Average Error: 34.2 → 8.7
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.16001008416394735 \cdot 10^{156}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le -5.18636062467436046 \cdot 10^{-242}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.18992965287049363 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(2 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\ \end{array}\]
\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 -5.16001008416394735 \cdot 10^{156}:\\
\;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\

\mathbf{elif}\;b \le -5.18636062467436046 \cdot 10^{-242}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.18992965287049363 \cdot 10^{140}:\\
\;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(2 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\

\end{array}
double f(double a, double b, double c) {
        double r48333 = b;
        double r48334 = -r48333;
        double r48335 = r48333 * r48333;
        double r48336 = 4.0;
        double r48337 = a;
        double r48338 = r48336 * r48337;
        double r48339 = c;
        double r48340 = r48338 * r48339;
        double r48341 = r48335 - r48340;
        double r48342 = sqrt(r48341);
        double r48343 = r48334 + r48342;
        double r48344 = 2.0;
        double r48345 = r48344 * r48337;
        double r48346 = r48343 / r48345;
        return r48346;
}
↓
double f(double a, double b, double c) {
        double r48347 = b;
        double r48348 = -5.160010084163947e+156;
        bool r48349 = r48347 <= r48348;
        double r48350 = -r48347;
        double r48351 = 2.0;
        double r48352 = a;
        double r48353 = c;
        double r48354 = r48352 * r48353;
        double r48355 = r48354 / r48347;
        double r48356 = r48351 * r48355;
        double r48357 = r48356 - r48347;
        double r48358 = r48350 + r48357;
        double r48359 = r48351 * r48352;
        double r48360 = r48358 / r48359;
        double r48361 = -5.1863606246743605e-242;
        bool r48362 = r48347 <= r48361;
        double r48363 = r48347 * r48347;
        double r48364 = 4.0;
        double r48365 = r48364 * r48352;
        double r48366 = r48365 * r48353;
        double r48367 = r48363 - r48366;
        double r48368 = sqrt(r48367);
        double r48369 = r48350 + r48368;
        double r48370 = r48369 / r48359;
        double r48371 = 1.1899296528704936e+140;
        bool r48372 = r48347 <= r48371;
        double r48373 = 0.5;
        double r48374 = r48353 / r48373;
        double r48375 = r48350 - r48368;
        double r48376 = r48374 / r48375;
        double r48377 = 1.0;
        double r48378 = r48373 / r48353;
        double r48379 = 2.0;
        double r48380 = r48379 * r48347;
        double r48381 = r48356 - r48380;
        double r48382 = r48378 * r48381;
        double r48383 = r48377 / r48382;
        double r48384 = r48372 ? r48376 : r48383;
        double r48385 = r48362 ? r48370 : r48384;
        double r48386 = r48349 ? r48360 : r48385;
        return r48386;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -5.160010084163947e+156

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 10.1

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

    if -5.160010084163947e+156 < b < -5.1863606246743605e-242

    1. Initial program 8.6

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

    if -5.1863606246743605e-242 < b < 1.1899296528704936e+140

    1. Initial program 32.3

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

    3. Applied flip-+32.4

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

    4. Simplified15.9

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

    6. Applied clear-num16.1

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

    7. Simplified15.1

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

    8. Taylor expanded around 0 9.7

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
    9. Using strategy rm

    10. Applied associate-/r*9.3

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{0.5}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]

    11. Simplified9.2

      \[\leadsto \frac{\color{blue}{\frac{c}{0.5}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

    if 1.1899296528704936e+140 < b

    1. Initial program 62.5

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

    3. Applied flip-+62.5

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

    4. Simplified35.1

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

    6. Applied clear-num35.1

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

    7. Simplified34.7

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

    8. Taylor expanded around 0 34.4

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

    9. Taylor expanded around inf 7.1

      \[\leadsto \frac{1}{\frac{0.5}{c} \cdot \color{blue}{\left(2 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.16001008416394735 \cdot 10^{156}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le -5.18636062467436046 \cdot 10^{-242}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.18992965287049363 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(2 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))