Average Error: 52.6 → 6.1
Time: 14.9s
Precision: 64
\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c}{b} \cdot -2}{2}\]
\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 r633541 = b;
        double r633542 = -r633541;
        double r633543 = r633541 * r633541;
        double r633544 = 4.0;
        double r633545 = a;
        double r633546 = r633544 * r633545;
        double r633547 = c;
        double r633548 = r633546 * r633547;
        double r633549 = r633543 - r633548;
        double r633550 = sqrt(r633549);
        double r633551 = r633542 + r633550;
        double r633552 = 2.0;
        double r633553 = r633552 * r633545;
        double r633554 = r633551 / r633553;
        return r633554;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r633555 = c;
        double r633556 = b;
        double r633557 = r633555 / r633556;
        double r633558 = -2.0;
        double r633559 = r633557 * r633558;
        double r633560 = 2.0;
        double r633561 = r633559 / r633560;
        return r633561;
}

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 52.6

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

  2. Simplified52.6

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

  3. Taylor expanded around inf 6.1

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

  4. Final simplification6.1

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))