Average Error: 1.7 → 0.9
Time: 18.2s
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le 2.1457672119140625 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0 - a \cdot c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a}\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le 2.1457672119140625 \cdot 10^{-05}:\\
\;\;\;\;\frac{\frac{\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{0.0 - a \cdot c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r1039572 = b_2;
        double r1039573 = -r1039572;
        double r1039574 = r1039572 * r1039572;
        double r1039575 = a;
        double r1039576 = c;
        double r1039577 = r1039575 * r1039576;
        double r1039578 = r1039574 - r1039577;
        double r1039579 = sqrt(r1039578);
        double r1039580 = r1039573 + r1039579;
        double r1039581 = r1039580 / r1039575;
        return r1039581;
}
↓
double f(double a, double b_2, double c) {
        double r1039582 = b_2;
        double r1039583 = 2.1457672119140625e-05;
        bool r1039584 = r1039582 <= r1039583;
        double r1039585 = r1039582 * r1039582;
        double r1039586 = c;
        double r1039587 = a;
        double r1039588 = r1039586 * r1039587;
        double r1039589 = r1039585 - r1039588;
        double r1039590 = sqrt(r1039589);
        double r1039591 = r1039590 + r1039582;
        double r1039592 = r1039590 - r1039582;
        double r1039593 = r1039591 * r1039592;
        double r1039594 = r1039593 / r1039591;
        double r1039595 = r1039594 / r1039587;
        double r1039596 = 0.0;
        double r1039597 = r1039587 * r1039586;
        double r1039598 = r1039596 - r1039597;
        double r1039599 = r1039598 / r1039591;
        double r1039600 = r1039599 / r1039587;
        double r1039601 = r1039584 ? r1039595 : r1039600;
        return r1039601;
}

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 < 2.1457672119140625e-05

    1. Initial program 0.4

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

    2. Simplified0.4

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

    4. Applied p16-flip--3.3

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

    6. Applied difference-of-squares1.2

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

    if 2.1457672119140625e-05 < b_2

    1. Initial program 3.0

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

    2. Simplified3.0

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

    4. Applied p16-flip--2.6

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

    6. Applied difference-of-squares3.0

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

    8. Applied associate-/l*3.2

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

    10. Applied p16-flip--3.4

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

    11. Applied associate-/r/3.4

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

    12. Applied associate-/r*3.3

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

    13. Simplified0.6

      \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(0.0\right) - \left(a \cdot c\right)\right)}}{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}\right)}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le 2.1457672119140625 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0 - a \cdot c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(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))