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 r3010217 = a;
        double r3010218 = b;
        double r3010219 = r3010217 + r3010218;
        double r3010220 = c;
        double r3010221 = r3010219 + r3010220;
        double r3010222 = 2.0;
        double r3010223 = /* ERROR: no posit support in C */;
        double r3010224 = r3010221 / r3010223;
        double r3010225 = r3010224 - r3010217;
        double r3010226 = r3010224 * r3010225;
        double r3010227 = r3010224 - r3010218;
        double r3010228 = r3010226 * r3010227;
        double r3010229 = r3010224 - r3010220;
        double r3010230 = r3010228 * r3010229;
        double r3010231 = sqrt(r3010230);
        return r3010231;
}
↓
double f(double a, double b, double c) {
        double r3010232 = a;
        double r3010233 = b;
        double r3010234 = r3010232 + r3010233;
        double r3010235 = c;
        double r3010236 = r3010234 + r3010235;
        double r3010237 = 2.0;
        double r3010238 = r3010236 / r3010237;
        double r3010239 = r3010238 - r3010232;
        double r3010240 = r3010238 * r3010239;
        double r3010241 = r3010238 + r3010233;
        double r3010242 = r3010238 - r3010233;
        double r3010243 = r3010241 * r3010242;
        double r3010244 = r3010240 * r3010243;
        double r3010245 = r3010244 / r3010241;
        double r3010246 = r3010238 - r3010235;
        double r3010247 = r3010245 * r3010246;
        double r3010248 = sqrt(r3010247);
        return r3010248;
}

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 +o rules:numerics
(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))))