Average Error: 1.5 → 1.5
Time: 21.2s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}\]
\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{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}
double f(double a, double b, double c) {
        double r1553192 = b;
        double r1553193 = -r1553192;
        double r1553194 = r1553192 * r1553192;
        double r1553195 = 4.0;
        double r1553196 = /* ERROR: no posit support in C */;
        double r1553197 = a;
        double r1553198 = c;
        double r1553199 = r1553197 * r1553198;
        double r1553200 = r1553196 * r1553199;
        double r1553201 = r1553194 - r1553200;
        double r1553202 = sqrt(r1553201);
        double r1553203 = r1553193 + r1553202;
        double r1553204 = 2.0;
        double r1553205 = /* ERROR: no posit support in C */;
        double r1553206 = r1553205 * r1553197;
        double r1553207 = r1553203 / r1553206;
        return r1553207;
}
↓
double f(double a, double b, double c) {
        double r1553208 = b;
        double r1553209 = r1553208 * r1553208;
        double r1553210 = c;
        double r1553211 = a;
        double r1553212 = r1553210 * r1553211;
        double r1553213 = 4.0;
        double r1553214 = r1553212 * r1553213;
        double r1553215 = r1553209 - r1553214;
        double r1553216 = sqrt(r1553215);
        double r1553217 = r1553216 - r1553208;
        double r1553218 = 2.0;
        double r1553219 = r1553217 / r1553218;
        double r1553220 = r1553219 / r1553211;
        return r1553220;
}

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. Final simplification1.5

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

Reproduce

herbie shell --seed 2019134 
(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)))