Average Error: 34.0 → 9.3
Time: 7.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.5069461462218695 \cdot 10^{125}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.2742398392973687 \cdot 10^{-86}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 9.16799708835065593 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.5069461462218695 \cdot 10^{125}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.2742398392973687 \cdot 10^{-86}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 9.16799708835065593 \cdot 10^{-7}:\\
\;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r146216 = b;
        double r146217 = -r146216;
        double r146218 = r146216 * r146216;
        double r146219 = 4.0;
        double r146220 = a;
        double r146221 = r146219 * r146220;
        double r146222 = c;
        double r146223 = r146221 * r146222;
        double r146224 = r146218 - r146223;
        double r146225 = sqrt(r146224);
        double r146226 = r146217 + r146225;
        double r146227 = 2.0;
        double r146228 = r146227 * r146220;
        double r146229 = r146226 / r146228;
        return r146229;
}
↓
double f(double a, double b, double c) {
        double r146230 = b;
        double r146231 = -8.50694614622187e+125;
        bool r146232 = r146230 <= r146231;
        double r146233 = 1.0;
        double r146234 = c;
        double r146235 = r146234 / r146230;
        double r146236 = a;
        double r146237 = r146230 / r146236;
        double r146238 = r146235 - r146237;
        double r146239 = r146233 * r146238;
        double r146240 = 2.2742398392973687e-86;
        bool r146241 = r146230 <= r146240;
        double r146242 = -r146230;
        double r146243 = r146230 * r146230;
        double r146244 = 4.0;
        double r146245 = r146244 * r146236;
        double r146246 = r146245 * r146234;
        double r146247 = r146243 - r146246;
        double r146248 = sqrt(r146247);
        double r146249 = r146242 + r146248;
        double r146250 = 1.0;
        double r146251 = 2.0;
        double r146252 = r146251 * r146236;
        double r146253 = r146250 / r146252;
        double r146254 = r146249 * r146253;
        double r146255 = 9.167997088350656e-07;
        bool r146256 = r146230 <= r146255;
        double r146257 = 0.0;
        double r146258 = r146236 * r146234;
        double r146259 = r146244 * r146258;
        double r146260 = r146257 + r146259;
        double r146261 = r146242 - r146248;
        double r146262 = r146260 / r146261;
        double r146263 = r146262 / r146252;
        double r146264 = -1.0;
        double r146265 = r146264 * r146235;
        double r146266 = r146256 ? r146263 : r146265;
        double r146267 = r146241 ? r146254 : r146266;
        double r146268 = r146232 ? r146239 : r146267;
        return r146268;
}

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

Original34.0
Target20.6
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -8.50694614622187e+125

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 2.7

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

    3. Simplified2.7

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

    if -8.50694614622187e+125 < b < 2.2742398392973687e-86

    1. Initial program 12.3

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

    3. Applied div-inv12.5

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

    if 2.2742398392973687e-86 < b < 9.167997088350656e-07

    1. Initial program 38.3

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

    3. Applied flip-+38.3

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

    4. Simplified18.8

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

    if 9.167997088350656e-07 < b

    1. Initial program 55.8

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

    2. Taylor expanded around inf 5.7

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.5069461462218695 \cdot 10^{125}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.2742398392973687 \cdot 10^{-86}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 9.16799708835065593 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))