Average Error: 28.8 → 0.4
Time: 8.6s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{4 \cdot \left(a \cdot c\right)}{-\left(b \cdot a + a \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{1}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{4 \cdot \left(a \cdot c\right)}{-\left(b \cdot a + a \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{1}{2}
double f(double a, double b, double c) {
        double r44062 = b;
        double r44063 = -r44062;
        double r44064 = r44062 * r44062;
        double r44065 = 4.0;
        double r44066 = a;
        double r44067 = r44065 * r44066;
        double r44068 = c;
        double r44069 = r44067 * r44068;
        double r44070 = r44064 - r44069;
        double r44071 = sqrt(r44070);
        double r44072 = r44063 + r44071;
        double r44073 = 2.0;
        double r44074 = r44073 * r44066;
        double r44075 = r44072 / r44074;
        return r44075;
}
↓
double f(double a, double b, double c) {
        double r44076 = 4.0;
        double r44077 = a;
        double r44078 = c;
        double r44079 = r44077 * r44078;
        double r44080 = r44076 * r44079;
        double r44081 = b;
        double r44082 = r44081 * r44077;
        double r44083 = r44081 * r44081;
        double r44084 = r44076 * r44077;
        double r44085 = r44084 * r44078;
        double r44086 = r44083 - r44085;
        double r44087 = sqrt(r44086);
        double r44088 = r44077 * r44087;
        double r44089 = r44082 + r44088;
        double r44090 = -r44089;
        double r44091 = r44080 / r44090;
        double r44092 = 1.0;
        double r44093 = 2.0;
        double r44094 = r44092 / r44093;
        double r44095 = r44091 * r44094;
        return r44095;
}

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 28.8

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

  3. Applied flip-+28.9

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

  4. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

  7. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

  8. Applied times-frac0.4

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  9. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}}\]

  10. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\]

  11. Simplified0.4

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]
  12. Using strategy rm

  13. Applied sub-neg0.4

    \[\leadsto \frac{1}{2} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{a \cdot \color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right)}}\]

  14. Applied distribute-lft-in0.4

    \[\leadsto \frac{1}{2} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{a \cdot \left(-b\right) + a \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]

  15. Simplified0.4

    \[\leadsto \frac{1}{2} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\left(-b\right) \cdot a} + a \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

  16. Final simplification0.4

    \[\leadsto \frac{4 \cdot \left(a \cdot c\right)}{-\left(b \cdot a + a \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))