Average Error: 33.6 → 10.0
Time: 24.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.2092322739463293 \cdot 10^{-86}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -3.2092322739463293 \cdot 10^{-86}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.891777552454845 \cdot 10^{+74}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r2773602 = b;
        double r2773603 = -r2773602;
        double r2773604 = r2773602 * r2773602;
        double r2773605 = 4.0;
        double r2773606 = a;
        double r2773607 = c;
        double r2773608 = r2773606 * r2773607;
        double r2773609 = r2773605 * r2773608;
        double r2773610 = r2773604 - r2773609;
        double r2773611 = sqrt(r2773610);
        double r2773612 = r2773603 - r2773611;
        double r2773613 = 2.0;
        double r2773614 = r2773613 * r2773606;
        double r2773615 = r2773612 / r2773614;
        return r2773615;
}
↓
double f(double a, double b, double c) {
        double r2773616 = b;
        double r2773617 = -3.2092322739463293e-86;
        bool r2773618 = r2773616 <= r2773617;
        double r2773619 = c;
        double r2773620 = r2773619 / r2773616;
        double r2773621 = -r2773620;
        double r2773622 = 2.891777552454845e+74;
        bool r2773623 = r2773616 <= r2773622;
        double r2773624 = -r2773616;
        double r2773625 = r2773616 * r2773616;
        double r2773626 = a;
        double r2773627 = r2773619 * r2773626;
        double r2773628 = 4.0;
        double r2773629 = r2773627 * r2773628;
        double r2773630 = r2773625 - r2773629;
        double r2773631 = sqrt(r2773630);
        double r2773632 = r2773624 - r2773631;
        double r2773633 = 2.0;
        double r2773634 = r2773626 * r2773633;
        double r2773635 = r2773632 / r2773634;
        double r2773636 = r2773624 / r2773626;
        double r2773637 = r2773623 ? r2773635 : r2773636;
        double r2773638 = r2773618 ? r2773621 : r2773637;
        return r2773638;
}

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

Original33.6
Target20.7
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.2092322739463293e-86

    1. Initial program 52.3

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

    3. Applied div-inv52.3

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

    4. Simplified52.3

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

    5. Taylor expanded around -inf 9.4

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

    6. Simplified9.4

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

    if -3.2092322739463293e-86 < b < 2.891777552454845e+74

    1. Initial program 13.1

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

    if 2.891777552454845e+74 < b

    1. Initial program 38.9

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

    3. Applied *-un-lft-identity38.9

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

    4. Applied associate-/l*39.0

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

    5. Taylor expanded around 0 4.4

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

    6. Simplified4.4

      \[\leadsto \color{blue}{-\frac{b}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.2092322739463293 \cdot 10^{-86}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(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)))