Average Error: 34.5 → 10.3
Time: 18.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -63362873442066488610789523456:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c \cdot \frac{1}{2}}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 6.484072051994263737451444554171174935457 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -63362873442066488610789523456:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c \cdot \frac{1}{2}}{b_2}\right)\\

\mathbf{elif}\;b_2 \le 6.484072051994263737451444554171174935457 \cdot 10^{-107}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r928538 = b_2;
        double r928539 = -r928538;
        double r928540 = r928538 * r928538;
        double r928541 = a;
        double r928542 = c;
        double r928543 = r928541 * r928542;
        double r928544 = r928540 - r928543;
        double r928545 = sqrt(r928544);
        double r928546 = r928539 + r928545;
        double r928547 = r928546 / r928541;
        return r928547;
}
↓
double f(double a, double b_2, double c) {
        double r928548 = b_2;
        double r928549 = -6.336287344206649e+28;
        bool r928550 = r928548 <= r928549;
        double r928551 = a;
        double r928552 = r928548 / r928551;
        double r928553 = -2.0;
        double r928554 = c;
        double r928555 = 0.5;
        double r928556 = r928554 * r928555;
        double r928557 = r928556 / r928548;
        double r928558 = fma(r928552, r928553, r928557);
        double r928559 = 6.484072051994264e-107;
        bool r928560 = r928548 <= r928559;
        double r928561 = 1.0;
        double r928562 = r928548 * r928548;
        double r928563 = r928551 * r928554;
        double r928564 = r928562 - r928563;
        double r928565 = sqrt(r928564);
        double r928566 = r928565 - r928548;
        double r928567 = r928551 / r928566;
        double r928568 = r928561 / r928567;
        double r928569 = -0.5;
        double r928570 = r928554 / r928548;
        double r928571 = r928569 * r928570;
        double r928572 = r928560 ? r928568 : r928571;
        double r928573 = r928550 ? r928558 : r928572;
        return r928573;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -6.336287344206649e+28

    1. Initial program 34.8

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

    2. Simplified34.8

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around -inf 7.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    4. Simplified7.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2} \cdot c}{b_2}\right)}\]

    if -6.336287344206649e+28 < b_2 < 6.484072051994264e-107

    1. Initial program 12.9

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

    2. Simplified12.9

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

    4. Applied clear-num12.9

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]

    if 6.484072051994264e-107 < b_2

    1. Initial program 52.5

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

    2. Simplified52.5

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf 9.7

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -63362873442066488610789523456:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c \cdot \frac{1}{2}}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 6.484072051994263737451444554171174935457 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))