Average Error: 34.3 → 13.2
Time: 17.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.304421310335197068961304849785779948437 \cdot 10^{-75}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.680510304999259194268524546555599685222 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -3.304421310335197068961304849785779948437 \cdot 10^{-75}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.680510304999259194268524546555599685222 \cdot 10^{-106}:\\
\;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\end{array}
double f(double a, double b, double c) {
        double r2089584 = b;
        double r2089585 = -r2089584;
        double r2089586 = r2089584 * r2089584;
        double r2089587 = 4.0;
        double r2089588 = a;
        double r2089589 = c;
        double r2089590 = r2089588 * r2089589;
        double r2089591 = r2089587 * r2089590;
        double r2089592 = r2089586 - r2089591;
        double r2089593 = sqrt(r2089592);
        double r2089594 = r2089585 - r2089593;
        double r2089595 = 2.0;
        double r2089596 = r2089595 * r2089588;
        double r2089597 = r2089594 / r2089596;
        return r2089597;
}
↓
double f(double a, double b, double c) {
        double r2089598 = b;
        double r2089599 = -3.304421310335197e-75;
        bool r2089600 = r2089598 <= r2089599;
        double r2089601 = -1.0;
        double r2089602 = c;
        double r2089603 = r2089602 / r2089598;
        double r2089604 = r2089601 * r2089603;
        double r2089605 = 1.6805103049992592e-106;
        bool r2089606 = r2089598 <= r2089605;
        double r2089607 = a;
        double r2089608 = r2089602 * r2089607;
        double r2089609 = 4.0;
        double r2089610 = r2089608 * r2089609;
        double r2089611 = -r2089598;
        double r2089612 = r2089598 * r2089598;
        double r2089613 = r2089612 - r2089610;
        double r2089614 = sqrt(r2089613);
        double r2089615 = r2089611 + r2089614;
        double r2089616 = r2089610 / r2089615;
        double r2089617 = 2.0;
        double r2089618 = r2089607 * r2089617;
        double r2089619 = r2089616 / r2089618;
        double r2089620 = r2089598 / r2089607;
        double r2089621 = r2089603 - r2089620;
        double r2089622 = 1.0;
        double r2089623 = r2089621 * r2089622;
        double r2089624 = r2089606 ? r2089619 : r2089623;
        double r2089625 = r2089600 ? r2089604 : r2089624;
        return r2089625;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.3
Target21.3
Herbie13.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.304421310335197e-75

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 9.8

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

    if -3.304421310335197e-75 < b < 1.6805103049992592e-106

    1. Initial program 17.6

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

    3. Applied flip--20.2

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

    4. Taylor expanded around inf 19.1

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

    if 1.6805103049992592e-106 < b

    1. Initial program 25.3

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

    3. Applied div-inv25.4

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

    4. Taylor expanded around inf 12.5

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

    5. Simplified12.5

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

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.304421310335197068961304849785779948437 \cdot 10^{-75}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.680510304999259194268524546555599685222 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))