Average Error: 34.5 → 9.9
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.52779163831840318 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.3062534203630095 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.52779163831840318 \cdot 10^{117}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.3062534203630095 \cdot 10^{-45}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r89125 = b;
        double r89126 = -r89125;
        double r89127 = r89125 * r89125;
        double r89128 = 4.0;
        double r89129 = a;
        double r89130 = c;
        double r89131 = r89129 * r89130;
        double r89132 = r89128 * r89131;
        double r89133 = r89127 - r89132;
        double r89134 = sqrt(r89133);
        double r89135 = r89126 + r89134;
        double r89136 = 2.0;
        double r89137 = r89136 * r89129;
        double r89138 = r89135 / r89137;
        return r89138;
}
↓
double f(double a, double b, double c) {
        double r89139 = b;
        double r89140 = -1.5277916383184032e+117;
        bool r89141 = r89139 <= r89140;
        double r89142 = 1.0;
        double r89143 = c;
        double r89144 = r89143 / r89139;
        double r89145 = a;
        double r89146 = r89139 / r89145;
        double r89147 = r89144 - r89146;
        double r89148 = r89142 * r89147;
        double r89149 = 4.3062534203630095e-45;
        bool r89150 = r89139 <= r89149;
        double r89151 = 1.0;
        double r89152 = 2.0;
        double r89153 = r89152 * r89145;
        double r89154 = -r89139;
        double r89155 = r89139 * r89139;
        double r89156 = 4.0;
        double r89157 = r89145 * r89143;
        double r89158 = r89156 * r89157;
        double r89159 = r89155 - r89158;
        double r89160 = sqrt(r89159);
        double r89161 = r89154 + r89160;
        double r89162 = r89153 / r89161;
        double r89163 = r89151 / r89162;
        double r89164 = -1.0;
        double r89165 = r89164 * r89144;
        double r89166 = r89150 ? r89163 : r89165;
        double r89167 = r89141 ? r89148 : r89166;
        return r89167;
}

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

Original34.5
Target21.4
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5277916383184032e+117

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 3.7

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

    3. Simplified3.7

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

    if -1.5277916383184032e+117 < b < 4.3062534203630095e-45

    1. Initial program 13.6

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

    3. Applied clear-num13.7

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

    if 4.3062534203630095e-45 < b

    1. Initial program 54.8

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

    2. Taylor expanded around inf 7.5

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.52779163831840318 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.3062534203630095 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))