Average Error: 44.2 → 11.8
Time: 55.2s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{\frac{-b}{c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{\frac{-b}{c}}
double f(double a, double b, double c) {
        double r8659999 = b;
        double r8660000 = -r8659999;
        double r8660001 = r8659999 * r8659999;
        double r8660002 = 4.0;
        double r8660003 = a;
        double r8660004 = r8660002 * r8660003;
        double r8660005 = c;
        double r8660006 = r8660004 * r8660005;
        double r8660007 = r8660001 - r8660006;
        double r8660008 = sqrt(r8660007);
        double r8660009 = r8660000 + r8660008;
        double r8660010 = 2.0;
        double r8660011 = r8660010 * r8660003;
        double r8660012 = r8660009 / r8660011;
        return r8660012;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r8660013 = 1.0;
        double r8660014 = b;
        double r8660015 = -r8660014;
        double r8660016 = c;
        double r8660017 = r8660015 / r8660016;
        double r8660018 = r8660013 / r8660017;
        return r8660018;
}

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 44.2

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

  2. Simplified44.2

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

  3. Taylor expanded around inf 11.8

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

  4. Taylor expanded around -inf 11.8

    \[\leadsto \frac{-2 \cdot \color{blue}{\frac{a \cdot c}{b}}}{2 \cdot a}\]
  5. Using strategy rm

  6. Applied clear-num11.9

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

  7. Simplified11.8

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

  8. Final simplification11.8

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

Reproduce

herbie shell --seed 2019125 
(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)))