Average Error: 33.8 → 10.1
Time: 9.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.12428337420519208 \cdot 10^{57}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\
\;\;\;\;\frac{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r78162 = b;
        double r78163 = -r78162;
        double r78164 = r78162 * r78162;
        double r78165 = 4.0;
        double r78166 = a;
        double r78167 = r78165 * r78166;
        double r78168 = c;
        double r78169 = r78167 * r78168;
        double r78170 = r78164 - r78169;
        double r78171 = sqrt(r78170);
        double r78172 = r78163 + r78171;
        double r78173 = 2.0;
        double r78174 = r78173 * r78166;
        double r78175 = r78172 / r78174;
        return r78175;
}
↓
double f(double a, double b, double c) {
        double r78176 = b;
        double r78177 = -3.124283374205192e+57;
        bool r78178 = r78176 <= r78177;
        double r78179 = 1.0;
        double r78180 = c;
        double r78181 = r78180 / r78176;
        double r78182 = a;
        double r78183 = r78176 / r78182;
        double r78184 = r78181 - r78183;
        double r78185 = r78179 * r78184;
        double r78186 = 3.84613441880261e-81;
        bool r78187 = r78176 <= r78186;
        double r78188 = 2.0;
        double r78189 = pow(r78176, r78188);
        double r78190 = 4.0;
        double r78191 = r78182 * r78180;
        double r78192 = r78190 * r78191;
        double r78193 = r78189 - r78192;
        double r78194 = sqrt(r78193);
        double r78195 = r78194 - r78176;
        double r78196 = 2.0;
        double r78197 = r78195 / r78196;
        double r78198 = r78197 / r78182;
        double r78199 = -1.0;
        double r78200 = r78199 * r78181;
        double r78201 = r78187 ? r78198 : r78200;
        double r78202 = r78178 ? r78185 : r78201;
        return r78202;
}

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
Target20.4
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.124283374205192e+57

    1. Initial program 39.5

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

    2. Simplified39.5

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

    3. Taylor expanded around 0 39.5

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

    4. Taylor expanded around -inf 5.4

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

    5. Simplified5.4

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

    if -3.124283374205192e+57 < b < 3.84613441880261e-81

    1. Initial program 12.7

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

    2. Simplified12.7

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

    3. Taylor expanded around 0 12.7

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

    if 3.84613441880261e-81 < b

    1. Initial program 53.0

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

    2. Simplified53.0

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

    3. Taylor expanded around 0 53.0

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

    4. Taylor expanded around inf 9.5

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))