Average Error: 34.5 → 9.9
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.52779163831840318 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.3062534203630095 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 r90151 = b;
        double r90152 = -r90151;
        double r90153 = r90151 * r90151;
        double r90154 = 4.0;
        double r90155 = a;
        double r90156 = c;
        double r90157 = r90155 * r90156;
        double r90158 = r90154 * r90157;
        double r90159 = r90153 - r90158;
        double r90160 = sqrt(r90159);
        double r90161 = r90152 + r90160;
        double r90162 = 2.0;
        double r90163 = r90162 * r90155;
        double r90164 = r90161 / r90163;
        return r90164;
}
↓
double f(double a, double b, double c) {
        double r90165 = b;
        double r90166 = -1.5277916383184032e+117;
        bool r90167 = r90165 <= r90166;
        double r90168 = 1.0;
        double r90169 = c;
        double r90170 = r90169 / r90165;
        double r90171 = a;
        double r90172 = r90165 / r90171;
        double r90173 = r90170 - r90172;
        double r90174 = r90168 * r90173;
        double r90175 = 4.3062534203630095e-45;
        bool r90176 = r90165 <= r90175;
        double r90177 = 1.0;
        double r90178 = 2.0;
        double r90179 = r90178 * r90171;
        double r90180 = -r90165;
        double r90181 = r90165 * r90165;
        double r90182 = 4.0;
        double r90183 = r90171 * r90169;
        double r90184 = r90182 * r90183;
        double r90185 = r90181 - r90184;
        double r90186 = sqrt(r90185);
        double r90187 = r90180 + r90186;
        double r90188 = r90179 / r90187;
        double r90189 = r90177 / r90188;
        double r90190 = -1.0;
        double r90191 = r90190 * r90170;
        double r90192 = r90176 ? r90189 : r90191;
        double r90193 = r90167 ? r90174 : r90192;
        return r90193;
}

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 +o rules:numerics
(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)))