Average Error: 0.2 → 0.2
Time: 42.1s
Precision: 64
\[\left(0\right) \lt a \land \left(0\right) \lt b \land \left(0\right) \lt c\]
\[\sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]
↓
\[\sqrt{\frac{\left(\frac{\left(a + b\right) + c}{2} \cdot \left(\frac{\left(a + b\right) + c}{2} - a\right)\right) \cdot \left(\left(\frac{\left(a + b\right) + c}{2} + b\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - b\right)\right)}{\frac{\left(a + b\right) + c}{2} + b} \cdot \left(\frac{\left(a + b\right) + c}{2} - c\right)}\]
\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{\frac{\left(\frac{\left(a + b\right) + c}{2} \cdot \left(\frac{\left(a + b\right) + c}{2} - a\right)\right) \cdot \left(\left(\frac{\left(a + b\right) + c}{2} + b\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - b\right)\right)}{\frac{\left(a + b\right) + c}{2} + b} \cdot \left(\frac{\left(a + b\right) + c}{2} - c\right)}
double f(double a, double b, double c) {
        double r2570606 = a;
        double r2570607 = b;
        double r2570608 = r2570606 + r2570607;
        double r2570609 = c;
        double r2570610 = r2570608 + r2570609;
        double r2570611 = 2.0;
        double r2570612 = /* ERROR: no posit support in C */;
        double r2570613 = r2570610 / r2570612;
        double r2570614 = r2570613 - r2570606;
        double r2570615 = r2570613 * r2570614;
        double r2570616 = r2570613 - r2570607;
        double r2570617 = r2570615 * r2570616;
        double r2570618 = r2570613 - r2570609;
        double r2570619 = r2570617 * r2570618;
        double r2570620 = sqrt(r2570619);
        return r2570620;
}
↓
double f(double a, double b, double c) {
        double r2570621 = a;
        double r2570622 = b;
        double r2570623 = r2570621 + r2570622;
        double r2570624 = c;
        double r2570625 = r2570623 + r2570624;
        double r2570626 = 2.0;
        double r2570627 = r2570625 / r2570626;
        double r2570628 = r2570627 - r2570621;
        double r2570629 = r2570627 * r2570628;
        double r2570630 = r2570627 + r2570622;
        double r2570631 = r2570627 - r2570622;
        double r2570632 = r2570630 * r2570631;
        double r2570633 = r2570629 * r2570632;
        double r2570634 = r2570633 / r2570630;
        double r2570635 = r2570627 - r2570624;
        double r2570636 = r2570634 * r2570635;
        double r2570637 = sqrt(r2570636);
        return r2570637;
}

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-flip--0.2

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

  4. Applied associate-*r/0.2

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

  6. Applied difference-of-squares0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 
(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))))