Average Error: 34.3 → 10.4
Time: 40.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.264659490877097952776006549579654784856 \cdot 10^{-67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.264659490877097952776006549579654784856 \cdot 10^{-67}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 0.173897874048477174557802982235443778336:\\
\;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r902495 = b_2;
        double r902496 = -r902495;
        double r902497 = r902495 * r902495;
        double r902498 = a;
        double r902499 = c;
        double r902500 = r902498 * r902499;
        double r902501 = r902497 - r902500;
        double r902502 = sqrt(r902501);
        double r902503 = r902496 - r902502;
        double r902504 = r902503 / r902498;
        return r902504;
}
↓
double f(double a, double b_2, double c) {
        double r902505 = b_2;
        double r902506 = -1.264659490877098e-67;
        bool r902507 = r902505 <= r902506;
        double r902508 = -0.5;
        double r902509 = c;
        double r902510 = r902509 / r902505;
        double r902511 = r902508 * r902510;
        double r902512 = 0.17389787404847717;
        bool r902513 = r902505 <= r902512;
        double r902514 = r902505 * r902505;
        double r902515 = a;
        double r902516 = r902509 * r902515;
        double r902517 = r902514 - r902516;
        double r902518 = sqrt(r902517);
        double r902519 = r902518 + r902505;
        double r902520 = -r902519;
        double r902521 = r902520 / r902515;
        double r902522 = -2.0;
        double r902523 = r902505 / r902515;
        double r902524 = 0.5;
        double r902525 = r902510 * r902524;
        double r902526 = fma(r902522, r902523, r902525);
        double r902527 = r902513 ? r902521 : r902526;
        double r902528 = r902507 ? r902511 : r902527;
        return r902528;
}

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 < -1.264659490877098e-67

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 8.1

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

    if -1.264659490877098e-67 < b_2 < 0.17389787404847717

    1. Initial program 15.2

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

    3. Applied *-un-lft-identity15.2

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

    4. Applied associate-/r*15.2

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

    5. Simplified15.2

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

    if 0.17389787404847717 < b_2

    1. Initial program 31.2

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

    3. Applied div-inv31.3

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

    4. Taylor expanded around inf 7.3

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

    5. Simplified7.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.264659490877097952776006549579654784856 \cdot 10^{-67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))