Average Error: 33.5 → 10.8
Time: 2.6m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.4077460761521334 \cdot 10^{+100}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 6.034138868458272 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{(b \cdot b + \left(\left(c \cdot a\right) \cdot -4\right))_*} - b}{2}}}\\ \mathbf{elif}\;b \le 2.0010050687059634 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 2.5479592607548863 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{(b \cdot b + \left(\left(c \cdot a\right) \cdot -4\right))_*} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\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 -5.4077460761521334 \cdot 10^{+100}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 6.034138868458272 \cdot 10^{-116}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{(b \cdot b + \left(\left(c \cdot a\right) \cdot -4\right))_*} - b}{2}}}\\

\mathbf{elif}\;b \le 2.0010050687059634 \cdot 10^{-66}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le 2.5479592607548863 \cdot 10^{-10}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{(b \cdot b + \left(\left(c \cdot a\right) \cdot -4\right))_*} - b}{2}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r7909255 = b;
        double r7909256 = -r7909255;
        double r7909257 = r7909255 * r7909255;
        double r7909258 = 4.0;
        double r7909259 = a;
        double r7909260 = c;
        double r7909261 = r7909259 * r7909260;
        double r7909262 = r7909258 * r7909261;
        double r7909263 = r7909257 - r7909262;
        double r7909264 = sqrt(r7909263);
        double r7909265 = r7909256 + r7909264;
        double r7909266 = 2.0;
        double r7909267 = r7909266 * r7909259;
        double r7909268 = r7909265 / r7909267;
        return r7909268;
}
↓
double f(double a, double b, double c) {
        double r7909269 = b;
        double r7909270 = -5.4077460761521334e+100;
        bool r7909271 = r7909269 <= r7909270;
        double r7909272 = c;
        double r7909273 = r7909272 / r7909269;
        double r7909274 = a;
        double r7909275 = r7909269 / r7909274;
        double r7909276 = r7909273 - r7909275;
        double r7909277 = 6.034138868458272e-116;
        bool r7909278 = r7909269 <= r7909277;
        double r7909279 = 1.0;
        double r7909280 = r7909272 * r7909274;
        double r7909281 = -4.0;
        double r7909282 = r7909280 * r7909281;
        double r7909283 = fma(r7909269, r7909269, r7909282);
        double r7909284 = sqrt(r7909283);
        double r7909285 = r7909284 - r7909269;
        double r7909286 = 2.0;
        double r7909287 = r7909285 / r7909286;
        double r7909288 = r7909274 / r7909287;
        double r7909289 = r7909279 / r7909288;
        double r7909290 = 2.0010050687059634e-66;
        bool r7909291 = r7909269 <= r7909290;
        double r7909292 = -r7909272;
        double r7909293 = r7909292 / r7909269;
        double r7909294 = 2.5479592607548863e-10;
        bool r7909295 = r7909269 <= r7909294;
        double r7909296 = r7909295 ? r7909289 : r7909293;
        double r7909297 = r7909291 ? r7909293 : r7909296;
        double r7909298 = r7909278 ? r7909289 : r7909297;
        double r7909299 = r7909271 ? r7909276 : r7909298;
        return r7909299;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.4
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.4077460761521334e+100

    1. Initial program 44.3

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

    2. Simplified44.3

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

    3. Taylor expanded around -inf 3.7

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

    if -5.4077460761521334e+100 < b < 6.034138868458272e-116 or 2.0010050687059634e-66 < b < 2.5479592607548863e-10

    1. Initial program 14.6

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

    2. Simplified14.5

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

    4. Applied *-un-lft-identity14.5

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

    5. Applied associate-/l*14.7

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

    if 6.034138868458272e-116 < b < 2.0010050687059634e-66 or 2.5479592607548863e-10 < b

    1. Initial program 51.7

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

    2. Simplified51.7

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

    3. Taylor expanded around inf 9.1

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

    4. Simplified9.1

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.4077460761521334 \cdot 10^{+100}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 6.034138868458272 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{(b \cdot b + \left(\left(c \cdot a\right) \cdot -4\right))_*} - b}{2}}}\\ \mathbf{elif}\;b \le 2.0010050687059634 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 2.5479592607548863 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{(b \cdot b + \left(\left(c \cdot a\right) \cdot -4\right))_*} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))