Average Error: 33.3 → 10.4
Time: 16.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.961198324014865 \cdot 10^{-88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 6.384705165981893 \cdot 10^{+101}:\\ \;\;\;\;\frac{-1}{\frac{a \cdot -2}{\left(-b\right) - \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -5.961198324014865 \cdot 10^{-88}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le 6.384705165981893 \cdot 10^{+101}:\\
\;\;\;\;\frac{-1}{\frac{a \cdot -2}{\left(-b\right) - \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r1626138 = b;
        double r1626139 = -r1626138;
        double r1626140 = r1626138 * r1626138;
        double r1626141 = 4.0;
        double r1626142 = a;
        double r1626143 = c;
        double r1626144 = r1626142 * r1626143;
        double r1626145 = r1626141 * r1626144;
        double r1626146 = r1626140 - r1626145;
        double r1626147 = sqrt(r1626146);
        double r1626148 = r1626139 - r1626147;
        double r1626149 = 2.0;
        double r1626150 = r1626149 * r1626142;
        double r1626151 = r1626148 / r1626150;
        return r1626151;
}
↓
double f(double a, double b, double c) {
        double r1626152 = b;
        double r1626153 = -5.961198324014865e-88;
        bool r1626154 = r1626152 <= r1626153;
        double r1626155 = c;
        double r1626156 = -r1626155;
        double r1626157 = r1626156 / r1626152;
        double r1626158 = 6.384705165981893e+101;
        bool r1626159 = r1626152 <= r1626158;
        double r1626160 = -1.0;
        double r1626161 = a;
        double r1626162 = -2.0;
        double r1626163 = r1626161 * r1626162;
        double r1626164 = -r1626152;
        double r1626165 = r1626152 * r1626152;
        double r1626166 = -4.0;
        double r1626167 = r1626166 * r1626161;
        double r1626168 = r1626167 * r1626155;
        double r1626169 = r1626165 + r1626168;
        double r1626170 = sqrt(r1626169);
        double r1626171 = r1626164 - r1626170;
        double r1626172 = r1626163 / r1626171;
        double r1626173 = r1626160 / r1626172;
        double r1626174 = r1626155 / r1626152;
        double r1626175 = r1626152 / r1626161;
        double r1626176 = r1626174 - r1626175;
        double r1626177 = r1626159 ? r1626173 : r1626176;
        double r1626178 = r1626154 ? r1626157 : r1626177;
        return r1626178;
}

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
Target20.2
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 < -5.961198324014865e-88

    1. Initial program 51.4

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

    2. Taylor expanded around -inf 9.8

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

    3. Simplified9.8

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

    if -5.961198324014865e-88 < b < 6.384705165981893e+101

    1. Initial program 13.1

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

    3. Applied clear-num13.3

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

    5. Applied frac-2neg13.3

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

    6. Simplified13.3

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

    7. Simplified13.3

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

    if 6.384705165981893e+101 < b

    1. Initial program 43.9

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.961198324014865 \cdot 10^{-88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 6.384705165981893 \cdot 10^{+101}:\\ \;\;\;\;\frac{-1}{\frac{a \cdot -2}{\left(-b\right) - \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))