Average Error: 1.5 → 1.5
Time: 27.8s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\]
\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}{a \cdot 2}
double f(double a, double b, double c) {
        double r1824859 = b;
        double r1824860 = -r1824859;
        double r1824861 = r1824859 * r1824859;
        double r1824862 = 4.0;
        double r1824863 = /* ERROR: no posit support in C */;
        double r1824864 = a;
        double r1824865 = c;
        double r1824866 = r1824864 * r1824865;
        double r1824867 = r1824863 * r1824866;
        double r1824868 = r1824861 - r1824867;
        double r1824869 = sqrt(r1824868);
        double r1824870 = r1824860 + r1824869;
        double r1824871 = 2.0;
        double r1824872 = /* ERROR: no posit support in C */;
        double r1824873 = r1824872 * r1824864;
        double r1824874 = r1824870 / r1824873;
        return r1824874;
}
↓
double f(double a, double b, double c) {
        double r1824875 = b;
        double r1824876 = r1824875 * r1824875;
        double r1824877 = c;
        double r1824878 = a;
        double r1824879 = 4.0;
        double r1824880 = r1824878 * r1824879;
        double r1824881 = r1824877 * r1824880;
        double r1824882 = r1824876 - r1824881;
        double r1824883 = sqrt(r1824882);
        double r1824884 = r1824883 - r1824875;
        double r1824885 = 2.0;
        double r1824886 = r1824878 * r1824885;
        double r1824887 = r1824884 / r1824886;
        return r1824887;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.5

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

  2. Simplified1.5

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

  4. Applied associate-/r*1.5

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

  6. Applied associate-*l*1.5

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

  8. Applied associate-/l/1.5

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

  9. Final simplification1.5

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(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)))