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 r1976940 = a;
        double r1976941 = b;
        double r1976942 = r1976940 + r1976941;
        double r1976943 = c;
        double r1976944 = r1976942 + r1976943;
        double r1976945 = 2.0;
        double r1976946 = /* ERROR: no posit support in C */;
        double r1976947 = r1976944 / r1976946;
        double r1976948 = r1976947 - r1976940;
        double r1976949 = r1976947 * r1976948;
        double r1976950 = r1976947 - r1976941;
        double r1976951 = r1976949 * r1976950;
        double r1976952 = r1976947 - r1976943;
        double r1976953 = r1976951 * r1976952;
        double r1976954 = sqrt(r1976953);
        return r1976954;
}
↓
double f(double a, double b, double c) {
        double r1976955 = a;
        double r1976956 = b;
        double r1976957 = r1976955 + r1976956;
        double r1976958 = c;
        double r1976959 = r1976957 + r1976958;
        double r1976960 = 2.0;
        double r1976961 = r1976959 / r1976960;
        double r1976962 = r1976958 + r1976956;
        double r1976963 = r1976962 + r1976955;
        double r1976964 = r1976963 / r1976960;
        double r1976965 = r1976964 - r1976956;
        double r1976966 = r1976961 * r1976965;
        double r1976967 = r1976964 - r1976955;
        double r1976968 = r1976966 * r1976967;
        double r1976969 = r1976961 - r1976958;
        double r1976970 = r1976968 * r1976969;
        double r1976971 = sqrt(r1976970);
        return r1976971;
}

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 +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))))