Average Error: 34.0 → 9.3
Time: 6.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -14858297.0087451544:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.46153957938955092 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.3845340503596435 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -14858297.0087451544:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.46153957938955092 \cdot 10^{-137}:\\
\;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.3845340503596435 \cdot 10^{70}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r82132 = b;
        double r82133 = -r82132;
        double r82134 = r82132 * r82132;
        double r82135 = 4.0;
        double r82136 = a;
        double r82137 = c;
        double r82138 = r82136 * r82137;
        double r82139 = r82135 * r82138;
        double r82140 = r82134 - r82139;
        double r82141 = sqrt(r82140);
        double r82142 = r82133 - r82141;
        double r82143 = 2.0;
        double r82144 = r82143 * r82136;
        double r82145 = r82142 / r82144;
        return r82145;
}
↓
double f(double a, double b, double c) {
        double r82146 = b;
        double r82147 = -14858297.008745154;
        bool r82148 = r82146 <= r82147;
        double r82149 = -1.0;
        double r82150 = c;
        double r82151 = r82150 / r82146;
        double r82152 = r82149 * r82151;
        double r82153 = -1.461539579389551e-137;
        bool r82154 = r82146 <= r82153;
        double r82155 = 2.0;
        double r82156 = pow(r82146, r82155);
        double r82157 = r82156 - r82156;
        double r82158 = 4.0;
        double r82159 = a;
        double r82160 = r82159 * r82150;
        double r82161 = r82158 * r82160;
        double r82162 = r82157 + r82161;
        double r82163 = r82146 * r82146;
        double r82164 = r82163 - r82161;
        double r82165 = sqrt(r82164);
        double r82166 = r82165 - r82146;
        double r82167 = r82162 / r82166;
        double r82168 = 2.0;
        double r82169 = r82168 * r82159;
        double r82170 = r82167 / r82169;
        double r82171 = 1.3845340503596435e+70;
        bool r82172 = r82146 <= r82171;
        double r82173 = -r82146;
        double r82174 = r82173 - r82165;
        double r82175 = r82174 / r82168;
        double r82176 = r82175 / r82159;
        double r82177 = r82146 / r82159;
        double r82178 = r82149 * r82177;
        double r82179 = r82172 ? r82176 : r82178;
        double r82180 = r82154 ? r82170 : r82179;
        double r82181 = r82148 ? r82152 : r82180;
        return r82181;
}

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

Original34.0
Target20.8
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -14858297.008745154

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 5.8

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

    if -14858297.008745154 < b < -1.461539579389551e-137

    1. Initial program 35.4

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

    3. Applied flip--35.5

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

    4. Simplified18.0

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

    5. Simplified18.0

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

    if -1.461539579389551e-137 < b < 1.3845340503596435e+70

    1. Initial program 11.6

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

    3. Applied associate-/r*11.6

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

    if 1.3845340503596435e+70 < b

    1. Initial program 41.5

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

    3. Applied associate-/r*41.5

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

    5. Applied *-un-lft-identity41.5

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

    6. Applied *-un-lft-identity41.5

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

    7. Applied times-frac41.5

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

    8. Applied associate-/l*41.6

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

    9. Taylor expanded around 0 5.7

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -14858297.0087451544:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.46153957938955092 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.3845340503596435 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))