r4025819 = c;
        double r4025820 = r4025818 + r4025819;
        double r4025821 = 2.0;
        double r4025822 = /* ERROR: no posit support in C */;
        double r4025823 = r4025820 / r4025822;
        double r4025824 = r4025823 - r4025816;
        double r4025825 = r4025823 * r4025824;
        double r4025826 = r4025823 - r4025817;
        double r4025827 = r4025825 * r4025826;
        double r4025828 = r4025823 - r4025819;
        double r4025829 = r4025827 * r4025828;
        double r4025830 = sqrt(r4025829);
        return r4025830;
}

↓

double f(double a, double b, double c) {
        double r4025831 = a;
        double r4025832 = b;
        double r4025833 = r4025831 + r4025832;
        double r4025834 = c;
        double r4025835 = r4025833 + r4025834;
        double r4025836 = 2.0;
        double r4025837 = r4025835 / r4025836;
        double r4025838 = r4025834 + r4025832;
        double r4025839 = r4025831 + r4025838;
        double r4025840 = r4025839 / r4025836;
        double r4025841 = r4025840 - r4025831;
        double r4025842 = r4025837 * r4025841;
        double r4025843 = r4025831 + r4025834;
        double r4025844 = r4025832 + r4025843;
        double r4025845 = r4025844 / r4025836;
        double r4025846 = r4025845 - r4025832;
        double r4025847 = /*Error: no posit support in C */;
        double r4025848 = /*Error: no posit support in C */;
        double r4025849 = r4025842 * r4025848;
        double r4025850 = r4025837 - r4025834;
        double r4025851 = r4025849 * r4025850;
        double r4025852 = sqrt(r4025851);
        return r4025852;
}

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))))