Average Error: 1.7 → 1.7
Time: 22.0s
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 r1353503 = b_2;
        double r1353504 = -r1353503;
        double r1353505 = r1353503 * r1353503;
        double r1353506 = a;
        double r1353507 = c;
        double r1353508 = r1353506 * r1353507;
        double r1353509 = r1353505 - r1353508;
        double r1353510 = sqrt(r1353509);
        double r1353511 = r1353504 - r1353510;
        double r1353512 = r1353511 / r1353506;
        return r1353512;
}
↓
double f(double a, double b_2, double c) {
        double r1353513 = b_2;
        double r1353514 = -r1353513;
        double r1353515 = r1353513 * r1353513;
        double r1353516 = a;
        double r1353517 = c;
        double r1353518 = r1353516 * r1353517;
        double r1353519 = r1353515 + r1353518;
        double r1353520 = r1353515 - r1353518;
        double r1353521 = r1353519 / r1353520;
        double r1353522 = r1353519 / r1353521;
        double r1353523 = sqrt(r1353522);
        double r1353524 = r1353514 - r1353523;
        double r1353525 = r1353524 / r1353516;
        return r1353525;
}

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--2.8

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

  5. Applied difference-of-squares2.7

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

  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 2019130 +o rules:numerics
(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))