Average Error: 33.8 → 9.3
Time: 21.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -0.03099989563658142946445117615894560003653:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.501186677105648684932975589367300398392 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + a \cdot \left(c \cdot 4\right)}{a \cdot 2}}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 63580190853209333432320:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -0.03099989563658142946445117615894560003653:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.501186677105648684932975589367300398392 \cdot 10^{-154}:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + a \cdot \left(c \cdot 4\right)}{a \cdot 2}}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} + \left(-b\right)}\\

\mathbf{elif}\;b \le 63580190853209333432320:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{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 r5228213 = b;
        double r5228214 = -r5228213;
        double r5228215 = r5228213 * r5228213;
        double r5228216 = 4.0;
        double r5228217 = a;
        double r5228218 = c;
        double r5228219 = r5228217 * r5228218;
        double r5228220 = r5228216 * r5228219;
        double r5228221 = r5228215 - r5228220;
        double r5228222 = sqrt(r5228221);
        double r5228223 = r5228214 - r5228222;
        double r5228224 = 2.0;
        double r5228225 = r5228224 * r5228217;
        double r5228226 = r5228223 / r5228225;
        return r5228226;
}
↓
double f(double a, double b, double c) {
        double r5228227 = b;
        double r5228228 = -0.03099989563658143;
        bool r5228229 = r5228227 <= r5228228;
        double r5228230 = -1.0;
        double r5228231 = c;
        double r5228232 = r5228231 / r5228227;
        double r5228233 = r5228230 * r5228232;
        double r5228234 = -2.5011866771056487e-154;
        bool r5228235 = r5228227 <= r5228234;
        double r5228236 = r5228227 * r5228227;
        double r5228237 = r5228236 - r5228236;
        double r5228238 = a;
        double r5228239 = 4.0;
        double r5228240 = r5228231 * r5228239;
        double r5228241 = r5228238 * r5228240;
        double r5228242 = r5228237 + r5228241;
        double r5228243 = 2.0;
        double r5228244 = r5228238 * r5228243;
        double r5228245 = r5228242 / r5228244;
        double r5228246 = r5228236 - r5228241;
        double r5228247 = sqrt(r5228246);
        double r5228248 = -r5228227;
        double r5228249 = r5228247 + r5228248;
        double r5228250 = r5228245 / r5228249;
        double r5228251 = 6.358019085320933e+22;
        bool r5228252 = r5228227 <= r5228251;
        double r5228253 = r5228248 - r5228247;
        double r5228254 = r5228253 / r5228244;
        double r5228255 = r5228227 / r5228238;
        double r5228256 = r5228232 - r5228255;
        double r5228257 = 1.0;
        double r5228258 = r5228256 * r5228257;
        double r5228259 = r5228252 ? r5228254 : r5228258;
        double r5228260 = r5228235 ? r5228250 : r5228259;
        double r5228261 = r5228229 ? r5228233 : r5228260;
        return r5228261;
}

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

Original33.8
Target20.8
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -0.03099989563658143

    1. Initial program 55.6

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

    2. Taylor expanded around -inf 6.4

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

    if -0.03099989563658143 < b < -2.5011866771056487e-154

    1. Initial program 31.3

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

    2. Taylor expanded around 0 31.3

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

    3. Simplified31.4

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

    5. Applied div-inv31.4

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

    7. Applied flip--31.5

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

    8. Applied associate-*l/31.5

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

    9. Simplified16.0

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

    if -2.5011866771056487e-154 < b < 6.358019085320933e+22

    1. Initial program 12.3

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

    2. Taylor expanded around 0 12.3

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

    3. Simplified12.4

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

    5. Applied div-inv12.5

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

    7. Applied *-un-lft-identity12.5

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

    8. Applied associate-*l*12.5

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

    9. Simplified12.4

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

    if 6.358019085320933e+22 < b

    1. Initial program 33.1

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

    2. Taylor expanded around inf 6.1

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

    3. Simplified6.1

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -0.03099989563658142946445117615894560003653:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.501186677105648684932975589367300398392 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + a \cdot \left(c \cdot 4\right)}{a \cdot 2}}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 63580190853209333432320:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +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)))