Average Error: 33.8 → 10.4
Time: 20.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.213216247196925388401125773743990732555 \cdot 10^{129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\ \mathbf{elif}\;b \le 6.088267304256603437292930310963869002155 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -8.213216247196925388401125773743990732555 \cdot 10^{129}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\

\mathbf{elif}\;b \le 6.088267304256603437292930310963869002155 \cdot 10^{-81}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r5256234 = b;
        double r5256235 = -r5256234;
        double r5256236 = r5256234 * r5256234;
        double r5256237 = 4.0;
        double r5256238 = a;
        double r5256239 = r5256237 * r5256238;
        double r5256240 = c;
        double r5256241 = r5256239 * r5256240;
        double r5256242 = r5256236 - r5256241;
        double r5256243 = sqrt(r5256242);
        double r5256244 = r5256235 + r5256243;
        double r5256245 = 2.0;
        double r5256246 = r5256245 * r5256238;
        double r5256247 = r5256244 / r5256246;
        return r5256247;
}
↓
double f(double a, double b, double c) {
        double r5256248 = b;
        double r5256249 = -8.213216247196925e+129;
        bool r5256250 = r5256248 <= r5256249;
        double r5256251 = a;
        double r5256252 = r5256248 / r5256251;
        double r5256253 = -2.0;
        double r5256254 = 2.0;
        double r5256255 = c;
        double r5256256 = r5256255 / r5256248;
        double r5256257 = r5256254 * r5256256;
        double r5256258 = fma(r5256252, r5256253, r5256257);
        double r5256259 = r5256258 / r5256254;
        double r5256260 = 6.088267304256603e-81;
        bool r5256261 = r5256248 <= r5256260;
        double r5256262 = 1.0;
        double r5256263 = r5256248 * r5256248;
        double r5256264 = 4.0;
        double r5256265 = r5256255 * r5256264;
        double r5256266 = r5256251 * r5256265;
        double r5256267 = r5256263 - r5256266;
        double r5256268 = sqrt(r5256267);
        double r5256269 = r5256268 - r5256248;
        double r5256270 = r5256251 / r5256269;
        double r5256271 = r5256262 / r5256270;
        double r5256272 = r5256271 / r5256254;
        double r5256273 = -2.0;
        double r5256274 = r5256256 * r5256273;
        double r5256275 = r5256274 / r5256254;
        double r5256276 = r5256261 ? r5256272 : r5256275;
        double r5256277 = r5256250 ? r5256259 : r5256276;
        return r5256277;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target20.6
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.213216247196925e+129

    1. Initial program 53.9

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

    2. Simplified53.9

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

    3. Taylor expanded around -inf 2.5

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified2.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}}{2}\]

    if -8.213216247196925e+129 < b < 6.088267304256603e-81

    1. Initial program 12.3

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

    2. Simplified12.4

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

    4. Applied clear-num12.5

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

    if 6.088267304256603e-81 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 10.5

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.213216247196925388401125773743990732555 \cdot 10^{129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\ \mathbf{elif}\;b \le 6.088267304256603437292930310963869002155 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

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

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