Average Error: 32.8 → 10.5
Time: 32.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3905772698079686 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 1.0937455763637174 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.3905772698079686 \cdot 10^{+104}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 1.0937455763637174 \cdot 10^{-150}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r11182063 = b;
        double r11182064 = -r11182063;
        double r11182065 = r11182063 * r11182063;
        double r11182066 = 3.0;
        double r11182067 = a;
        double r11182068 = r11182066 * r11182067;
        double r11182069 = c;
        double r11182070 = r11182068 * r11182069;
        double r11182071 = r11182065 - r11182070;
        double r11182072 = sqrt(r11182071);
        double r11182073 = r11182064 + r11182072;
        double r11182074 = r11182073 / r11182068;
        return r11182074;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r11182075 = b;
        double r11182076 = -1.3905772698079686e+104;
        bool r11182077 = r11182075 <= r11182076;
        double r11182078 = 0.5;
        double r11182079 = c;
        double r11182080 = r11182079 / r11182075;
        double r11182081 = r11182078 * r11182080;
        double r11182082 = a;
        double r11182083 = r11182075 / r11182082;
        double r11182084 = 0.6666666666666666;
        double r11182085 = r11182083 * r11182084;
        double r11182086 = r11182081 - r11182085;
        double r11182087 = 1.0937455763637174e-150;
        bool r11182088 = r11182075 <= r11182087;
        double r11182089 = r11182075 * r11182075;
        double r11182090 = 3.0;
        double r11182091 = r11182090 * r11182082;
        double r11182092 = r11182091 * r11182079;
        double r11182093 = r11182089 - r11182092;
        double r11182094 = sqrt(r11182093);
        double r11182095 = r11182094 - r11182075;
        double r11182096 = r11182095 / r11182090;
        double r11182097 = r11182096 / r11182082;
        double r11182098 = -0.5;
        double r11182099 = r11182098 * r11182080;
        double r11182100 = r11182088 ? r11182097 : r11182099;
        double r11182101 = r11182077 ? r11182086 : r11182100;
        return r11182101;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3905772698079686e+104

    1. Initial program 44.4

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

    2. Simplified44.4

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

    3. Taylor expanded around 0 44.4

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

    4. Simplified44.5

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

    5. Taylor expanded around -inf 4.0

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

    if -1.3905772698079686e+104 < b < 1.0937455763637174e-150

    1. Initial program 10.3

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

    2. Simplified10.3

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

    4. Applied associate-/r*10.3

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

    if 1.0937455763637174e-150 < b

    1. Initial program 49.4

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

    2. Simplified49.4

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

    3. Taylor expanded around 0 49.4

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

    4. Simplified49.4

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

    5. Taylor expanded around inf 12.8

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3905772698079686 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 1.0937455763637174 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))