Average Error: 1.5 → 1.5
Time: 26.2s
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 r1800156 = b;
        double r1800157 = -r1800156;
        double r1800158 = r1800156 * r1800156;
        double r1800159 = 4.0;
        double r1800160 = /* ERROR: no posit support in C */;
        double r1800161 = a;
        double r1800162 = c;
        double r1800163 = r1800161 * r1800162;
        double r1800164 = r1800160 * r1800163;
        double r1800165 = r1800158 - r1800164;
        double r1800166 = sqrt(r1800165);
        double r1800167 = r1800157 + r1800166;
        double r1800168 = 2.0;
        double r1800169 = /* ERROR: no posit support in C */;
        double r1800170 = r1800169 * r1800161;
        double r1800171 = r1800167 / r1800170;
        return r1800171;
}
↓
double f(double a, double b, double c) {
        double r1800172 = b;
        double r1800173 = r1800172 * r1800172;
        double r1800174 = c;
        double r1800175 = a;
        double r1800176 = 4.0;
        double r1800177 = r1800175 * r1800176;
        double r1800178 = r1800174 * r1800177;
        double r1800179 = r1800173 - r1800178;
        double r1800180 = sqrt(r1800179);
        double r1800181 = r1800180 - r1800172;
        double r1800182 = r1800181 / r1800175;
        double r1800183 = 2.0;
        double r1800184 = r1800182 / r1800183;
        return r1800184;
}

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 +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)))