Average Error: 34.2 → 9.5
Time: 20.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{c}{b} \cdot 2\right)}{2}\\ \mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{c}{b} \cdot 2\right)}{2}\\

\mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}{a} - \frac{b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2706263 = b;
        double r2706264 = -r2706263;
        double r2706265 = r2706263 * r2706263;
        double r2706266 = 4.0;
        double r2706267 = a;
        double r2706268 = r2706266 * r2706267;
        double r2706269 = c;
        double r2706270 = r2706268 * r2706269;
        double r2706271 = r2706265 - r2706270;
        double r2706272 = sqrt(r2706271);
        double r2706273 = r2706264 + r2706272;
        double r2706274 = 2.0;
        double r2706275 = r2706274 * r2706267;
        double r2706276 = r2706273 / r2706275;
        return r2706276;
}
↓
double f(double a, double b, double c) {
        double r2706277 = b;
        double r2706278 = -3.7108875578650606e+138;
        bool r2706279 = r2706277 <= r2706278;
        double r2706280 = -2.0;
        double r2706281 = a;
        double r2706282 = r2706277 / r2706281;
        double r2706283 = c;
        double r2706284 = r2706283 / r2706277;
        double r2706285 = 2.0;
        double r2706286 = r2706284 * r2706285;
        double r2706287 = fma(r2706280, r2706282, r2706286);
        double r2706288 = r2706287 / r2706285;
        double r2706289 = 4.626043257219638e-62;
        bool r2706290 = r2706277 <= r2706289;
        double r2706291 = r2706277 * r2706277;
        double r2706292 = 4.0;
        double r2706293 = r2706292 * r2706283;
        double r2706294 = r2706293 * r2706281;
        double r2706295 = r2706291 - r2706294;
        double r2706296 = sqrt(r2706295);
        double r2706297 = r2706296 / r2706281;
        double r2706298 = r2706297 - r2706282;
        double r2706299 = r2706298 / r2706285;
        double r2706300 = -2.0;
        double r2706301 = r2706300 * r2706284;
        double r2706302 = r2706301 / r2706285;
        double r2706303 = r2706290 ? r2706299 : r2706302;
        double r2706304 = r2706279 ? r2706288 : r2706303;
        return r2706304;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.2
Target21.0
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.7108875578650606e+138

    1. Initial program 58.5

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

    2. Simplified58.5

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

    3. Taylor expanded around -inf 2.0

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

    4. Simplified2.0

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

    if -3.7108875578650606e+138 < b < 4.626043257219638e-62

    1. Initial program 12.3

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

    2. Simplified12.3

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

    4. Applied div-sub12.3

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

    if 4.626043257219638e-62 < b

    1. Initial program 53.7

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

    2. Simplified53.7

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

    3. Taylor expanded around inf 8.5

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{c}{b} \cdot 2\right)}{2}\\ \mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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)))