Average Error: 34.3 → 10.2
Time: 17.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.202443222624254327680309207854310362882 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 5.202443222624254327680309207854310362882 \cdot 10^{-45}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r85312 = b;
        double r85313 = -r85312;
        double r85314 = r85312 * r85312;
        double r85315 = 4.0;
        double r85316 = a;
        double r85317 = r85315 * r85316;
        double r85318 = c;
        double r85319 = r85317 * r85318;
        double r85320 = r85314 - r85319;
        double r85321 = sqrt(r85320);
        double r85322 = r85313 + r85321;
        double r85323 = 2.0;
        double r85324 = r85323 * r85316;
        double r85325 = r85322 / r85324;
        return r85325;
}
↓
double f(double a, double b, double c) {
        double r85326 = b;
        double r85327 = -1.569310777886352e+111;
        bool r85328 = r85326 <= r85327;
        double r85329 = 1.0;
        double r85330 = c;
        double r85331 = r85330 / r85326;
        double r85332 = a;
        double r85333 = r85326 / r85332;
        double r85334 = r85331 - r85333;
        double r85335 = r85329 * r85334;
        double r85336 = 5.2024432226242543e-45;
        bool r85337 = r85326 <= r85336;
        double r85338 = 1.0;
        double r85339 = 2.0;
        double r85340 = r85339 * r85332;
        double r85341 = r85326 * r85326;
        double r85342 = 4.0;
        double r85343 = r85342 * r85332;
        double r85344 = r85343 * r85330;
        double r85345 = r85341 - r85344;
        double r85346 = sqrt(r85345);
        double r85347 = r85346 - r85326;
        double r85348 = r85340 / r85347;
        double r85349 = r85338 / r85348;
        double r85350 = -1.0;
        double r85351 = r85350 * r85331;
        double r85352 = r85337 ? r85349 : r85351;
        double r85353 = r85328 ? r85335 : r85352;
        return r85353;
}

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

Original34.3
Target21.1
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 < -1.569310777886352e+111

    1. Initial program 50.4

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

    2. Simplified50.4

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

    3. Taylor expanded around -inf 3.9

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

    4. Simplified3.9

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

    if -1.569310777886352e+111 < b < 5.2024432226242543e-45

    1. Initial program 14.0

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

    2. Simplified14.0

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

    4. Applied clear-num14.1

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

    if 5.2024432226242543e-45 < b

    1. Initial program 54.5

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

    2. Simplified54.5

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

    3. Taylor expanded around inf 7.4

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.202443222624254327680309207854310362882 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))