Average Error: 0.2 → 0.2
Time: 44.1s
Precision: 64
\[\left(0\right) \lt a \land \left(0\right) \lt b \land \left(0\right) \lt c\]
\[\sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]
↓
\[\sqrt{\left(\left(\frac{\left(a + b\right) + c}{2} \cdot \left(\frac{\left(a + b\right) + c}{2} - a\right)\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - b\right)\right) \cdot \frac{\frac{\left(a + b\right) + c}{2} + c}{\frac{\frac{\left(a + b\right) + c}{2} + c}{\frac{\left(a + b\right) + c}{2} - c}}}\]
\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(\frac{\left(a + b\right) + c}{2} \cdot \left(\frac{\left(a + b\right) + c}{2} - a\right)\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - b\right)\right) \cdot \frac{\frac{\left(a + b\right) + c}{2} + c}{\frac{\frac{\left(a + b\right) + c}{2} + c}{\frac{\left(a + b\right) + c}{2} - c}}}
double f(double a, double b, double c) {
        double r1712602 = a;
        double r1712603 = b;
        double r1712604 = r1712602 + r1712603;
        double r1712605 = c;
        double r1712606 = r1712604 + r1712605;
        double r1712607 = 2.0;
        double r1712608 = /* ERROR: no posit support in C */;
        double r1712609 = r1712606 / r1712608;
        double r1712610 = r1712609 - r1712602;
        double r1712611 = r1712609 * r1712610;
        double r1712612 = r1712609 - r1712603;
        double r1712613 = r1712611 * r1712612;
        double r1712614 = r1712609 - r1712605;
        double r1712615 = r1712613 * r1712614;
        double r1712616 = sqrt(r1712615);
        return r1712616;
}
↓
double f(double a, double b, double c) {
        double r1712617 = a;
        double r1712618 = b;
        double r1712619 = r1712617 + r1712618;
        double r1712620 = c;
        double r1712621 = r1712619 + r1712620;
        double r1712622 = 2.0;
        double r1712623 = r1712621 / r1712622;
        double r1712624 = r1712623 - r1712617;
        double r1712625 = r1712623 * r1712624;
        double r1712626 = r1712623 - r1712618;
        double r1712627 = r1712625 * r1712626;
        double r1712628 = r1712623 + r1712620;
        double r1712629 = r1712623 - r1712620;
        double r1712630 = r1712628 / r1712629;
        double r1712631 = r1712628 / r1712630;
        double r1712632 = r1712627 * r1712631;
        double r1712633 = sqrt(r1712632);
        return r1712633;
}

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

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

  5. Applied difference-of-squares0.2

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

  6. Applied associate-/l*0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019120 
(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))))