Average Error: 34.4 → 6.8
Time: 5.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.2709120954995131 \cdot 10^{127}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3916403728232559 \cdot 10^{-271}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 9.70708453026941506 \cdot 10^{92}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.2709120954995131 \cdot 10^{127}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.3916403728232559 \cdot 10^{-271}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 9.70708453026941506 \cdot 10^{92}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r105196 = b;
        double r105197 = -r105196;
        double r105198 = r105196 * r105196;
        double r105199 = 4.0;
        double r105200 = a;
        double r105201 = c;
        double r105202 = r105200 * r105201;
        double r105203 = r105199 * r105202;
        double r105204 = r105198 - r105203;
        double r105205 = sqrt(r105204);
        double r105206 = r105197 - r105205;
        double r105207 = 2.0;
        double r105208 = r105207 * r105200;
        double r105209 = r105206 / r105208;
        return r105209;
}
↓
double f(double a, double b, double c) {
        double r105210 = b;
        double r105211 = -3.270912095499513e+127;
        bool r105212 = r105210 <= r105211;
        double r105213 = -1.0;
        double r105214 = c;
        double r105215 = r105214 / r105210;
        double r105216 = r105213 * r105215;
        double r105217 = 1.391640372823256e-271;
        bool r105218 = r105210 <= r105217;
        double r105219 = 2.0;
        double r105220 = r105219 * r105214;
        double r105221 = -r105210;
        double r105222 = r105210 * r105210;
        double r105223 = 4.0;
        double r105224 = a;
        double r105225 = r105224 * r105214;
        double r105226 = r105223 * r105225;
        double r105227 = r105222 - r105226;
        double r105228 = sqrt(r105227);
        double r105229 = r105221 + r105228;
        double r105230 = r105220 / r105229;
        double r105231 = 9.707084530269415e+92;
        bool r105232 = r105210 <= r105231;
        double r105233 = 1.0;
        double r105234 = r105219 * r105224;
        double r105235 = r105221 - r105228;
        double r105236 = r105234 / r105235;
        double r105237 = r105233 / r105236;
        double r105238 = 1.0;
        double r105239 = r105210 / r105224;
        double r105240 = r105215 - r105239;
        double r105241 = r105238 * r105240;
        double r105242 = r105232 ? r105237 : r105241;
        double r105243 = r105218 ? r105230 : r105242;
        double r105244 = r105212 ? r105216 : r105243;
        return r105244;
}

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
Target20.8
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.270912095499513e+127

    1. Initial program 61.2

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

    2. Taylor expanded around -inf 2.1

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

    if -3.270912095499513e+127 < b < 1.391640372823256e-271

    1. Initial program 33.4

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

    3. Applied div-inv33.4

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

    5. Applied flip--33.5

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

    6. Applied associate-*l/33.5

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

    7. Simplified14.9

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

    8. Taylor expanded around 0 8.7

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

    if 1.391640372823256e-271 < b < 9.707084530269415e+92

    1. Initial program 9.1

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

    3. Applied clear-num9.2

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

    if 9.707084530269415e+92 < b

    1. Initial program 45.7

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

    2. Taylor expanded around inf 4.0

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

    3. Simplified4.0

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.2709120954995131 \cdot 10^{127}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3916403728232559 \cdot 10^{-271}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 9.70708453026941506 \cdot 10^{92}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))