Average Error: 33.6 → 10.4
Time: 21.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.264659490877098 \cdot 10^{-67}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 0.17389787404847717:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.264659490877098 \cdot 10^{-67}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 0.17389787404847717:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r2946550 = b;
        double r2946551 = -r2946550;
        double r2946552 = r2946550 * r2946550;
        double r2946553 = 4.0;
        double r2946554 = a;
        double r2946555 = c;
        double r2946556 = r2946554 * r2946555;
        double r2946557 = r2946553 * r2946556;
        double r2946558 = r2946552 - r2946557;
        double r2946559 = sqrt(r2946558);
        double r2946560 = r2946551 - r2946559;
        double r2946561 = 2.0;
        double r2946562 = r2946561 * r2946554;
        double r2946563 = r2946560 / r2946562;
        return r2946563;
}
↓
double f(double a, double b, double c) {
        double r2946564 = b;
        double r2946565 = -1.264659490877098e-67;
        bool r2946566 = r2946564 <= r2946565;
        double r2946567 = -2.0;
        double r2946568 = c;
        double r2946569 = r2946568 / r2946564;
        double r2946570 = r2946567 * r2946569;
        double r2946571 = 2.0;
        double r2946572 = r2946570 / r2946571;
        double r2946573 = 0.17389787404847717;
        bool r2946574 = r2946564 <= r2946573;
        double r2946575 = 1.0;
        double r2946576 = a;
        double r2946577 = r2946575 / r2946576;
        double r2946578 = -r2946564;
        double r2946579 = -4.0;
        double r2946580 = r2946576 * r2946579;
        double r2946581 = r2946564 * r2946564;
        double r2946582 = fma(r2946580, r2946568, r2946581);
        double r2946583 = sqrt(r2946582);
        double r2946584 = r2946578 - r2946583;
        double r2946585 = r2946577 * r2946584;
        double r2946586 = r2946585 / r2946571;
        double r2946587 = r2946564 / r2946576;
        double r2946588 = r2946569 - r2946587;
        double r2946589 = r2946588 * r2946571;
        double r2946590 = r2946589 / r2946571;
        double r2946591 = r2946574 ? r2946586 : r2946590;
        double r2946592 = r2946566 ? r2946572 : r2946591;
        return r2946592;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.9
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.264659490877098e-67

    1. Initial program 52.9

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified53.0

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{a}}{2}}\]

    3. Taylor expanded around -inf 8.1

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

    if -1.264659490877098e-67 < b < 0.17389787404847717

    1. Initial program 15.0

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified15.0

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{a}}{2}}\]
    3. Using strategy rm

    4. Applied div-inv15.1

      \[\leadsto \frac{\color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{2}\]

    if 0.17389787404847717 < b

    1. Initial program 29.8

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified29.8

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{a}}{2}}\]

    3. Taylor expanded around inf 7.3

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

    4. Simplified7.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.264659490877098 \cdot 10^{-67}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 0.17389787404847717:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))