Average Error: 28.4 → 16.1
Time: 29.1s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1984.600261148631:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 1984.600261148631:\\
\;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r3911051 = b;
        double r3911052 = -r3911051;
        double r3911053 = r3911051 * r3911051;
        double r3911054 = 3.0;
        double r3911055 = a;
        double r3911056 = r3911054 * r3911055;
        double r3911057 = c;
        double r3911058 = r3911056 * r3911057;
        double r3911059 = r3911053 - r3911058;
        double r3911060 = sqrt(r3911059);
        double r3911061 = r3911052 + r3911060;
        double r3911062 = r3911061 / r3911056;
        return r3911062;
}
↓
double f(double a, double b, double c) {
        double r3911063 = b;
        double r3911064 = 1984.600261148631;
        bool r3911065 = r3911063 <= r3911064;
        double r3911066 = a;
        double r3911067 = c;
        double r3911068 = r3911066 * r3911067;
        double r3911069 = -3.0;
        double r3911070 = r3911068 * r3911069;
        double r3911071 = r3911063 * r3911063;
        double r3911072 = r3911070 + r3911071;
        double r3911073 = sqrt(r3911072);
        double r3911074 = r3911072 * r3911073;
        double r3911075 = r3911063 * r3911071;
        double r3911076 = r3911074 - r3911075;
        double r3911077 = r3911063 * r3911073;
        double r3911078 = r3911071 + r3911077;
        double r3911079 = r3911072 + r3911078;
        double r3911080 = r3911076 / r3911079;
        double r3911081 = 3.0;
        double r3911082 = r3911066 * r3911081;
        double r3911083 = r3911080 / r3911082;
        double r3911084 = -0.5;
        double r3911085 = r3911067 / r3911063;
        double r3911086 = r3911084 * r3911085;
        double r3911087 = r3911065 ? r3911083 : r3911086;
        return r3911087;
}

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 < 1984.600261148631

    1. Initial program 17.3

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

    2. Simplified17.3

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

    4. Applied flip3--17.4

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

    5. Simplified16.7

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

    6. Simplified16.7

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

    if 1984.600261148631 < b

    1. Initial program 37.0

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

    2. Simplified37.0

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

    3. Taylor expanded around inf 15.8

      \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied div-inv15.8

      \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \frac{1}{b}\right)}}{3 \cdot a}\]

    6. Applied associate-*r*15.8

      \[\leadsto \frac{\color{blue}{\left(\frac{-3}{2} \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{b}}}{3 \cdot a}\]

    7. Taylor expanded around -inf 15.7

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1984.600261148631:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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 a) c)))) (* 3 a)))