Average Error: 34.1 → 8.1
Time: 7.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.257476678127677856918278287038350045718 \cdot 10^{107}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.496744794133526836762101371765290843051 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 5.354608489416471204042085887246325611474 \cdot 10^{62}:\\ \;\;\;\;\frac{\frac{a}{\sqrt[3]{a}}}{\sqrt[3]{a}} \cdot \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.257476678127677856918278287038350045718 \cdot 10^{107}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 2.496744794133526836762101371765290843051 \cdot 10^{-135}:\\
\;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\

\mathbf{elif}\;b \le 5.354608489416471204042085887246325611474 \cdot 10^{62}:\\
\;\;\;\;\frac{\frac{a}{\sqrt[3]{a}}}{\sqrt[3]{a}} \cdot \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}{\sqrt[3]{a}}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r133183 = b;
        double r133184 = -r133183;
        double r133185 = r133183 * r133183;
        double r133186 = 3.0;
        double r133187 = a;
        double r133188 = r133186 * r133187;
        double r133189 = c;
        double r133190 = r133188 * r133189;
        double r133191 = r133185 - r133190;
        double r133192 = sqrt(r133191);
        double r133193 = r133184 + r133192;
        double r133194 = r133193 / r133188;
        return r133194;
}
↓
double f(double a, double b, double c) {
        double r133195 = b;
        double r133196 = -1.2574766781276779e+107;
        bool r133197 = r133195 <= r133196;
        double r133198 = 0.5;
        double r133199 = c;
        double r133200 = r133199 / r133195;
        double r133201 = r133198 * r133200;
        double r133202 = 0.6666666666666666;
        double r133203 = a;
        double r133204 = r133195 / r133203;
        double r133205 = r133202 * r133204;
        double r133206 = r133201 - r133205;
        double r133207 = 2.4967447941335268e-135;
        bool r133208 = r133195 <= r133207;
        double r133209 = -r133195;
        double r133210 = r133195 * r133195;
        double r133211 = 3.0;
        double r133212 = r133211 * r133203;
        double r133213 = r133212 * r133199;
        double r133214 = r133210 - r133213;
        double r133215 = sqrt(r133214);
        double r133216 = r133209 + r133215;
        double r133217 = r133216 / r133211;
        double r133218 = r133217 / r133203;
        double r133219 = 5.354608489416471e+62;
        bool r133220 = r133195 <= r133219;
        double r133221 = cbrt(r133203);
        double r133222 = r133203 / r133221;
        double r133223 = r133222 / r133221;
        double r133224 = 1.0;
        double r133225 = r133209 - r133215;
        double r133226 = r133225 / r133199;
        double r133227 = r133224 / r133226;
        double r133228 = r133227 / r133221;
        double r133229 = r133223 * r133228;
        double r133230 = -0.5;
        double r133231 = r133230 * r133200;
        double r133232 = r133220 ? r133229 : r133231;
        double r133233 = r133208 ? r133218 : r133232;
        double r133234 = r133197 ? r133206 : r133233;
        return r133234;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -1.2574766781276779e+107

    1. Initial program 48.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around -inf 3.7

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}}\]

    if -1.2574766781276779e+107 < b < 2.4967447941335268e-135

    1. Initial program 11.7

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

    3. Applied associate-/r*11.7

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}}\]

    if 2.4967447941335268e-135 < b < 5.354608489416471e+62

    1. Initial program 39.0

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

    3. Applied flip-+39.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

    4. Simplified16.1

      \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
    5. Using strategy rm

    6. Applied associate-/r*16.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3}}{a}}\]

    7. Simplified16.1

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

    9. Applied clear-num16.4

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{3 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{3 \cdot \left(a \cdot c\right)}}}}{a}\]

    10. Simplified16.3

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a \cdot c}}}}{a}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt17.0

      \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a \cdot c}}}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

    13. Applied *-un-lft-identity17.0

      \[\leadsto \frac{\frac{1}{\frac{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{a \cdot c}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    14. Applied times-frac14.3

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    15. Applied add-sqr-sqrt14.3

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    16. Applied times-frac13.7

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{1}}{\frac{1}{a}} \cdot \frac{\sqrt{1}}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    17. Applied times-frac10.2

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{1}}{\frac{1}{a}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\sqrt{1}}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}{\sqrt[3]{a}}}\]

    18. Simplified10.2

      \[\leadsto \color{blue}{\frac{\frac{a}{\sqrt[3]{a}}}{\sqrt[3]{a}}} \cdot \frac{\frac{\sqrt{1}}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}{\sqrt[3]{a}}\]

    19. Simplified10.2

      \[\leadsto \frac{\frac{a}{\sqrt[3]{a}}}{\sqrt[3]{a}} \cdot \color{blue}{\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}{\sqrt[3]{a}}}\]

    if 5.354608489416471e+62 < b

    1. Initial program 57.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around inf 3.8

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.257476678127677856918278287038350045718 \cdot 10^{107}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.496744794133526836762101371765290843051 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 5.354608489416471204042085887246325611474 \cdot 10^{62}:\\ \;\;\;\;\frac{\frac{a}{\sqrt[3]{a}}}{\sqrt[3]{a}} \cdot \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))