Average Error: 52.5 → 6.2
Time: 19.1s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c}{b} \cdot -1\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{c}{b} \cdot -1
double f(double a, double b, double c) {
        double r1074042 = b;
        double r1074043 = -r1074042;
        double r1074044 = r1074042 * r1074042;
        double r1074045 = 4.0;
        double r1074046 = a;
        double r1074047 = r1074045 * r1074046;
        double r1074048 = c;
        double r1074049 = r1074047 * r1074048;
        double r1074050 = r1074044 - r1074049;
        double r1074051 = sqrt(r1074050);
        double r1074052 = r1074043 + r1074051;
        double r1074053 = 2.0;
        double r1074054 = r1074053 * r1074046;
        double r1074055 = r1074052 / r1074054;
        return r1074055;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r1074056 = c;
        double r1074057 = b;
        double r1074058 = r1074056 / r1074057;
        double r1074059 = -1.0;
        double r1074060 = r1074058 * r1074059;
        return r1074060;
}

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 52.5

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

  2. Simplified52.5

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

  3. Taylor expanded around inf 6.2

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

  4. Final simplification6.2

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

Reproduce

herbie shell --seed 2019170 +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) (* (* 4.0 a) c)))) (* 2.0 a)))