Average Error: 33.3 → 6.6
Time: 41.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.411206454162785 \cdot 10^{+120}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 8.142093116881289 \cdot 10^{-248}:\\ \;\;\;\;\frac{4}{\sqrt{c \cdot \left(-4 \cdot a\right) + b \cdot b} - b} \cdot \left(\frac{1}{2} \cdot c\right)\\ \mathbf{elif}\;b \le 5.419916601733116 \cdot 10^{+77}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -3.411206454162785 \cdot 10^{+120}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 8.142093116881289 \cdot 10^{-248}:\\
\;\;\;\;\frac{4}{\sqrt{c \cdot \left(-4 \cdot a\right) + b \cdot b} - b} \cdot \left(\frac{1}{2} \cdot c\right)\\

\mathbf{elif}\;b \le 5.419916601733116 \cdot 10^{+77}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4931249 = b;
        double r4931250 = -r4931249;
        double r4931251 = r4931249 * r4931249;
        double r4931252 = 4.0;
        double r4931253 = a;
        double r4931254 = c;
        double r4931255 = r4931253 * r4931254;
        double r4931256 = r4931252 * r4931255;
        double r4931257 = r4931251 - r4931256;
        double r4931258 = sqrt(r4931257);
        double r4931259 = r4931250 - r4931258;
        double r4931260 = 2.0;
        double r4931261 = r4931260 * r4931253;
        double r4931262 = r4931259 / r4931261;
        return r4931262;
}
↓
double f(double a, double b, double c) {
        double r4931263 = b;
        double r4931264 = -3.411206454162785e+120;
        bool r4931265 = r4931263 <= r4931264;
        double r4931266 = c;
        double r4931267 = r4931266 / r4931263;
        double r4931268 = -r4931267;
        double r4931269 = 8.142093116881289e-248;
        bool r4931270 = r4931263 <= r4931269;
        double r4931271 = 4.0;
        double r4931272 = -4.0;
        double r4931273 = a;
        double r4931274 = r4931272 * r4931273;
        double r4931275 = r4931266 * r4931274;
        double r4931276 = r4931263 * r4931263;
        double r4931277 = r4931275 + r4931276;
        double r4931278 = sqrt(r4931277);
        double r4931279 = r4931278 - r4931263;
        double r4931280 = r4931271 / r4931279;
        double r4931281 = 0.5;
        double r4931282 = r4931281 * r4931266;
        double r4931283 = r4931280 * r4931282;
        double r4931284 = 5.419916601733116e+77;
        bool r4931285 = r4931263 <= r4931284;
        double r4931286 = -r4931263;
        double r4931287 = r4931273 * r4931266;
        double r4931288 = r4931287 * r4931271;
        double r4931289 = r4931276 - r4931288;
        double r4931290 = sqrt(r4931289);
        double r4931291 = r4931286 - r4931290;
        double r4931292 = 1.0;
        double r4931293 = 2.0;
        double r4931294 = r4931293 * r4931273;
        double r4931295 = r4931292 / r4931294;
        double r4931296 = r4931291 * r4931295;
        double r4931297 = r4931263 / r4931273;
        double r4931298 = r4931267 - r4931297;
        double r4931299 = r4931285 ? r4931296 : r4931298;
        double r4931300 = r4931270 ? r4931283 : r4931299;
        double r4931301 = r4931265 ? r4931268 : r4931300;
        return r4931301;
}

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

Original33.3
Target20.6
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.411206454162785e+120

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 1.9

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

    3. Simplified1.9

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

    if -3.411206454162785e+120 < b < 8.142093116881289e-248

    1. Initial program 30.9

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

    3. Applied flip--31.1

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

    4. Applied associate-/l/36.0

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

    5. Simplified20.1

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

    7. Applied times-frac14.3

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

    8. Simplified8.9

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

    9. Simplified9.0

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

    if 8.142093116881289e-248 < b < 5.419916601733116e+77

    1. Initial program 8.5

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

    3. Applied div-inv8.7

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

    if 5.419916601733116e+77 < b

    1. Initial program 40.5

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

    3. Applied div-inv40.6

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

    4. Taylor expanded around inf 4.7

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.411206454162785 \cdot 10^{+120}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 8.142093116881289 \cdot 10^{-248}:\\ \;\;\;\;\frac{4}{\sqrt{c \cdot \left(-4 \cdot a\right) + b \cdot b} - b} \cdot \left(\frac{1}{2} \cdot c\right)\\ \mathbf{elif}\;b \le 5.419916601733116 \cdot 10^{+77}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))