Average Error: 44.2 → 11.8
Time: 19.4s
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 r3229114 = b;
        double r3229115 = -r3229114;
        double r3229116 = r3229114 * r3229114;
        double r3229117 = 3.0;
        double r3229118 = a;
        double r3229119 = r3229117 * r3229118;
        double r3229120 = c;
        double r3229121 = r3229119 * r3229120;
        double r3229122 = r3229116 - r3229121;
        double r3229123 = sqrt(r3229122);
        double r3229124 = r3229115 + r3229123;
        double r3229125 = r3229124 / r3229119;
        return r3229125;
}
↓
double f(double __attribute__((unused)) a, double b, double c, double __attribute__((unused)) d) {
        double r3229126 = c;
        double r3229127 = b;
        double r3229128 = r3229126 / r3229127;
        double r3229129 = -0.5;
        double r3229130 = r3229128 * r3229129;
        return r3229130;
}

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 44.2

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

  2. Simplified44.2

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

  3. Taylor expanded around inf 11.8

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

  4. Final simplification11.8

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

Reproduce

herbie shell --seed 2019133 
(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)))