Average Error: 34.1 → 10.5
Time: 4.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r68247 = b;
        double r68248 = -r68247;
        double r68249 = r68247 * r68247;
        double r68250 = 4.0;
        double r68251 = a;
        double r68252 = c;
        double r68253 = r68251 * r68252;
        double r68254 = r68250 * r68253;
        double r68255 = r68249 - r68254;
        double r68256 = sqrt(r68255);
        double r68257 = r68248 - r68256;
        double r68258 = 2.0;
        double r68259 = r68258 * r68251;
        double r68260 = r68257 / r68259;
        return r68260;
}
↓
double f(double a, double b, double c) {
        double r68261 = b;
        double r68262 = -5.6439472304372656e-71;
        bool r68263 = r68261 <= r68262;
        double r68264 = -1.0;
        double r68265 = c;
        double r68266 = r68265 / r68261;
        double r68267 = r68264 * r68266;
        double r68268 = 1.4983382189642058e+54;
        bool r68269 = r68261 <= r68268;
        double r68270 = 1.0;
        double r68271 = 2.0;
        double r68272 = r68270 / r68271;
        double r68273 = -r68261;
        double r68274 = r68261 * r68261;
        double r68275 = 4.0;
        double r68276 = a;
        double r68277 = r68276 * r68265;
        double r68278 = r68275 * r68277;
        double r68279 = r68274 - r68278;
        double r68280 = sqrt(r68279);
        double r68281 = r68273 - r68280;
        double r68282 = r68281 / r68276;
        double r68283 = r68272 * r68282;
        double r68284 = 1.0;
        double r68285 = r68261 / r68276;
        double r68286 = r68266 - r68285;
        double r68287 = r68284 * r68286;
        double r68288 = r68269 ? r68283 : r68287;
        double r68289 = r68263 ? r68267 : r68288;
        return r68289;
}

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

Original34.1
Target20.8
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.6439472304372656e-71

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 9.2

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

    if -5.6439472304372656e-71 < b < 1.4983382189642058e+54

    1. Initial program 14.3

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

    3. Applied *-un-lft-identity14.3

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

    4. Applied times-frac14.2

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

    if 1.4983382189642058e+54 < b

    1. Initial program 37.9

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

    2. Taylor expanded around inf 5.0

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

    3. Simplified5.0

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))