Average Error: 1.7 → 1.7
Time: 24.1s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\frac{\left(-b_2\right) - \sqrt{\frac{b_2 \cdot b_2 + a \cdot c}{\frac{b_2 \cdot b_2 + a \cdot c}{b_2 \cdot b_2 - a \cdot c}}}}{a}\]
\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}
↓
\frac{\left(-b_2\right) - \sqrt{\frac{b_2 \cdot b_2 + a \cdot c}{\frac{b_2 \cdot b_2 + a \cdot c}{b_2 \cdot b_2 - a \cdot c}}}}{a}
double f(double a, double b_2, double c) {
        double r1551445 = b_2;
        double r1551446 = -r1551445;
        double r1551447 = r1551445 * r1551445;
        double r1551448 = a;
        double r1551449 = c;
        double r1551450 = r1551448 * r1551449;
        double r1551451 = r1551447 - r1551450;
        double r1551452 = sqrt(r1551451);
        double r1551453 = r1551446 - r1551452;
        double r1551454 = r1551453 / r1551448;
        return r1551454;
}
↓
double f(double a, double b_2, double c) {
        double r1551455 = b_2;
        double r1551456 = -r1551455;
        double r1551457 = r1551455 * r1551455;
        double r1551458 = a;
        double r1551459 = c;
        double r1551460 = r1551458 * r1551459;
        double r1551461 = r1551457 + r1551460;
        double r1551462 = r1551457 - r1551460;
        double r1551463 = r1551461 / r1551462;
        double r1551464 = r1551461 / r1551463;
        double r1551465 = sqrt(r1551464);
        double r1551466 = r1551456 - r1551465;
        double r1551467 = r1551466 / r1551458;
        return r1551467;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.7

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

  3. Applied p16-flip--2.7

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

  5. Applied difference-of-squares2.6

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

  6. Applied associate-/l*1.7

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

  7. Final simplification1.7

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

Reproduce

herbie shell --seed 2019134 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/.p16 (-.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))