Average Error: 33.7 → 8.3
Time: 18.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.3209183644448 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\ \mathbf{elif}\;b \le 2.026128983134594 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -3.234164035284793 \cdot 10^{+22}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -6.3209183644448 \cdot 10^{-115}:\\
\;\;\;\;\frac{\frac{1}{2}}{a} \cdot \frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\

\mathbf{elif}\;b \le 2.026128983134594 \cdot 10^{+103}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3819232 = b;
        double r3819233 = -r3819232;
        double r3819234 = r3819232 * r3819232;
        double r3819235 = 4.0;
        double r3819236 = a;
        double r3819237 = c;
        double r3819238 = r3819236 * r3819237;
        double r3819239 = r3819235 * r3819238;
        double r3819240 = r3819234 - r3819239;
        double r3819241 = sqrt(r3819240);
        double r3819242 = r3819233 - r3819241;
        double r3819243 = 2.0;
        double r3819244 = r3819243 * r3819236;
        double r3819245 = r3819242 / r3819244;
        return r3819245;
}
↓
double f(double a, double b, double c) {
        double r3819246 = b;
        double r3819247 = -3.234164035284793e+22;
        bool r3819248 = r3819246 <= r3819247;
        double r3819249 = c;
        double r3819250 = r3819249 / r3819246;
        double r3819251 = -r3819250;
        double r3819252 = -6.3209183644448e-115;
        bool r3819253 = r3819246 <= r3819252;
        double r3819254 = 0.5;
        double r3819255 = a;
        double r3819256 = r3819254 / r3819255;
        double r3819257 = r3819255 * r3819249;
        double r3819258 = 4.0;
        double r3819259 = r3819257 * r3819258;
        double r3819260 = r3819246 * r3819246;
        double r3819261 = r3819260 - r3819259;
        double r3819262 = sqrt(r3819261);
        double r3819263 = r3819262 - r3819246;
        double r3819264 = r3819259 / r3819263;
        double r3819265 = r3819256 * r3819264;
        double r3819266 = 2.026128983134594e+103;
        bool r3819267 = r3819246 <= r3819266;
        double r3819268 = -r3819246;
        double r3819269 = r3819268 - r3819262;
        double r3819270 = r3819254 * r3819269;
        double r3819271 = r3819270 / r3819255;
        double r3819272 = r3819246 / r3819255;
        double r3819273 = r3819250 - r3819272;
        double r3819274 = r3819267 ? r3819271 : r3819273;
        double r3819275 = r3819253 ? r3819265 : r3819274;
        double r3819276 = r3819248 ? r3819251 : r3819275;
        return r3819276;
}

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

Original33.7
Target20.9
Herbie8.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.234164035284793e+22

    1. Initial program 55.9

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

    2. Taylor expanded around -inf 4.6

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

    3. Simplified4.6

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

    if -3.234164035284793e+22 < b < -6.3209183644448e-115

    1. Initial program 38.4

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

    3. Applied div-inv38.4

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

    4. Simplified38.4

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

    6. Applied flip--38.5

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

    7. Simplified15.7

      \[\leadsto \frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \frac{\frac{1}{2}}{a}\]

    8. Simplified15.7

      \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{\frac{1}{2}}{a}\]

    if -6.3209183644448e-115 < b < 2.026128983134594e+103

    1. Initial program 11.3

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

    3. Applied div-inv11.4

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

    4. Simplified11.4

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

    6. Applied associate-*r/11.3

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

    if 2.026128983134594e+103 < b

    1. Initial program 45.2

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

    2. Taylor expanded around inf 3.2

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.3209183644448 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\ \mathbf{elif}\;b \le 2.026128983134594 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))