Average Error: 34.6 → 10.7
Time: 30.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\
\;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -1\\

\end{array}
double f(double a, double b, double c) {
        double r5162199 = b;
        double r5162200 = -r5162199;
        double r5162201 = r5162199 * r5162199;
        double r5162202 = 4.0;
        double r5162203 = a;
        double r5162204 = r5162202 * r5162203;
        double r5162205 = c;
        double r5162206 = r5162204 * r5162205;
        double r5162207 = r5162201 - r5162206;
        double r5162208 = sqrt(r5162207);
        double r5162209 = r5162200 + r5162208;
        double r5162210 = 2.0;
        double r5162211 = r5162210 * r5162203;
        double r5162212 = r5162209 / r5162211;
        return r5162212;
}
↓
double f(double a, double b, double c) {
        double r5162213 = b;
        double r5162214 = -2.7668189408748547e+100;
        bool r5162215 = r5162213 <= r5162214;
        double r5162216 = c;
        double r5162217 = r5162216 / r5162213;
        double r5162218 = a;
        double r5162219 = r5162213 / r5162218;
        double r5162220 = r5162217 - r5162219;
        double r5162221 = 1.0;
        double r5162222 = r5162220 * r5162221;
        double r5162223 = 7.923524897992037e-153;
        bool r5162224 = r5162213 <= r5162223;
        double r5162225 = 1.0;
        double r5162226 = 2.0;
        double r5162227 = r5162218 * r5162226;
        double r5162228 = r5162225 / r5162227;
        double r5162229 = r5162213 * r5162213;
        double r5162230 = r5162218 * r5162216;
        double r5162231 = 4.0;
        double r5162232 = r5162230 * r5162231;
        double r5162233 = r5162229 - r5162232;
        double r5162234 = sqrt(r5162233);
        double r5162235 = r5162234 - r5162213;
        double r5162236 = r5162225 / r5162235;
        double r5162237 = r5162228 / r5162236;
        double r5162238 = -1.0;
        double r5162239 = r5162217 * r5162238;
        double r5162240 = r5162224 ? r5162237 : r5162239;
        double r5162241 = r5162215 ? r5162222 : r5162240;
        return r5162241;
}

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

Original34.6
Target21.1
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.7668189408748547e+100

    1. Initial program 47.2

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

    2. Simplified47.2

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

    3. Taylor expanded around -inf 4.0

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

    4. Simplified4.0

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

    if -2.7668189408748547e+100 < b < 7.923524897992037e-153

    1. Initial program 10.8

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

    2. Simplified10.9

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

    4. Applied clear-num11.0

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

    6. Applied div-inv11.1

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

    7. Applied associate-/r*11.0

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

    if 7.923524897992037e-153 < b

    1. Initial program 50.5

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

    2. Simplified50.5

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

    3. Taylor expanded around inf 12.7

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))