Average Error: 28.1 → 0.4
Time: 24.3s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\frac{1}{\frac{2}{4}}}{\frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{\frac{1}{\frac{2}{4}}}{\frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}
double f(double a, double b, double c) {
        double r124107 = b;
        double r124108 = -r124107;
        double r124109 = r124107 * r124107;
        double r124110 = 4.0;
        double r124111 = a;
        double r124112 = r124110 * r124111;
        double r124113 = c;
        double r124114 = r124112 * r124113;
        double r124115 = r124109 - r124114;
        double r124116 = sqrt(r124115);
        double r124117 = r124108 + r124116;
        double r124118 = 2.0;
        double r124119 = r124118 * r124111;
        double r124120 = r124117 / r124119;
        return r124120;
}
↓
double f(double a, double b, double c) {
        double r124121 = 1.0;
        double r124122 = 2.0;
        double r124123 = 4.0;
        double r124124 = r124122 / r124123;
        double r124125 = r124121 / r124124;
        double r124126 = c;
        double r124127 = r124121 / r124126;
        double r124128 = r124125 / r124127;
        double r124129 = b;
        double r124130 = -r124129;
        double r124131 = r124129 * r124129;
        double r124132 = a;
        double r124133 = r124123 * r124132;
        double r124134 = r124133 * r124126;
        double r124135 = r124131 - r124134;
        double r124136 = sqrt(r124135);
        double r124137 = r124130 - r124136;
        double r124138 = r124128 / r124137;
        return r124138;
}

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 28.1

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

  3. Applied flip-+28.2

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

  4. Simplified0.4

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

  6. Applied clear-num0.5

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

  7. Simplified0.5

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

  9. Applied associate-/r*0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020035 
(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)))