Average Error: 1.6 → 1.5
Time: 1.2m
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\]
\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}
↓
\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}
double f(double a, double b, double c) {
        double r1385822 = b;
        double r1385823 = -r1385822;
        double r1385824 = r1385822 * r1385822;
        double r1385825 = 4.0;
        double r1385826 = /* ERROR: no posit support in C */;
        double r1385827 = a;
        double r1385828 = c;
        double r1385829 = r1385827 * r1385828;
        double r1385830 = r1385826 * r1385829;
        double r1385831 = r1385824 - r1385830;
        double r1385832 = sqrt(r1385831);
        double r1385833 = r1385823 + r1385832;
        double r1385834 = 2.0;
        double r1385835 = /* ERROR: no posit support in C */;
        double r1385836 = r1385835 * r1385827;
        double r1385837 = r1385833 / r1385836;
        return r1385837;
}
↓
double f(double a, double b, double c) {
        double r1385838 = b;
        double r1385839 = r1385838 * r1385838;
        double r1385840 = c;
        double r1385841 = a;
        double r1385842 = 4.0;
        double r1385843 = r1385841 * r1385842;
        double r1385844 = r1385840 * r1385843;
        double r1385845 = r1385839 - r1385844;
        double r1385846 = sqrt(r1385845);
        double r1385847 = r1385846 - r1385838;
        double r1385848 = 2.0;
        double r1385849 = r1385848 * r1385841;
        double r1385850 = r1385847 / r1385849;
        return r1385850;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.6

    \[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]

  2. Simplified1.6

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

  4. Applied associate-*l*1.5

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

  5. Final simplification1.5

    \[\leadsto \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  (/.p16 (+.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))