Average Error: 33.4 → 6.6
Time: 5.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.060976138917674342180206539993786896862 \cdot 10^{65}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -5.269237409668851033931621618066544269281 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\\ \mathbf{elif}\;b \le 4.059864939717054460242095466484063430125 \cdot 10^{128}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.060976138917674342180206539993786896862 \cdot 10^{65}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -5.269237409668851033931621618066544269281 \cdot 10^{-239}:\\
\;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\\

\mathbf{elif}\;b \le 4.059864939717054460242095466484063430125 \cdot 10^{128}:\\
\;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r85197 = b;
        double r85198 = -r85197;
        double r85199 = r85197 * r85197;
        double r85200 = 4.0;
        double r85201 = a;
        double r85202 = r85200 * r85201;
        double r85203 = c;
        double r85204 = r85202 * r85203;
        double r85205 = r85199 - r85204;
        double r85206 = sqrt(r85205);
        double r85207 = r85198 + r85206;
        double r85208 = 2.0;
        double r85209 = r85208 * r85201;
        double r85210 = r85207 / r85209;
        return r85210;
}
↓
double f(double a, double b, double c) {
        double r85211 = b;
        double r85212 = -3.0609761389176743e+65;
        bool r85213 = r85211 <= r85212;
        double r85214 = 1.0;
        double r85215 = c;
        double r85216 = r85215 / r85211;
        double r85217 = a;
        double r85218 = r85211 / r85217;
        double r85219 = r85216 - r85218;
        double r85220 = r85214 * r85219;
        double r85221 = -5.269237409668851e-239;
        bool r85222 = r85211 <= r85221;
        double r85223 = -r85211;
        double r85224 = r85211 * r85211;
        double r85225 = 4.0;
        double r85226 = r85225 * r85217;
        double r85227 = r85226 * r85215;
        double r85228 = r85224 - r85227;
        double r85229 = sqrt(r85228);
        double r85230 = r85223 + r85229;
        double r85231 = sqrt(r85230);
        double r85232 = 2.0;
        double r85233 = r85231 / r85232;
        double r85234 = r85231 / r85217;
        double r85235 = r85233 * r85234;
        double r85236 = 4.0598649397170545e+128;
        bool r85237 = r85211 <= r85236;
        double r85238 = 1.0;
        double r85239 = 0.5;
        double r85240 = r85239 / r85215;
        double r85241 = r85223 - r85229;
        double r85242 = r85240 * r85241;
        double r85243 = r85238 / r85242;
        double r85244 = -1.0;
        double r85245 = r85244 * r85216;
        double r85246 = r85237 ? r85243 : r85245;
        double r85247 = r85222 ? r85235 : r85246;
        double r85248 = r85213 ? r85220 : r85247;
        return r85248;
}

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

Original33.4
Target20.5
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.0609761389176743e+65

    1. Initial program 40.1

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

    2. Taylor expanded around -inf 4.4

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

    3. Simplified4.4

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

    if -3.0609761389176743e+65 < b < -5.269237409668851e-239

    1. Initial program 7.6

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

    3. Applied add-sqr-sqrt8.0

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

    4. Applied times-frac8.0

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

    if -5.269237409668851e-239 < b < 4.0598649397170545e+128

    1. Initial program 31.1

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

    3. Applied flip-+31.2

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

    4. Simplified15.4

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num15.6

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

    7. Simplified14.7

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

    8. Taylor expanded around 0 9.3

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

    if 4.0598649397170545e+128 < b

    1. Initial program 61.3

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

    2. Taylor expanded around inf 2.0

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.060976138917674342180206539993786896862 \cdot 10^{65}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -5.269237409668851033931621618066544269281 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\\ \mathbf{elif}\;b \le 4.059864939717054460242095466484063430125 \cdot 10^{128}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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)))