Average Error: 52.3 → 6.4
Time: 28.3s
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 r909662 = b;
        double r909663 = -r909662;
        double r909664 = r909662 * r909662;
        double r909665 = 4.0;
        double r909666 = a;
        double r909667 = r909665 * r909666;
        double r909668 = c;
        double r909669 = r909667 * r909668;
        double r909670 = r909664 - r909669;
        double r909671 = sqrt(r909670);
        double r909672 = r909663 + r909671;
        double r909673 = 2.0;
        double r909674 = r909673 * r909666;
        double r909675 = r909672 / r909674;
        return r909675;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r909676 = c;
        double r909677 = b;
        double r909678 = r909676 / r909677;
        double r909679 = -2.0;
        double r909680 = r909678 * r909679;
        double r909681 = 2.0;
        double r909682 = r909680 / r909681;
        return r909682;
}

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 52.3

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

  2. Simplified52.3

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

  3. Taylor expanded around inf 6.4

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

  4. Final simplification6.4

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

Reproduce

herbie shell --seed 2019142 +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)))