Average Error: 44.0 → 12.0
Time: 15.7s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c}{b} \cdot -2}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{c}{b} \cdot -2}{2}
double f(double a, double b, double c) {
        double r587464 = b;
        double r587465 = -r587464;
        double r587466 = r587464 * r587464;
        double r587467 = 4.0;
        double r587468 = a;
        double r587469 = r587467 * r587468;
        double r587470 = c;
        double r587471 = r587469 * r587470;
        double r587472 = r587466 - r587471;
        double r587473 = sqrt(r587472);
        double r587474 = r587465 + r587473;
        double r587475 = 2.0;
        double r587476 = r587475 * r587468;
        double r587477 = r587474 / r587476;
        return r587477;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r587478 = c;
        double r587479 = b;
        double r587480 = r587478 / r587479;
        double r587481 = -2.0;
        double r587482 = r587480 * r587481;
        double r587483 = 2.0;
        double r587484 = r587482 / r587483;
        return r587484;
}

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. Initial program 44.0

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Simplified44.0

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]

  3. Taylor expanded around inf 12.0

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

  4. Final simplification12.0

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))