Average Error: 43.9 → 0.3
Time: 6.6s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{-\left(b + \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{-\left(b + \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}\right)}
double f(double a, double b, double c) {
        double r48125 = b;
        double r48126 = -r48125;
        double r48127 = r48125 * r48125;
        double r48128 = 4.0;
        double r48129 = a;
        double r48130 = r48128 * r48129;
        double r48131 = c;
        double r48132 = r48130 * r48131;
        double r48133 = r48127 - r48132;
        double r48134 = sqrt(r48133);
        double r48135 = r48126 + r48134;
        double r48136 = 2.0;
        double r48137 = r48136 * r48129;
        double r48138 = r48135 / r48137;
        return r48138;
}
↓
double f(double a, double b, double c) {
        double r48139 = 1.0;
        double r48140 = 2.0;
        double r48141 = r48139 / r48140;
        double r48142 = 4.0;
        double r48143 = a;
        double r48144 = c;
        double r48145 = r48143 * r48144;
        double r48146 = r48142 * r48145;
        double r48147 = r48146 / r48143;
        double r48148 = b;
        double r48149 = r48148 * r48148;
        double r48150 = 3.0;
        double r48151 = pow(r48149, r48150);
        double r48152 = r48142 * r48143;
        double r48153 = r48152 * r48144;
        double r48154 = pow(r48153, r48150);
        double r48155 = r48151 - r48154;
        double r48156 = 2.0;
        double r48157 = pow(r48148, r48156);
        double r48158 = r48153 + r48157;
        double r48159 = r48153 * r48158;
        double r48160 = r48149 * r48149;
        double r48161 = r48159 + r48160;
        double r48162 = r48155 / r48161;
        double r48163 = sqrt(r48162);
        double r48164 = r48148 + r48163;
        double r48165 = -r48164;
        double r48166 = r48147 / r48165;
        double r48167 = r48141 * r48166;
        return r48167;
}

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 43.9

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

  3. Applied flip-+43.9

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

  4. Simplified0.4

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

  6. Applied add-cbrt-cube0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}}}}{2 \cdot a}\]

  7. Simplified0.4

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{\color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}}}{2 \cdot a}\]
  8. Using strategy rm

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

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}\right)}}}{2 \cdot a}\]

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

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}\right)}}{2 \cdot a}\]

  11. Applied times-frac0.4

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}}}{2 \cdot a}\]

  12. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}}{a}}\]

  13. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}}{a}\]

  14. Simplified0.2

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

  16. Applied flip3--0.3

    \[\leadsto \frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{-\left(b + \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}\right)}\]

  17. Simplified0.3

    \[\leadsto \frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{-\left(b + \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\color{blue}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}}\right)}\]

  18. Final simplification0.3

    \[\leadsto \frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{-\left(b + \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}\right)}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))