Average Error: 34.4 → 9.0
Time: 15.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.890025456402396757167722705339283465851 \cdot 10^{59}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -8.752414306529149001923320350234914308904 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + 4 \cdot \left(c \cdot a\right)}{a \cdot 2}}{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} + \left(-b\right)}\\ \mathbf{elif}\;b \le 3.424685282990076228564514143307324629132 \cdot 10^{98}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -7.890025456402396757167722705339283465851 \cdot 10^{59}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -8.752414306529149001923320350234914308904 \cdot 10^{-148}:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + 4 \cdot \left(c \cdot a\right)}{a \cdot 2}}{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} + \left(-b\right)}\\

\mathbf{elif}\;b \le 3.424685282990076228564514143307324629132 \cdot 10^{98}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}\right) \cdot \frac{1}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r64310 = b;
        double r64311 = -r64310;
        double r64312 = r64310 * r64310;
        double r64313 = 4.0;
        double r64314 = a;
        double r64315 = c;
        double r64316 = r64314 * r64315;
        double r64317 = r64313 * r64316;
        double r64318 = r64312 - r64317;
        double r64319 = sqrt(r64318);
        double r64320 = r64311 - r64319;
        double r64321 = 2.0;
        double r64322 = r64321 * r64314;
        double r64323 = r64320 / r64322;
        return r64323;
}
↓
double f(double a, double b, double c) {
        double r64324 = b;
        double r64325 = -7.890025456402397e+59;
        bool r64326 = r64324 <= r64325;
        double r64327 = -1.0;
        double r64328 = c;
        double r64329 = r64328 / r64324;
        double r64330 = r64327 * r64329;
        double r64331 = -8.752414306529149e-148;
        bool r64332 = r64324 <= r64331;
        double r64333 = r64324 * r64324;
        double r64334 = r64333 - r64333;
        double r64335 = 4.0;
        double r64336 = a;
        double r64337 = r64328 * r64336;
        double r64338 = r64335 * r64337;
        double r64339 = r64334 + r64338;
        double r64340 = 2.0;
        double r64341 = r64336 * r64340;
        double r64342 = r64339 / r64341;
        double r64343 = r64335 * r64328;
        double r64344 = r64343 * r64336;
        double r64345 = r64333 - r64344;
        double r64346 = sqrt(r64345);
        double r64347 = -r64324;
        double r64348 = r64346 + r64347;
        double r64349 = r64342 / r64348;
        double r64350 = 3.424685282990076e+98;
        bool r64351 = r64324 <= r64350;
        double r64352 = r64347 - r64346;
        double r64353 = 1.0;
        double r64354 = r64353 / r64341;
        double r64355 = r64352 * r64354;
        double r64356 = r64324 / r64336;
        double r64357 = r64329 - r64356;
        double r64358 = 1.0;
        double r64359 = r64357 * r64358;
        double r64360 = r64351 ? r64355 : r64359;
        double r64361 = r64332 ? r64349 : r64360;
        double r64362 = r64326 ? r64330 : r64361;
        return r64362;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -7.890025456402397e+59

    1. Initial program 57.6

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

    2. Simplified57.6

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

    3. Taylor expanded around -inf 3.3

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

    if -7.890025456402397e+59 < b < -8.752414306529149e-148

    1. Initial program 37.6

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

    2. Simplified37.6

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

    4. Applied div-inv37.7

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

    6. Applied flip--37.7

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

    7. Applied associate-*l/37.7

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

    8. Simplified16.4

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

    if -8.752414306529149e-148 < b < 3.424685282990076e+98

    1. Initial program 11.7

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

    2. Simplified11.8

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

    4. Applied div-inv11.9

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

    if 3.424685282990076e+98 < b

    1. Initial program 47.8

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

    2. Simplified47.8

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

    3. Taylor expanded around inf 3.7

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

    4. Simplified3.7

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.890025456402396757167722705339283465851 \cdot 10^{59}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -8.752414306529149001923320350234914308904 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + 4 \cdot \left(c \cdot a\right)}{a \cdot 2}}{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} + \left(-b\right)}\\ \mathbf{elif}\;b \le 3.424685282990076228564514143307324629132 \cdot 10^{98}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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)))