Average Error: 0.2 → 0.2
Time: 53.1s
Precision: 64
\[\left(0\right) \lt a \land \left(0\right) \lt b \land \left(0\right) \lt c\]
\[\sqrt{\left(\left(\left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - a\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - b\right)\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{a}{b}\right)}{c}\right)}{\left(2\right)}\right) - c\right)\right)}\]
↓
\[\sqrt{\left(\left(\frac{\left(a + b\right) + c}{2} \cdot \left(\frac{\left(a + b\right) + c}{2} - a\right)\right) \cdot \left(\frac{1.0}{\frac{2}{a + \left(c + b\right)}} - b\right)\right) \cdot \frac{\frac{\left(c + b\right) + a}{2} \cdot \frac{\left(c + b\right) + a}{2} - c \cdot c}{\frac{\left(c + b\right) + a}{2} + c}}\]
\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{1.0}{\frac{2}{a + \left(c + b\right)}} - b\right)\right) \cdot \frac{\frac{\left(c + b\right) + a}{2} \cdot \frac{\left(c + b\right) + a}{2} - c \cdot c}{\frac{\left(c + b\right) + a}{2} + c}}
double f(double a, double b, double c) {
        double r1632802 = a;
        double r1632803 = b;
        double r1632804 = r1632802 + r1632803;
        double r1632805 = c;
        double r1632806 = r1632804 + r1632805;
        double r1632807 = 2.0;
        double r1632808 = /* ERROR: no posit support in C */;
        double r1632809 = r1632806 / r1632808;
        double r1632810 = r1632809 - r1632802;
        double r1632811 = r1632809 * r1632810;
        double r1632812 = r1632809 - r1632803;
        double r1632813 = r1632811 * r1632812;
        double r1632814 = r1632809 - r1632805;
        double r1632815 = r1632813 * r1632814;
        double r1632816 = sqrt(r1632815);
        return r1632816;
}
↓
double f(double a, double b, double c) {
        double r1632817 = a;
        double r1632818 = b;
        double r1632819 = r1632817 + r1632818;
        double r1632820 = c;
        double r1632821 = r1632819 + r1632820;
        double r1632822 = 2.0;
        double r1632823 = r1632821 / r1632822;
        double r1632824 = r1632823 - r1632817;
        double r1632825 = r1632823 * r1632824;
        double r1632826 = 1.0;
        double r1632827 = r1632820 + r1632818;
        double r1632828 = r1632817 + r1632827;
        double r1632829 = r1632822 / r1632828;
        double r1632830 = r1632826 / r1632829;
        double r1632831 = r1632830 - r1632818;
        double r1632832 = r1632825 * r1632831;
        double r1632833 = r1632827 + r1632817;
        double r1632834 = r1632833 / r1632822;
        double r1632835 = r1632834 * r1632834;
        double r1632836 = r1632820 * r1632820;
        double r1632837 = r1632835 - r1632836;
        double r1632838 = r1632834 + r1632820;
        double r1632839 = r1632837 / r1632838;
        double r1632840 = r1632832 * r1632839;
        double r1632841 = sqrt(r1632840);
        return r1632841;
}

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-*-un-lft-identity0.2

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

  4. Applied associate-/r*0.2

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

  5. Simplified0.2

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

  7. Applied p16-*-un-lft-identity0.2

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

  8. Applied associate-/r*0.2

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

  9. Simplified0.2

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

  11. Applied p16-*-un-lft-identity0.2

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

  12. Applied associate-/l*0.2

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

  14. Applied p16-flip--0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019154 
(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))))