Average Error: 33.6 → 10.2
Time: 14.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\
\;\;\;\;\frac{\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 r75327 = b;
        double r75328 = -r75327;
        double r75329 = r75327 * r75327;
        double r75330 = 4.0;
        double r75331 = a;
        double r75332 = r75330 * r75331;
        double r75333 = c;
        double r75334 = r75332 * r75333;
        double r75335 = r75329 - r75334;
        double r75336 = sqrt(r75335);
        double r75337 = r75328 + r75336;
        double r75338 = 2.0;
        double r75339 = r75338 * r75331;
        double r75340 = r75337 / r75339;
        return r75340;
}
↓
double f(double a, double b, double c) {
        double r75341 = b;
        double r75342 = -1.2609617020890706e+118;
        bool r75343 = r75341 <= r75342;
        double r75344 = 1.0;
        double r75345 = c;
        double r75346 = r75345 / r75341;
        double r75347 = a;
        double r75348 = r75341 / r75347;
        double r75349 = r75346 - r75348;
        double r75350 = r75344 * r75349;
        double r75351 = 5.81843322574321e-115;
        bool r75352 = r75341 <= r75351;
        double r75353 = -r75341;
        double r75354 = r75341 * r75341;
        double r75355 = 4.0;
        double r75356 = r75355 * r75347;
        double r75357 = r75356 * r75345;
        double r75358 = r75354 - r75357;
        double r75359 = sqrt(r75358);
        double r75360 = r75353 + r75359;
        double r75361 = 2.0;
        double r75362 = r75361 * r75347;
        double r75363 = r75360 / r75362;
        double r75364 = -1.0;
        double r75365 = r75364 * r75346;
        double r75366 = r75352 ? r75363 : r75365;
        double r75367 = r75343 ? r75350 : r75366;
        return r75367;
}

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
Target20.5
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2609617020890706e+118

    1. Initial program 51.6

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

    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 -1.2609617020890706e+118 < b < 5.81843322574321e-115

    1. Initial program 11.5

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

    if 5.81843322574321e-115 < b

    1. Initial program 51.3

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

    2. Taylor expanded around inf 11.3

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(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)))