Average Error: 28.3 → 16.9
Time: 16.0s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 22041.35648233499887282960116863250732422:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1.5 \cdot a}{\frac{b}{c}}}{3 \cdot a}\\ \end{array}\]
\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 22041.35648233499887282960116863250732422:\\
\;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-1.5 \cdot a}{\frac{b}{c}}}{3 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r63075 = b;
        double r63076 = -r63075;
        double r63077 = r63075 * r63075;
        double r63078 = 3.0;
        double r63079 = a;
        double r63080 = r63078 * r63079;
        double r63081 = c;
        double r63082 = r63080 * r63081;
        double r63083 = r63077 - r63082;
        double r63084 = sqrt(r63083);
        double r63085 = r63076 + r63084;
        double r63086 = r63085 / r63080;
        return r63086;
}
↓
double f(double a, double b, double c) {
        double r63087 = b;
        double r63088 = 22041.356482335;
        bool r63089 = r63087 <= r63088;
        double r63090 = r63087 * r63087;
        double r63091 = 3.0;
        double r63092 = a;
        double r63093 = c;
        double r63094 = r63092 * r63093;
        double r63095 = r63091 * r63094;
        double r63096 = r63090 - r63095;
        double r63097 = r63090 - r63096;
        double r63098 = -r63087;
        double r63099 = r63091 * r63092;
        double r63100 = r63099 * r63093;
        double r63101 = r63090 - r63100;
        double r63102 = sqrt(r63101);
        double r63103 = r63098 - r63102;
        double r63104 = r63097 / r63103;
        double r63105 = r63104 / r63099;
        double r63106 = -1.5;
        double r63107 = r63106 * r63092;
        double r63108 = r63087 / r63093;
        double r63109 = r63107 / r63108;
        double r63110 = r63109 / r63099;
        double r63111 = r63089 ? r63105 : r63110;
        return r63111;
}

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 2 regimes

  2. if b < 22041.356482335

    1. Initial program 19.9

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

    3. Applied flip-+20.0

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

    4. Simplified18.9

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

    if 22041.356482335 < b

    1. Initial program 38.7

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

    2. Taylor expanded around inf 14.5

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    3. Using strategy rm

    4. Applied associate-/l*14.5

      \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a}\]
    5. Using strategy rm

    6. Applied associate-*r/14.5

      \[\leadsto \frac{\color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}}}}{3 \cdot a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 22041.35648233499887282960116863250732422:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1.5 \cdot a}{\frac{b}{c}}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.05367121277235087e-8 a 94906265.6242515594) (< 1.05367121277235087e-8 b 94906265.6242515594) (< 1.05367121277235087e-8 c 94906265.6242515594))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))