Average Error: 33.7 → 8.8
Time: 19.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6733539759872003 \cdot 10^{120}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.34481359587178172 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 3.36279139322822572 \cdot 10^{22}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.6733539759872003 \cdot 10^{120}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.34481359587178172 \cdot 10^{-114}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\

\mathbf{elif}\;b \le 3.36279139322822572 \cdot 10^{22}:\\
\;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r49170 = b;
        double r49171 = -r49170;
        double r49172 = r49170 * r49170;
        double r49173 = 4.0;
        double r49174 = a;
        double r49175 = c;
        double r49176 = r49174 * r49175;
        double r49177 = r49173 * r49176;
        double r49178 = r49172 - r49177;
        double r49179 = sqrt(r49178);
        double r49180 = r49171 + r49179;
        double r49181 = 2.0;
        double r49182 = r49181 * r49174;
        double r49183 = r49180 / r49182;
        return r49183;
}
↓
double f(double a, double b, double c) {
        double r49184 = b;
        double r49185 = -1.6733539759872003e+120;
        bool r49186 = r49184 <= r49185;
        double r49187 = 1.0;
        double r49188 = c;
        double r49189 = r49188 / r49184;
        double r49190 = a;
        double r49191 = r49184 / r49190;
        double r49192 = r49189 - r49191;
        double r49193 = r49187 * r49192;
        double r49194 = 2.3448135958717817e-114;
        bool r49195 = r49184 <= r49194;
        double r49196 = 1.0;
        double r49197 = 2.0;
        double r49198 = r49197 * r49190;
        double r49199 = r49184 * r49184;
        double r49200 = 4.0;
        double r49201 = r49190 * r49188;
        double r49202 = r49200 * r49201;
        double r49203 = r49199 - r49202;
        double r49204 = sqrt(r49203);
        double r49205 = r49204 - r49184;
        double r49206 = r49198 / r49205;
        double r49207 = r49196 / r49206;
        double r49208 = 3.3627913932282257e+22;
        bool r49209 = r49184 <= r49208;
        double r49210 = r49200 * r49188;
        double r49211 = r49210 * r49190;
        double r49212 = -r49184;
        double r49213 = sqrt(r49204);
        double r49214 = r49213 * r49213;
        double r49215 = r49212 - r49214;
        double r49216 = r49211 / r49215;
        double r49217 = r49216 / r49198;
        double r49218 = -1.0;
        double r49219 = r49218 * r49189;
        double r49220 = r49209 ? r49217 : r49219;
        double r49221 = r49195 ? r49207 : r49220;
        double r49222 = r49186 ? r49193 : r49221;
        return r49222;
}

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

Original33.7
Target20.8
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6733539759872003e+120

    1. Initial program 52.5

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

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

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

    if -1.6733539759872003e+120 < b < 2.3448135958717817e-114

    1. Initial program 11.0

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

    3. Applied clear-num11.2

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

    4. Simplified11.2

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

    if 2.3448135958717817e-114 < b < 3.3627913932282257e+22

    1. Initial program 36.9

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

    3. Applied flip-+36.9

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

    4. Simplified16.0

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(4 \cdot c\right) \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt16.0

      \[\leadsto \frac{\frac{0 + \left(4 \cdot c\right) \cdot a}{\left(-b\right) - \sqrt{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{2 \cdot a}\]

    7. Applied sqrt-prod16.1

      \[\leadsto \frac{\frac{0 + \left(4 \cdot c\right) \cdot a}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{2 \cdot a}\]

    if 3.3627913932282257e+22 < b

    1. Initial program 56.3

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

    2. Taylor expanded around inf 5.1

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6733539759872003 \cdot 10^{120}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.34481359587178172 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 3.36279139322822572 \cdot 10^{22}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))