Average Error: 33.6 → 10.3
Time: 9.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.16908657181932359 \cdot 10^{-104}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\ \end{array}\]
\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 -4.16908657181932359 \cdot 10^{-104}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r35270 = b;
        double r35271 = -r35270;
        double r35272 = r35270 * r35270;
        double r35273 = 4.0;
        double r35274 = a;
        double r35275 = c;
        double r35276 = r35274 * r35275;
        double r35277 = r35273 * r35276;
        double r35278 = r35272 - r35277;
        double r35279 = sqrt(r35278);
        double r35280 = r35271 - r35279;
        double r35281 = 2.0;
        double r35282 = r35281 * r35274;
        double r35283 = r35280 / r35282;
        return r35283;
}
↓
double f(double a, double b, double c) {
        double r35284 = b;
        double r35285 = -4.1690865718193236e-104;
        bool r35286 = r35284 <= r35285;
        double r35287 = -1.0;
        double r35288 = c;
        double r35289 = r35288 / r35284;
        double r35290 = r35287 * r35289;
        double r35291 = 1.3316184968738608e+61;
        bool r35292 = r35284 <= r35291;
        double r35293 = 1.0;
        double r35294 = 2.0;
        double r35295 = r35293 / r35294;
        double r35296 = -r35284;
        double r35297 = 2.0;
        double r35298 = pow(r35284, r35297);
        double r35299 = 4.0;
        double r35300 = a;
        double r35301 = r35300 * r35288;
        double r35302 = r35299 * r35301;
        double r35303 = r35298 - r35302;
        double r35304 = sqrt(r35303);
        double r35305 = r35296 - r35304;
        double r35306 = r35305 / r35300;
        double r35307 = r35295 * r35306;
        double r35308 = -2.0;
        double r35309 = r35284 / r35300;
        double r35310 = r35308 * r35309;
        double r35311 = r35295 * r35310;
        double r35312 = r35292 ? r35307 : r35311;
        double r35313 = r35286 ? r35290 : r35312;
        return r35313;
}

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

Original33.6
Target20.6
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.1690865718193236e-104

    1. Initial program 51.5

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

    2. Taylor expanded around -inf 11.0

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

    if -4.1690865718193236e-104 < b < 1.3316184968738608e+61

    1. Initial program 12.3

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

    3. Applied *-un-lft-identity12.3

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

    4. Applied times-frac12.3

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

    6. Applied clear-num12.4

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

    8. Applied div-inv12.4

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

    9. Applied add-cube-cbrt12.4

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{a \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    10. Applied times-frac12.4

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{a} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\right)}\]

    11. Simplified12.4

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{1}{a}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\right)\]

    12. Simplified12.4

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

    14. Applied associate-*l/12.3

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

    15. Simplified12.3

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

    if 1.3316184968738608e+61 < b

    1. Initial program 39.6

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

    3. Applied *-un-lft-identity39.6

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

    4. Applied times-frac39.5

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

    6. Applied clear-num39.6

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

    7. Taylor expanded around 0 4.6

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(-2 \cdot \frac{b}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.16908657181932359 \cdot 10^{-104}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

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