Average Error: 28.7 → 16.8
Time: 17.0s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 53.72772989939689125549193704500794410706:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{elif}\;b \le 250.8081548127311179996468126773834228516:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2125.07914601017046152264811098575592041:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 53.72772989939689125549193704500794410706:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\

\mathbf{elif}\;b \le 250.8081548127311179996468126773834228516:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2125.07914601017046152264811098575592041:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4460083 = b;
        double r4460084 = -r4460083;
        double r4460085 = r4460083 * r4460083;
        double r4460086 = 3.0;
        double r4460087 = a;
        double r4460088 = r4460086 * r4460087;
        double r4460089 = c;
        double r4460090 = r4460088 * r4460089;
        double r4460091 = r4460085 - r4460090;
        double r4460092 = sqrt(r4460091);
        double r4460093 = r4460084 + r4460092;
        double r4460094 = r4460093 / r4460088;
        return r4460094;
}
↓
double f(double a, double b, double c) {
        double r4460095 = b;
        double r4460096 = 53.72772989939689;
        bool r4460097 = r4460095 <= r4460096;
        double r4460098 = r4460095 * r4460095;
        double r4460099 = 3.0;
        double r4460100 = a;
        double r4460101 = r4460099 * r4460100;
        double r4460102 = c;
        double r4460103 = r4460101 * r4460102;
        double r4460104 = r4460098 - r4460103;
        double r4460105 = r4460104 - r4460098;
        double r4460106 = sqrt(r4460104);
        double r4460107 = r4460106 + r4460095;
        double r4460108 = r4460105 / r4460107;
        double r4460109 = r4460108 / r4460101;
        double r4460110 = 250.80815481273112;
        bool r4460111 = r4460095 <= r4460110;
        double r4460112 = -0.5;
        double r4460113 = r4460102 / r4460095;
        double r4460114 = r4460112 * r4460113;
        double r4460115 = 2125.0791460101705;
        bool r4460116 = r4460095 <= r4460115;
        double r4460117 = r4460116 ? r4460109 : r4460114;
        double r4460118 = r4460111 ? r4460114 : r4460117;
        double r4460119 = r4460097 ? r4460109 : r4460118;
        return r4460119;
}

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 < 53.72772989939689 or 250.80815481273112 < b < 2125.0791460101705

    1. Initial program 17.4

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

    2. Simplified17.4

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

    4. Applied flip--17.3

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

    5. Simplified16.3

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

    if 53.72772989939689 < b < 250.80815481273112 or 2125.0791460101705 < b

    1. Initial program 35.4

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

    2. Simplified35.4

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

    3. Taylor expanded around inf 17.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 53.72772989939689125549193704500794410706:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{elif}\;b \le 250.8081548127311179996468126773834228516:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2125.07914601017046152264811098575592041:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))