Average Error: 1.5 → 1.5
Time: 43.6s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{1.0}{2} \cdot \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a}\]
\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{1.0}{2} \cdot \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a}
double f(double a, double b, double c) {
        double r1138964 = b;
        double r1138965 = -r1138964;
        double r1138966 = r1138964 * r1138964;
        double r1138967 = 4.0;
        double r1138968 = /* ERROR: no posit support in C */;
        double r1138969 = a;
        double r1138970 = c;
        double r1138971 = r1138969 * r1138970;
        double r1138972 = r1138968 * r1138971;
        double r1138973 = r1138966 - r1138972;
        double r1138974 = sqrt(r1138973);
        double r1138975 = r1138965 + r1138974;
        double r1138976 = 2.0;
        double r1138977 = /* ERROR: no posit support in C */;
        double r1138978 = r1138977 * r1138969;
        double r1138979 = r1138975 / r1138978;
        return r1138979;
}
↓
double f(double a, double b, double c) {
        double r1138980 = 1.0;
        double r1138981 = 2.0;
        double r1138982 = r1138980 / r1138981;
        double r1138983 = b;
        double r1138984 = r1138983 * r1138983;
        double r1138985 = c;
        double r1138986 = a;
        double r1138987 = 4.0;
        double r1138988 = r1138986 * r1138987;
        double r1138989 = r1138985 * r1138988;
        double r1138990 = r1138984 - r1138989;
        double r1138991 = sqrt(r1138990);
        double r1138992 = r1138991 - r1138983;
        double r1138993 = r1138992 / r1138986;
        double r1138994 = r1138982 * r1138993;
        return r1138994;
}

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-*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. Using strategy rm

  6. Applied p16-*-un-lft-identity1.5

    \[\leadsto \frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(c \cdot \left(a \cdot \left(4\right)\right)\right)\right)}\right) - b\right)\right)}}{\left(\left(2\right) \cdot a\right)}\]

  7. Applied p16-times-frac1.5

    \[\leadsto \color{blue}{\left(\frac{\left(1.0\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(c \cdot \left(a \cdot \left(4\right)\right)\right)\right)}\right) - b\right)}{a}\right)}\]

  8. Final simplification1.5

    \[\leadsto \frac{1.0}{2} \cdot \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a}\]

Reproduce

herbie shell --seed 2019158 +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)))