Average Error: 0.2 → 0.2
Time: 59.6s
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 r7630189 = a;
        double r7630190 = b;
        double r7630191 = r7630189 + r7630190;
        double r7630192 = c;
        double r7630193 = r7630191 + r7630192;
        double r7630194 = 2.0;
        double r7630195 = /* ERROR: no posit support in C */;
        double r7630196 = r7630193 / r7630195;
        double r7630197 = r7630196 - r7630189;
        double r7630198 = r7630196 * r7630197;
        double r7630199 = r7630196 - r7630190;
        double r7630200 = r7630198 * r7630199;
        double r7630201 = r7630196 - r7630192;
        double r7630202 = r7630200 * r7630201;
        double r7630203 = sqrt(r7630202);
        return r7630203;
}
↓
double f(double a, double b, double c) {
        double r7630204 = a;
        double r7630205 = b;
        double r7630206 = r7630204 + r7630205;
        double r7630207 = c;
        double r7630208 = r7630206 + r7630207;
        double r7630209 = 2.0;
        double r7630210 = r7630208 / r7630209;
        double r7630211 = r7630210 - r7630204;
        double r7630212 = r7630210 * r7630211;
        double r7630213 = r7630210 - r7630205;
        double r7630214 = r7630212 * r7630213;
        double r7630215 = r7630210 + r7630207;
        double r7630216 = r7630210 - r7630207;
        double r7630217 = r7630215 / r7630216;
        double r7630218 = r7630215 / r7630217;
        double r7630219 = r7630214 * r7630218;
        double r7630220 = sqrt(r7630219);
        return r7630220;
}

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 2019135 +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))))