Average Error: 33.3 → 10.4
Time: 25.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.179137486378021 \cdot 10^{-24}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -4.179137486378021 \cdot 10^{-24}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3005292 = b;
        double r3005293 = -r3005292;
        double r3005294 = r3005292 * r3005292;
        double r3005295 = 4.0;
        double r3005296 = a;
        double r3005297 = c;
        double r3005298 = r3005296 * r3005297;
        double r3005299 = r3005295 * r3005298;
        double r3005300 = r3005294 - r3005299;
        double r3005301 = sqrt(r3005300);
        double r3005302 = r3005293 - r3005301;
        double r3005303 = 2.0;
        double r3005304 = r3005303 * r3005296;
        double r3005305 = r3005302 / r3005304;
        return r3005305;
}
↓
double f(double a, double b, double c) {
        double r3005306 = b;
        double r3005307 = -4.179137486378021e-24;
        bool r3005308 = r3005306 <= r3005307;
        double r3005309 = c;
        double r3005310 = r3005309 / r3005306;
        double r3005311 = -r3005310;
        double r3005312 = 2.3648644896474148e+52;
        bool r3005313 = r3005306 <= r3005312;
        double r3005314 = -r3005306;
        double r3005315 = r3005306 * r3005306;
        double r3005316 = a;
        double r3005317 = r3005309 * r3005316;
        double r3005318 = 4.0;
        double r3005319 = r3005317 * r3005318;
        double r3005320 = r3005315 - r3005319;
        double r3005321 = sqrt(r3005320);
        double r3005322 = r3005314 - r3005321;
        double r3005323 = 2.0;
        double r3005324 = r3005316 * r3005323;
        double r3005325 = r3005322 / r3005324;
        double r3005326 = r3005314 / r3005316;
        double r3005327 = r3005313 ? r3005325 : r3005326;
        double r3005328 = r3005308 ? r3005311 : r3005327;
        return r3005328;
}

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

Original33.3
Target20.7
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.179137486378021e-24

    1. Initial program 54.6

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

    3. Applied div-inv54.6

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

    4. Simplified54.6

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

    5. Taylor expanded around -inf 7.1

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

    6. Simplified7.1

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

    if -4.179137486378021e-24 < b < 2.3648644896474148e+52

    1. Initial program 15.1

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

    if 2.3648644896474148e+52 < b

    1. Initial program 36.7

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

    3. Applied *-un-lft-identity36.7

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

    4. Applied associate-/l*36.8

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

    5. Taylor expanded around 0 5.5

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

    6. Simplified5.5

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.179137486378021 \cdot 10^{-24}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))