Average Error: 0.2 → 0.2
Time: 1.1m
Precision: 64
\[\left(0\right) \lt a \land \left(0\right) \lt b \land \left(0\right) \lt c\]
\[\sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]
↓
\[\sqrt{\left(\left(\left(\frac{\left(a + b\right) + c}{2} - b\right) \cdot \frac{\left(a + b\right) + c}{2}\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - a\right)\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - c\right)}\]
\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{\left(\left(\left(\frac{\left(a + b\right) + c}{2} - b\right) \cdot \frac{\left(a + b\right) + c}{2}\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - a\right)\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - c\right)}
double f(double a, double b, double c) {
        double r6652592 = a;
        double r6652593 = b;
        double r6652594 = r6652592 + r6652593;
        double r6652595 = c;
        double r6652596 = r6652594 + r6652595;
        double r6652597 = 2.0;
        double r6652598 = /* ERROR: no posit support in C */;
        double r6652599 = r6652596 / r6652598;
        double r6652600 = r6652599 - r6652592;
        double r6652601 = r6652599 * r6652600;
        double r6652602 = r6652599 - r6652593;
        double r6652603 = r6652601 * r6652602;
        double r6652604 = r6652599 - r6652595;
        double r6652605 = r6652603 * r6652604;
        double r6652606 = sqrt(r6652605);
        return r6652606;
}
↓
double f(double a, double b, double c) {
        double r6652607 = a;
        double r6652608 = b;
        double r6652609 = r6652607 + r6652608;
        double r6652610 = c;
        double r6652611 = r6652609 + r6652610;
        double r6652612 = 2.0;
        double r6652613 = r6652611 / r6652612;
        double r6652614 = r6652613 - r6652608;
        double r6652615 = r6652614 * r6652613;
        double r6652616 = r6652613 - r6652607;
        double r6652617 = r6652615 * r6652616;
        double r6652618 = r6652613 - r6652610;
        double r6652619 = r6652617 * r6652618;
        double r6652620 = sqrt(r6652619);
        return r6652620;
}

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 *-commutative0.2

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

  5. Applied associate-*r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 
(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))))