Average Error: 34.4 → 10.6
Time: 6.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.9644058459680186 \cdot 10^{71}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.05029242402897421 \cdot 10^{-108}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.9644058459680186 \cdot 10^{71}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.05029242402897421 \cdot 10^{-108}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r91280 = b;
        double r91281 = -r91280;
        double r91282 = r91280 * r91280;
        double r91283 = 4.0;
        double r91284 = a;
        double r91285 = c;
        double r91286 = r91284 * r91285;
        double r91287 = r91283 * r91286;
        double r91288 = r91282 - r91287;
        double r91289 = sqrt(r91288);
        double r91290 = r91281 + r91289;
        double r91291 = 2.0;
        double r91292 = r91291 * r91284;
        double r91293 = r91290 / r91292;
        return r91293;
}
↓
double f(double a, double b, double c) {
        double r91294 = b;
        double r91295 = -2.9644058459680186e+71;
        bool r91296 = r91294 <= r91295;
        double r91297 = 1.0;
        double r91298 = c;
        double r91299 = r91298 / r91294;
        double r91300 = a;
        double r91301 = r91294 / r91300;
        double r91302 = r91299 - r91301;
        double r91303 = r91297 * r91302;
        double r91304 = 1.0502924240289742e-108;
        bool r91305 = r91294 <= r91304;
        double r91306 = -r91294;
        double r91307 = r91294 * r91294;
        double r91308 = 4.0;
        double r91309 = r91300 * r91298;
        double r91310 = r91308 * r91309;
        double r91311 = r91307 - r91310;
        double r91312 = sqrt(r91311);
        double r91313 = r91306 + r91312;
        double r91314 = 1.0;
        double r91315 = 2.0;
        double r91316 = r91315 * r91300;
        double r91317 = r91314 / r91316;
        double r91318 = r91313 * r91317;
        double r91319 = -1.0;
        double r91320 = r91319 * r91299;
        double r91321 = r91305 ? r91318 : r91320;
        double r91322 = r91296 ? r91303 : r91321;
        return r91322;
}

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

Original34.4
Target21.4
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.9644058459680186e+71

    1. Initial program 42.3

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

    2. Taylor expanded around -inf 4.7

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

    3. Simplified4.7

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

    if -2.9644058459680186e+71 < b < 1.0502924240289742e-108

    1. Initial program 13.1

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

    3. Applied div-inv13.2

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

    if 1.0502924240289742e-108 < b

    1. Initial program 51.6

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

    2. Taylor expanded around inf 10.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.9644058459680186 \cdot 10^{71}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.05029242402897421 \cdot 10^{-108}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))