Average Error: 43.9 → 11.9
Time: 54.8s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{b} \cdot \frac{-1}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{b} \cdot \frac{-1}{2}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r26484193 = b;
        double r26484194 = -r26484193;
        double r26484195 = r26484193 * r26484193;
        double r26484196 = 3.0;
        double r26484197 = a;
        double r26484198 = r26484196 * r26484197;
        double r26484199 = c;
        double r26484200 = r26484198 * r26484199;
        double r26484201 = r26484195 - r26484200;
        double r26484202 = sqrt(r26484201);
        double r26484203 = r26484194 + r26484202;
        double r26484204 = r26484203 / r26484198;
        return r26484204;
}
↓
double f(double __attribute__((unused)) a, double b, double c, double __attribute__((unused)) d) {
        double r26484205 = c;
        double r26484206 = b;
        double r26484207 = r26484205 / r26484206;
        double r26484208 = -0.5;
        double r26484209 = r26484207 * r26484208;
        return r26484209;
}

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. Initial program 43.9

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

  2. Simplified43.9

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

  3. Taylor expanded around inf 11.9

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

  4. Final simplification11.9

    \[\leadsto \frac{c}{b} \cdot \frac{-1}{2}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (a b c d)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))