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 r5437126 = a;
        double r5437127 = b;
        double r5437128 = r5437126 + r5437127;
        double r5437129 = c;
        double r5437130 = r5437128 + r5437129;
        double r5437131 = 2.0;
        double r5437132 = /* ERROR: no posit support in C */;
        double r5437133 = r5437130 / r5437132;
        double r5437134 = r5437133 - r5437126;
        double r5437135 = r5437133 * r5437134;
        double r5437136 = r5437133 - r5437127;
        double r5437137 = r5437135 * r5437136;
        double r5437138 = r5437133 - r5437129;
        double r5437139 = r5437137 * r5437138;
        double r5437140 = sqrt(r5437139);
        return r5437140;
}
↓
double f(double a, double b, double c) {
        double r5437141 = a;
        double r5437142 = b;
        double r5437143 = r5437141 + r5437142;
        double r5437144 = c;
        double r5437145 = r5437143 + r5437144;
        double r5437146 = 2.0;
        double r5437147 = r5437145 / r5437146;
        double r5437148 = r5437147 - r5437142;
        double r5437149 = r5437148 * r5437147;
        double r5437150 = r5437147 - r5437141;
        double r5437151 = r5437149 * r5437150;
        double r5437152 = r5437147 - r5437144;
        double r5437153 = r5437151 * r5437152;
        double r5437154 = sqrt(r5437153);
        return r5437154;
}

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))))