Average Error: 34.3 → 7.1
Time: 38.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.05669002671933381232315467688999002364 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.411004807395853104361669776563711353544 \cdot 10^{-303}:\\ \;\;\;\;\frac{\frac{c \cdot 4}{2}}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{\left(-a\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -6.05669002671933381232315467688999002364 \cdot 10^{153}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.411004807395853104361669776563711353544 \cdot 10^{-303}:\\
\;\;\;\;\frac{\frac{c \cdot 4}{2}}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + \left(-b\right)}\\

\mathbf{elif}\;b \le 0.173897874048477174557802982235443778336:\\
\;\;\;\;\frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{\left(-a\right) \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3340233 = b;
        double r3340234 = -r3340233;
        double r3340235 = r3340233 * r3340233;
        double r3340236 = 4.0;
        double r3340237 = a;
        double r3340238 = c;
        double r3340239 = r3340237 * r3340238;
        double r3340240 = r3340236 * r3340239;
        double r3340241 = r3340235 - r3340240;
        double r3340242 = sqrt(r3340241);
        double r3340243 = r3340234 - r3340242;
        double r3340244 = 2.0;
        double r3340245 = r3340244 * r3340237;
        double r3340246 = r3340243 / r3340245;
        return r3340246;
}
↓
double f(double a, double b, double c) {
        double r3340247 = b;
        double r3340248 = -6.056690026719334e+153;
        bool r3340249 = r3340247 <= r3340248;
        double r3340250 = -1.0;
        double r3340251 = c;
        double r3340252 = r3340251 / r3340247;
        double r3340253 = r3340250 * r3340252;
        double r3340254 = 3.411004807395853e-303;
        bool r3340255 = r3340247 <= r3340254;
        double r3340256 = 4.0;
        double r3340257 = r3340251 * r3340256;
        double r3340258 = 2.0;
        double r3340259 = r3340257 / r3340258;
        double r3340260 = r3340247 * r3340247;
        double r3340261 = a;
        double r3340262 = r3340251 * r3340261;
        double r3340263 = r3340256 * r3340262;
        double r3340264 = r3340260 - r3340263;
        double r3340265 = sqrt(r3340264);
        double r3340266 = -r3340247;
        double r3340267 = r3340265 + r3340266;
        double r3340268 = r3340259 / r3340267;
        double r3340269 = 0.17389787404847717;
        bool r3340270 = r3340247 <= r3340269;
        double r3340271 = r3340247 + r3340265;
        double r3340272 = -r3340261;
        double r3340273 = r3340272 * r3340258;
        double r3340274 = r3340271 / r3340273;
        double r3340275 = 1.0;
        double r3340276 = r3340247 / r3340261;
        double r3340277 = r3340252 - r3340276;
        double r3340278 = r3340275 * r3340277;
        double r3340279 = r3340270 ? r3340274 : r3340278;
        double r3340280 = r3340255 ? r3340268 : r3340279;
        double r3340281 = r3340249 ? r3340253 : r3340280;
        return r3340281;
}

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
Target21.3
Herbie7.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6.056690026719334e+153

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 1.1

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

    if -6.056690026719334e+153 < b < 3.411004807395853e-303

    1. Initial program 34.0

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

    3. Applied div-inv34.1

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

    5. Applied flip--34.1

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

    6. Applied associate-*l/34.1

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

    7. Simplified13.2

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

    8. Taylor expanded around 0 7.6

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

    if 3.411004807395853e-303 < b < 0.17389787404847717

    1. Initial program 11.3

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

    3. Applied frac-2neg11.3

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

    4. Simplified11.3

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

    if 0.17389787404847717 < b

    1. Initial program 31.2

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

    2. Taylor expanded around inf 7.3

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

    3. Simplified7.3

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

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.05669002671933381232315467688999002364 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.411004807395853104361669776563711353544 \cdot 10^{-303}:\\ \;\;\;\;\frac{\frac{c \cdot 4}{2}}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{\left(-a\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))