Average Error: 34.5 → 10.2
Time: 4.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.3645547041066157 \cdot 10^{-80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -8.3645547041066157 \cdot 10^{-80}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\
\;\;\;\;\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r77649 = b;
        double r77650 = -r77649;
        double r77651 = r77649 * r77649;
        double r77652 = 4.0;
        double r77653 = a;
        double r77654 = c;
        double r77655 = r77653 * r77654;
        double r77656 = r77652 * r77655;
        double r77657 = r77651 - r77656;
        double r77658 = sqrt(r77657);
        double r77659 = r77650 - r77658;
        double r77660 = 2.0;
        double r77661 = r77660 * r77653;
        double r77662 = r77659 / r77661;
        return r77662;
}
↓
double f(double a, double b, double c) {
        double r77663 = b;
        double r77664 = -8.364554704106616e-80;
        bool r77665 = r77663 <= r77664;
        double r77666 = -1.0;
        double r77667 = c;
        double r77668 = r77667 / r77663;
        double r77669 = r77666 * r77668;
        double r77670 = 4.1199128263687574e+46;
        bool r77671 = r77663 <= r77670;
        double r77672 = 1.0;
        double r77673 = -r77663;
        double r77674 = r77663 * r77663;
        double r77675 = 4.0;
        double r77676 = a;
        double r77677 = r77676 * r77667;
        double r77678 = r77675 * r77677;
        double r77679 = r77674 - r77678;
        double r77680 = sqrt(r77679);
        double r77681 = r77673 - r77680;
        double r77682 = 2.0;
        double r77683 = r77682 * r77676;
        double r77684 = r77681 / r77683;
        double r77685 = r77672 / r77684;
        double r77686 = r77672 / r77685;
        double r77687 = 1.0;
        double r77688 = r77663 / r77676;
        double r77689 = r77668 - r77688;
        double r77690 = r77687 * r77689;
        double r77691 = r77671 ? r77686 : r77690;
        double r77692 = r77665 ? r77669 : r77691;
        return r77692;
}

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

Original34.5
Target21.1
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.364554704106616e-80

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 9.1

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

    if -8.364554704106616e-80 < b < 4.1199128263687574e+46

    1. Initial program 13.8

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

    3. Applied clear-num13.9

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

    5. Applied clear-num13.8

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

    if 4.1199128263687574e+46 < b

    1. Initial program 36.8

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

    2. Taylor expanded around inf 5.2

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

    3. Simplified5.2

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.3645547041066157 \cdot 10^{-80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))