Average Error: 1.7 → 0.4
Time: 20.6s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -0.07232666015625:\\ \;\;\;\;\frac{1.0}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -0.07232666015625:\\
\;\;\;\;\frac{1.0}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r1169485 = b_2;
        double r1169486 = -r1169485;
        double r1169487 = r1169485 * r1169485;
        double r1169488 = a;
        double r1169489 = c;
        double r1169490 = r1169488 * r1169489;
        double r1169491 = r1169487 - r1169490;
        double r1169492 = sqrt(r1169491);
        double r1169493 = r1169486 - r1169492;
        double r1169494 = r1169493 / r1169488;
        return r1169494;
}
↓
double f(double a, double b_2, double c) {
        double r1169495 = b_2;
        double r1169496 = -0.07232666015625;
        bool r1169497 = r1169495 <= r1169496;
        double r1169498 = 1.0;
        double r1169499 = r1169495 * r1169495;
        double r1169500 = c;
        double r1169501 = a;
        double r1169502 = r1169500 * r1169501;
        double r1169503 = r1169499 - r1169502;
        double r1169504 = sqrt(r1169503);
        double r1169505 = r1169504 - r1169495;
        double r1169506 = r1169505 / r1169500;
        double r1169507 = r1169498 / r1169506;
        double r1169508 = -r1169495;
        double r1169509 = r1169501 * r1169500;
        double r1169510 = r1169499 - r1169509;
        double r1169511 = sqrt(r1169510);
        double r1169512 = r1169508 - r1169511;
        double r1169513 = r1169512 / r1169501;
        double r1169514 = r1169497 ? r1169507 : r1169513;
        return r1169514;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b_2 < -0.07232666015625

    1. Initial program 3.4

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

    3. Applied p16-flip--3.0

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

    4. Simplified0.7

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

    5. Simplified0.7

      \[\leadsto \color{blue}{\frac{\left(\frac{\left(c \cdot a\right)}{\left(0.0\right)}\right)}{\left(a \cdot \left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)\right)}}\]
    6. Using strategy rm

    7. Applied associate-/r*0.5

      \[\leadsto \color{blue}{\frac{\left(\frac{\left(\frac{\left(c \cdot a\right)}{\left(0.0\right)}\right)}{a}\right)}{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}}\]

    8. Simplified0.4

      \[\leadsto \frac{\color{blue}{\left(\left(1.0\right) \cdot c\right)}}{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}\]
    9. Using strategy rm

    10. Applied associate-/l*0.5

      \[\leadsto \color{blue}{\frac{\left(1.0\right)}{\left(\frac{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}{c}\right)}}\]

    if -0.07232666015625 < b_2

    1. Initial program 0.4

      \[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -0.07232666015625:\\ \;\;\;\;\frac{1.0}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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))