Average Error: 34.2 → 9.5
Time: 19.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{c \cdot 2}{b}\right)}{2}\\ \mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, \frac{c \cdot 2}{b}\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{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3627391 = b;
        double r3627392 = -r3627391;
        double r3627393 = r3627391 * r3627391;
        double r3627394 = 4.0;
        double r3627395 = a;
        double r3627396 = c;
        double r3627397 = r3627395 * r3627396;
        double r3627398 = r3627394 * r3627397;
        double r3627399 = r3627393 - r3627398;
        double r3627400 = sqrt(r3627399);
        double r3627401 = r3627392 + r3627400;
        double r3627402 = 2.0;
        double r3627403 = r3627402 * r3627395;
        double r3627404 = r3627401 / r3627403;
        return r3627404;
}
↓
double f(double a, double b, double c) {
        double r3627405 = b;
        double r3627406 = -3.7108875578650606e+138;
        bool r3627407 = r3627405 <= r3627406;
        double r3627408 = -2.0;
        double r3627409 = a;
        double r3627410 = r3627405 / r3627409;
        double r3627411 = c;
        double r3627412 = 2.0;
        double r3627413 = r3627411 * r3627412;
        double r3627414 = r3627413 / r3627405;
        double r3627415 = fma(r3627408, r3627410, r3627414);
        double r3627416 = r3627415 / r3627412;
        double r3627417 = 4.626043257219638e-62;
        bool r3627418 = r3627405 <= r3627417;
        double r3627419 = r3627405 * r3627405;
        double r3627420 = 4.0;
        double r3627421 = r3627420 * r3627411;
        double r3627422 = r3627421 * r3627409;
        double r3627423 = r3627419 - r3627422;
        double r3627424 = sqrt(r3627423);
        double r3627425 = r3627424 / r3627409;
        double r3627426 = r3627425 - r3627410;
        double r3627427 = r3627426 / r3627412;
        double r3627428 = r3627411 / r3627405;
        double r3627429 = -2.0;
        double r3627430 = r3627428 * r3627429;
        double r3627431 = r3627430 / r3627412;
        double r3627432 = r3627418 ? r3627427 : r3627431;
        double r3627433 = r3627407 ? r3627416 : r3627432;
        return r3627433;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

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

Derivation

  1. Split input into 3 regimes

  2. if b < -3.7108875578650606e+138

    1. Initial program 58.5

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

    2. Simplified58.5

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

    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

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

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

    1. Initial program 12.3

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

    2. Simplified12.3

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

    4. Applied div-sub12.3

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

    if 4.626043257219638e-62 < b

    1. Initial program 53.7

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

    2. Simplified53.7

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

    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

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))