Average Error: 1.7 → 1.7
Time: 13.8s
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 r743371 = b_2;
        double r743372 = -r743371;
        double r743373 = r743371 * r743371;
        double r743374 = a;
        double r743375 = c;
        double r743376 = r743374 * r743375;
        double r743377 = r743373 - r743376;
        double r743378 = sqrt(r743377);
        double r743379 = r743372 + r743378;
        double r743380 = r743379 / r743374;
        return r743380;
}
↓
double f(double a, double b_2, double c) {
        double r743381 = b_2;
        double r743382 = r743381 * r743381;
        double r743383 = c;
        double r743384 = a;
        double r743385 = r743383 * r743384;
        double r743386 = r743382 + r743385;
        double r743387 = r743382 - r743385;
        double r743388 = r743386 / r743387;
        double r743389 = r743386 / r743388;
        double r743390 = sqrt(r743389);
        double r743391 = r743390 - r743381;
        double r743392 = r743391 / r743384;
        return r743392;
}

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 2019121 
(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))