Average Error: 0.2 → 0.2
Time: 1.1m
Precision: 64
\[\left(0\right) \lt a \land \left(0\right) \lt b \land \left(0\right) \lt c\]
\[\sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]
↓
\[\sqrt{\left(\left(\frac{\left(a + b\right) + c}{2} \cdot \left(\frac{\left(c + b\right) + a}{2} - b\right)\right) \cdot \left(\frac{\left(c + b\right) + a}{2} - a\right)\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - c\right)}\]
\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(\frac{\left(a + b\right) + c}{2} \cdot \left(\frac{\left(c + b\right) + a}{2} - b\right)\right) \cdot \left(\frac{\left(c + b\right) + a}{2} - a\right)\right) \cdot \left(\frac{\left(a + b\right) + c}{2} - c\right)}
double f(double a, double b, double c) {
        double r2554662 = a;
        double r2554663 = b;
        double r2554664 = r2554662 + r2554663;
        double r2554665 = c;
        double r2554666 = r2554664 + r2554665;
        double r2554667 = 2.0;
        double r2554668 = /* ERROR: no posit support in C */;
        double r2554669 = r2554666 / r2554668;
        double r2554670 = r2554669 - r2554662;
        double r2554671 = r2554669 * r2554670;
        double r2554672 = r2554669 - r2554663;
        double r2554673 = r2554671 * r2554672;
        double r2554674 = r2554669 - r2554665;
        double r2554675 = r2554673 * r2554674;
        double r2554676 = sqrt(r2554675);
        return r2554676;
}
↓
double f(double a, double b, double c) {
        double r2554677 = a;
        double r2554678 = b;
        double r2554679 = r2554677 + r2554678;
        double r2554680 = c;
        double r2554681 = r2554679 + r2554680;
        double r2554682 = 2.0;
        double r2554683 = r2554681 / r2554682;
        double r2554684 = r2554680 + r2554678;
        double r2554685 = r2554684 + r2554677;
        double r2554686 = r2554685 / r2554682;
        double r2554687 = r2554686 - r2554678;
        double r2554688 = r2554683 * r2554687;
        double r2554689 = r2554686 - r2554677;
        double r2554690 = r2554688 * r2554689;
        double r2554691 = r2554683 - r2554680;
        double r2554692 = r2554690 * r2554691;
        double r2554693 = sqrt(r2554692);
        return r2554693;
}

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 associate-*l*0.2

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

  5. Applied *p16-rgt-identity-expand0.2

    \[\leadsto \sqrt{\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\color{blue}{\left(\left(2\right) \cdot \left(1.0\right)\right)}}\right) \cdot \left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

  6. Applied *p16-rgt-identity-expand0.2

    \[\leadsto \sqrt{\left(\left(\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{a}{b}\right)}{c}\right) \cdot \left(1.0\right)\right)}}{\left(\left(2\right) \cdot \left(1.0\right)\right)}\right) \cdot \left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

  7. Applied p16-times-frac0.2

    \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(1.0\right)}\right)\right)} \cdot \left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

  8. Applied associate-*l*0.2

    \[\leadsto \sqrt{\left(\color{blue}{\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(1.0\right)}{\left(1.0\right)}\right) \cdot \left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right)\right)\right)} \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]

  9. Simplified0.2

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

  11. Applied associate-*r*0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 
(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))))