Average Error: 1.7 → 1.7
Time: 3.9m
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\frac{\sqrt{\frac{b_2 \cdot b_2 + c \cdot a}{\frac{b_2 \cdot b_2 + c \cdot a}{b_2 \cdot b_2 - c \cdot a}}} - b_2}{a}\]
\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}
↓
\frac{\sqrt{\frac{b_2 \cdot b_2 + c \cdot a}{\frac{b_2 \cdot b_2 + c \cdot a}{b_2 \cdot b_2 - c \cdot a}}} - b_2}{a}
double f(double a, double b_2, double c) {
        double r893289 = b_2;
        double r893290 = -r893289;
        double r893291 = r893289 * r893289;
        double r893292 = a;
        double r893293 = c;
        double r893294 = r893292 * r893293;
        double r893295 = r893291 - r893294;
        double r893296 = sqrt(r893295);
        double r893297 = r893290 + r893296;
        double r893298 = r893297 / r893292;
        return r893298;
}
↓
double f(double a, double b_2, double c) {
        double r893299 = b_2;
        double r893300 = r893299 * r893299;
        double r893301 = c;
        double r893302 = a;
        double r893303 = r893301 * r893302;
        double r893304 = r893300 + r893303;
        double r893305 = r893300 - r893303;
        double r893306 = r893304 / r893305;
        double r893307 = r893304 / r893306;
        double r893308 = sqrt(r893307);
        double r893309 = r893308 - r893299;
        double r893310 = r893309 / r893302;
        return r893310;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.7

    \[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]

  2. Simplified1.7

    \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}{a}}\]
  3. Using strategy rm

  4. Applied p16-flip--2.7

    \[\leadsto \frac{\left(\left(\sqrt{\color{blue}{\left(\frac{\left(\left(\left(b_2 \cdot b_2\right) \cdot \left(b_2 \cdot b_2\right)\right) - \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)\right)}{\left(\frac{\left(b_2 \cdot b_2\right)}{\left(c \cdot a\right)}\right)}\right)}}\right) - b_2\right)}{a}\]
  5. Using strategy rm

  6. Applied difference-of-squares2.6

    \[\leadsto \frac{\left(\left(\sqrt{\left(\frac{\color{blue}{\left(\left(\frac{\left(b_2 \cdot b_2\right)}{\left(c \cdot a\right)}\right) \cdot \left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)\right)}}{\left(\frac{\left(b_2 \cdot b_2\right)}{\left(c \cdot a\right)}\right)}\right)}\right) - b_2\right)}{a}\]

  7. Applied associate-/l*1.7

    \[\leadsto \frac{\left(\left(\sqrt{\color{blue}{\left(\frac{\left(\frac{\left(b_2 \cdot b_2\right)}{\left(c \cdot a\right)}\right)}{\left(\frac{\left(\frac{\left(b_2 \cdot b_2\right)}{\left(c \cdot a\right)}\right)}{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}\right)}}\right) - b_2\right)}{a}\]

  8. Final simplification1.7

    \[\leadsto \frac{\sqrt{\frac{b_2 \cdot b_2 + c \cdot a}{\frac{b_2 \cdot b_2 + c \cdot a}{b_2 \cdot b_2 - c \cdot a}}} - b_2}{a}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/.p16 (+.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))