Average Error: 34.2 → 9.1
Time: 18.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -763129212434271441067123993682640896:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -763129212434271441067123993682640896:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\
\;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\
\;\;\;\;\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 \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r72345 = b;
        double r72346 = -r72345;
        double r72347 = r72345 * r72345;
        double r72348 = 4.0;
        double r72349 = a;
        double r72350 = c;
        double r72351 = r72349 * r72350;
        double r72352 = r72348 * r72351;
        double r72353 = r72347 - r72352;
        double r72354 = sqrt(r72353);
        double r72355 = r72346 - r72354;
        double r72356 = 2.0;
        double r72357 = r72356 * r72349;
        double r72358 = r72355 / r72357;
        return r72358;
}
↓
double f(double a, double b, double c) {
        double r72359 = b;
        double r72360 = -7.631292124342714e+35;
        bool r72361 = r72359 <= r72360;
        double r72362 = -1.0;
        double r72363 = c;
        double r72364 = r72363 / r72359;
        double r72365 = r72362 * r72364;
        double r72366 = 9.580019013081131e-278;
        bool r72367 = r72359 <= r72366;
        double r72368 = 4.0;
        double r72369 = a;
        double r72370 = r72368 * r72369;
        double r72371 = r72363 * r72370;
        double r72372 = 2.0;
        double r72373 = pow(r72359, r72372);
        double r72374 = r72369 * r72363;
        double r72375 = r72368 * r72374;
        double r72376 = r72373 - r72375;
        double r72377 = sqrt(r72376);
        double r72378 = r72377 - r72359;
        double r72379 = r72371 / r72378;
        double r72380 = 2.0;
        double r72381 = r72380 * r72369;
        double r72382 = r72379 / r72381;
        double r72383 = 5.031608061939103e+53;
        bool r72384 = r72359 <= r72383;
        double r72385 = -r72359;
        double r72386 = r72359 * r72359;
        double r72387 = r72386 - r72375;
        double r72388 = sqrt(r72387);
        double r72389 = r72385 - r72388;
        double r72390 = 1.0;
        double r72391 = r72390 / r72381;
        double r72392 = r72389 * r72391;
        double r72393 = 1.0;
        double r72394 = r72359 / r72369;
        double r72395 = r72364 - r72394;
        double r72396 = r72393 * r72395;
        double r72397 = r72384 ? r72392 : r72396;
        double r72398 = r72367 ? r72382 : r72397;
        double r72399 = r72361 ? r72365 : r72398;
        return r72399;
}

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

Original34.2
Target21.3
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.631292124342714e+35

    1. Initial program 56.2

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

    2. Taylor expanded around -inf 4.5

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

    if -7.631292124342714e+35 < b < 9.580019013081131e-278

    1. Initial program 27.7

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

    3. Applied flip--27.7

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

    4. Simplified16.7

      \[\leadsto \frac{\frac{\color{blue}{0 + c \cdot \left(4 \cdot a\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified16.7

      \[\leadsto \frac{\frac{0 + c \cdot \left(4 \cdot a\right)}{\color{blue}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}}{2 \cdot a}\]

    if 9.580019013081131e-278 < b < 5.031608061939103e+53

    1. Initial program 9.4

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

    3. Applied div-inv9.6

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

    if 5.031608061939103e+53 < b

    1. Initial program 39.6

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

    2. Taylor expanded around inf 5.7

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

    3. Simplified5.7

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -763129212434271441067123993682640896:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(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)))