Average Error: 34.4 → 10.2
Time: 17.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\

\mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r4004117 = b;
        double r4004118 = -r4004117;
        double r4004119 = r4004117 * r4004117;
        double r4004120 = 4.0;
        double r4004121 = a;
        double r4004122 = r4004120 * r4004121;
        double r4004123 = c;
        double r4004124 = r4004122 * r4004123;
        double r4004125 = r4004119 - r4004124;
        double r4004126 = sqrt(r4004125);
        double r4004127 = r4004118 + r4004126;
        double r4004128 = 2.0;
        double r4004129 = r4004128 * r4004121;
        double r4004130 = r4004127 / r4004129;
        return r4004130;
}
↓
double f(double a, double b, double c) {
        double r4004131 = b;
        double r4004132 = -1.7633154797394035e+89;
        bool r4004133 = r4004131 <= r4004132;
        double r4004134 = 2.0;
        double r4004135 = c;
        double r4004136 = r4004135 / r4004131;
        double r4004137 = r4004134 * r4004136;
        double r4004138 = a;
        double r4004139 = r4004131 / r4004138;
        double r4004140 = 2.0;
        double r4004141 = r4004139 * r4004140;
        double r4004142 = r4004137 - r4004141;
        double r4004143 = r4004142 / r4004134;
        double r4004144 = 9.136492990928292e-23;
        bool r4004145 = r4004131 <= r4004144;
        double r4004146 = r4004131 * r4004131;
        double r4004147 = r4004135 * r4004138;
        double r4004148 = 4.0;
        double r4004149 = r4004147 * r4004148;
        double r4004150 = r4004146 - r4004149;
        double r4004151 = sqrt(r4004150);
        double r4004152 = r4004151 - r4004131;
        double r4004153 = r4004152 / r4004138;
        double r4004154 = r4004153 / r4004134;
        double r4004155 = -2.0;
        double r4004156 = r4004155 * r4004136;
        double r4004157 = r4004156 / r4004134;
        double r4004158 = r4004145 ? r4004154 : r4004157;
        double r4004159 = r4004133 ? r4004143 : r4004158;
        return r4004159;
}

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

Original34.4
Target21.3
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7633154797394035e+89

    1. Initial program 45.7

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

    2. Simplified45.7

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

    3. Taylor expanded around -inf 3.9

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    if -1.7633154797394035e+89 < b < 9.136492990928292e-23

    1. Initial program 15.0

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

    2. Simplified15.0

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

    4. Applied div-inv15.1

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

    6. Applied un-div-inv15.0

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

    if 9.136492990928292e-23 < b

    1. Initial program 55.5

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

    2. Simplified55.4

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

    3. Taylor expanded around inf 6.7

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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)))