Average Error: 34.4 → 10.1
Time: 17.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.635925081143504476780080161813975782827 \cdot 10^{-66}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.206904744652339671334892722279467095293 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -8.635925081143504476780080161813975782827 \cdot 10^{-66}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.206904744652339671334892722279467095293 \cdot 10^{101}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4041260 = b;
        double r4041261 = -r4041260;
        double r4041262 = r4041260 * r4041260;
        double r4041263 = 4.0;
        double r4041264 = a;
        double r4041265 = c;
        double r4041266 = r4041264 * r4041265;
        double r4041267 = r4041263 * r4041266;
        double r4041268 = r4041262 - r4041267;
        double r4041269 = sqrt(r4041268);
        double r4041270 = r4041261 - r4041269;
        double r4041271 = 2.0;
        double r4041272 = r4041271 * r4041264;
        double r4041273 = r4041270 / r4041272;
        return r4041273;
}
↓
double f(double a, double b, double c) {
        double r4041274 = b;
        double r4041275 = -8.635925081143504e-66;
        bool r4041276 = r4041274 <= r4041275;
        double r4041277 = -1.0;
        double r4041278 = c;
        double r4041279 = r4041278 / r4041274;
        double r4041280 = r4041277 * r4041279;
        double r4041281 = 3.2069047446523397e+101;
        bool r4041282 = r4041274 <= r4041281;
        double r4041283 = -r4041274;
        double r4041284 = r4041274 * r4041274;
        double r4041285 = 4.0;
        double r4041286 = a;
        double r4041287 = r4041278 * r4041286;
        double r4041288 = r4041285 * r4041287;
        double r4041289 = r4041284 - r4041288;
        double r4041290 = sqrt(r4041289);
        double r4041291 = r4041283 - r4041290;
        double r4041292 = 2.0;
        double r4041293 = r4041292 * r4041286;
        double r4041294 = r4041291 / r4041293;
        double r4041295 = 1.0;
        double r4041296 = r4041274 / r4041286;
        double r4041297 = r4041279 - r4041296;
        double r4041298 = r4041295 * r4041297;
        double r4041299 = r4041282 ? r4041294 : r4041298;
        double r4041300 = r4041276 ? r4041280 : r4041299;
        return r4041300;
}

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

Original34.4
Target20.9
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.635925081143504e-66

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 8.4

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

    if -8.635925081143504e-66 < b < 3.2069047446523397e+101

    1. Initial program 13.4

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

    3. Applied div-inv13.5

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

    5. Applied un-div-inv13.4

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

    if 3.2069047446523397e+101 < b

    1. Initial program 46.8

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

    2. Taylor expanded around inf 4.4

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

    3. Simplified4.4

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.635925081143504476780080161813975782827 \cdot 10^{-66}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.206904744652339671334892722279467095293 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))