Average Error: 34.6 → 10.7
Time: 29.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\
\;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -1\\

\end{array}
double f(double a, double b, double c) {
        double r5858691 = b;
        double r5858692 = -r5858691;
        double r5858693 = r5858691 * r5858691;
        double r5858694 = 4.0;
        double r5858695 = a;
        double r5858696 = r5858694 * r5858695;
        double r5858697 = c;
        double r5858698 = r5858696 * r5858697;
        double r5858699 = r5858693 - r5858698;
        double r5858700 = sqrt(r5858699);
        double r5858701 = r5858692 + r5858700;
        double r5858702 = 2.0;
        double r5858703 = r5858702 * r5858695;
        double r5858704 = r5858701 / r5858703;
        return r5858704;
}
↓
double f(double a, double b, double c) {
        double r5858705 = b;
        double r5858706 = -2.7668189408748547e+100;
        bool r5858707 = r5858705 <= r5858706;
        double r5858708 = c;
        double r5858709 = r5858708 / r5858705;
        double r5858710 = a;
        double r5858711 = r5858705 / r5858710;
        double r5858712 = r5858709 - r5858711;
        double r5858713 = 1.0;
        double r5858714 = r5858712 * r5858713;
        double r5858715 = 7.923524897992037e-153;
        bool r5858716 = r5858705 <= r5858715;
        double r5858717 = 1.0;
        double r5858718 = 2.0;
        double r5858719 = r5858710 * r5858718;
        double r5858720 = r5858717 / r5858719;
        double r5858721 = r5858705 * r5858705;
        double r5858722 = r5858710 * r5858708;
        double r5858723 = 4.0;
        double r5858724 = r5858722 * r5858723;
        double r5858725 = r5858721 - r5858724;
        double r5858726 = sqrt(r5858725);
        double r5858727 = r5858726 - r5858705;
        double r5858728 = r5858717 / r5858727;
        double r5858729 = r5858720 / r5858728;
        double r5858730 = -1.0;
        double r5858731 = r5858709 * r5858730;
        double r5858732 = r5858716 ? r5858729 : r5858731;
        double r5858733 = r5858707 ? r5858714 : r5858732;
        return r5858733;
}

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

Original34.6
Target21.1
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.7668189408748547e+100

    1. Initial program 47.2

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

    2. Simplified47.2

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

    3. Taylor expanded around -inf 4.0

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

    4. Simplified4.0

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

    if -2.7668189408748547e+100 < b < 7.923524897992037e-153

    1. Initial program 10.8

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

    2. Simplified10.9

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

    4. Applied clear-num11.0

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

    6. Applied div-inv11.1

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

    7. Applied associate-/r*11.0

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

    if 7.923524897992037e-153 < b

    1. Initial program 50.5

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

    2. Simplified50.5

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

    3. Taylor expanded around inf 12.7

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

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

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