Average Error: 28.7 → 0.3
Time: 5.9s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{2} \cdot \frac{\frac{c \cdot 4}{1}}{\left(-b\right) - \frac{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{2} \cdot \frac{\frac{c \cdot 4}{1}}{\left(-b\right) - \frac{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}}
double f(double a, double b, double c) {
        double r33100 = b;
        double r33101 = -r33100;
        double r33102 = r33100 * r33100;
        double r33103 = 4.0;
        double r33104 = a;
        double r33105 = r33103 * r33104;
        double r33106 = c;
        double r33107 = r33105 * r33106;
        double r33108 = r33102 - r33107;
        double r33109 = sqrt(r33108);
        double r33110 = r33101 + r33109;
        double r33111 = 2.0;
        double r33112 = r33111 * r33104;
        double r33113 = r33110 / r33112;
        return r33113;
}
↓
double f(double a, double b, double c) {
        double r33114 = 1.0;
        double r33115 = 2.0;
        double r33116 = r33114 / r33115;
        double r33117 = c;
        double r33118 = 4.0;
        double r33119 = r33117 * r33118;
        double r33120 = r33119 / r33114;
        double r33121 = b;
        double r33122 = -r33121;
        double r33123 = r33121 * r33121;
        double r33124 = r33123 * r33123;
        double r33125 = a;
        double r33126 = r33118 * r33125;
        double r33127 = r33126 * r33117;
        double r33128 = r33127 * r33127;
        double r33129 = r33124 - r33128;
        double r33130 = sqrt(r33129);
        double r33131 = r33123 + r33127;
        double r33132 = sqrt(r33131);
        double r33133 = r33130 / r33132;
        double r33134 = r33122 - r33133;
        double r33135 = r33120 / r33134;
        double r33136 = r33116 * r33135;
        return r33136;
}

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 28.7

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

  3. Applied flip-+28.8

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

  4. Simplified0.5

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

  6. Applied *-un-lft-identity0.5

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

  7. Applied *-un-lft-identity0.5

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

  8. Applied times-frac0.5

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

  9. Applied times-frac0.5

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

  10. Simplified0.5

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

  11. Simplified0.5

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

  13. Applied associate-/r*0.3

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

  14. Simplified0.3

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

  16. Applied flip--0.3

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

  17. Applied sqrt-div0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(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)))