Average Error: 33.3 → 10.4
Time: 24.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.179137486378021 \cdot 10^{-24}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -4.179137486378021 \cdot 10^{-24}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\
\;\;\;\;\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 r2315820 = b;
        double r2315821 = -r2315820;
        double r2315822 = r2315820 * r2315820;
        double r2315823 = 4.0;
        double r2315824 = a;
        double r2315825 = c;
        double r2315826 = r2315824 * r2315825;
        double r2315827 = r2315823 * r2315826;
        double r2315828 = r2315822 - r2315827;
        double r2315829 = sqrt(r2315828);
        double r2315830 = r2315821 - r2315829;
        double r2315831 = 2.0;
        double r2315832 = r2315831 * r2315824;
        double r2315833 = r2315830 / r2315832;
        return r2315833;
}
↓
double f(double a, double b, double c) {
        double r2315834 = b;
        double r2315835 = -4.179137486378021e-24;
        bool r2315836 = r2315834 <= r2315835;
        double r2315837 = c;
        double r2315838 = r2315837 / r2315834;
        double r2315839 = -r2315838;
        double r2315840 = 2.3648644896474148e+52;
        bool r2315841 = r2315834 <= r2315840;
        double r2315842 = -r2315834;
        double r2315843 = r2315834 * r2315834;
        double r2315844 = a;
        double r2315845 = r2315837 * r2315844;
        double r2315846 = 4.0;
        double r2315847 = r2315845 * r2315846;
        double r2315848 = r2315843 - r2315847;
        double r2315849 = sqrt(r2315848);
        double r2315850 = r2315842 - r2315849;
        double r2315851 = 2.0;
        double r2315852 = r2315844 * r2315851;
        double r2315853 = r2315850 / r2315852;
        double r2315854 = r2315842 / r2315844;
        double r2315855 = r2315841 ? r2315853 : r2315854;
        double r2315856 = r2315836 ? r2315839 : r2315855;
        return r2315856;
}

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

Original33.3
Target20.7
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 < -4.179137486378021e-24

    1. Initial program 54.6

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

    2. Taylor expanded around -inf 7.1

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

    3. Simplified7.1

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

    if -4.179137486378021e-24 < b < 2.3648644896474148e+52

    1. Initial program 15.1

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

    if 2.3648644896474148e+52 < b

    1. Initial program 36.7

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

    3. Applied *-un-lft-identity36.7

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

    4. Applied *-un-lft-identity36.7

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

    5. Applied distribute-rgt-neg-in36.7

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

    6. Applied distribute-lft-out--36.7

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

    7. Applied associate-/l*36.8

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

    8. Taylor expanded around 0 5.5

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

    9. Simplified5.5

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.179137486378021 \cdot 10^{-24}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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)))