Average Error: 1.5 → 1.5
Time: 27.2s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a}}{2}\]
\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{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a}}{2}
double f(double a, double b, double c) {
        double r1825072 = b;
        double r1825073 = -r1825072;
        double r1825074 = r1825072 * r1825072;
        double r1825075 = 4.0;
        double r1825076 = /* ERROR: no posit support in C */;
        double r1825077 = a;
        double r1825078 = c;
        double r1825079 = r1825077 * r1825078;
        double r1825080 = r1825076 * r1825079;
        double r1825081 = r1825074 - r1825080;
        double r1825082 = sqrt(r1825081);
        double r1825083 = r1825073 + r1825082;
        double r1825084 = 2.0;
        double r1825085 = /* ERROR: no posit support in C */;
        double r1825086 = r1825085 * r1825077;
        double r1825087 = r1825083 / r1825086;
        return r1825087;
}
↓
double f(double a, double b, double c) {
        double r1825088 = b;
        double r1825089 = r1825088 * r1825088;
        double r1825090 = c;
        double r1825091 = a;
        double r1825092 = 4.0;
        double r1825093 = r1825091 * r1825092;
        double r1825094 = r1825090 * r1825093;
        double r1825095 = r1825089 - r1825094;
        double r1825096 = sqrt(r1825095);
        double r1825097 = r1825096 - r1825088;
        double r1825098 = r1825097 / r1825091;
        double r1825099 = 2.0;
        double r1825100 = r1825098 / r1825099;
        return r1825100;
}

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. Using strategy rm

  10. Applied associate-/r*1.5

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

  11. Final simplification1.5

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

Reproduce

herbie shell --seed 2019128 
(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)))