Average Error: 33.6 → 10.4
Time: 15.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.1144981103869975 \cdot 10^{+131}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r7455228 = b;
        double r7455229 = -r7455228;
        double r7455230 = r7455228 * r7455228;
        double r7455231 = 4.0;
        double r7455232 = a;
        double r7455233 = r7455231 * r7455232;
        double r7455234 = c;
        double r7455235 = r7455233 * r7455234;
        double r7455236 = r7455230 - r7455235;
        double r7455237 = sqrt(r7455236);
        double r7455238 = r7455229 + r7455237;
        double r7455239 = 2.0;
        double r7455240 = r7455239 * r7455232;
        double r7455241 = r7455238 / r7455240;
        return r7455241;
}
↓
double f(double a, double b, double c) {
        double r7455242 = b;
        double r7455243 = -2.1144981103869975e+131;
        bool r7455244 = r7455242 <= r7455243;
        double r7455245 = c;
        double r7455246 = r7455245 / r7455242;
        double r7455247 = a;
        double r7455248 = r7455242 / r7455247;
        double r7455249 = r7455246 - r7455248;
        double r7455250 = 4.5810084990875205e-68;
        bool r7455251 = r7455242 <= r7455250;
        double r7455252 = 1.0;
        double r7455253 = 2.0;
        double r7455254 = r7455247 * r7455253;
        double r7455255 = -4.0;
        double r7455256 = r7455247 * r7455255;
        double r7455257 = r7455256 * r7455245;
        double r7455258 = r7455242 * r7455242;
        double r7455259 = r7455257 + r7455258;
        double r7455260 = sqrt(r7455259);
        double r7455261 = r7455260 - r7455242;
        double r7455262 = r7455254 / r7455261;
        double r7455263 = r7455252 / r7455262;
        double r7455264 = -r7455246;
        double r7455265 = r7455251 ? r7455263 : r7455264;
        double r7455266 = r7455244 ? r7455249 : r7455265;
        return r7455266;
}

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

Original33.6
Target21.0
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.1144981103869975e+131

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 2.6

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

    if -2.1144981103869975e+131 < b < 4.5810084990875205e-68

    1. Initial program 13.3

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

    3. Applied div-inv13.5

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

    4. Simplified13.5

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

    6. Applied associate-*r/13.3

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

    7. Simplified13.3

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

    9. Applied associate-/l/13.3

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

    11. Applied clear-num13.4

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

    if 4.5810084990875205e-68 < b

    1. Initial program 52.0

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified9.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))