Average Error: 43.8 → 0.4
Time: 7.5s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{0 + 4 \cdot \left(a \cdot c\right)}{a} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{0 + 4 \cdot \left(a \cdot c\right)}{a} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}
double f(double a, double b, double c) {
        double r36056 = b;
        double r36057 = -r36056;
        double r36058 = r36056 * r36056;
        double r36059 = 4.0;
        double r36060 = a;
        double r36061 = r36059 * r36060;
        double r36062 = c;
        double r36063 = r36061 * r36062;
        double r36064 = r36058 - r36063;
        double r36065 = sqrt(r36064);
        double r36066 = r36057 + r36065;
        double r36067 = 2.0;
        double r36068 = r36067 * r36060;
        double r36069 = r36066 / r36068;
        return r36069;
}
↓
double f(double a, double b, double c) {
        double r36070 = 0.0;
        double r36071 = 4.0;
        double r36072 = a;
        double r36073 = c;
        double r36074 = r36072 * r36073;
        double r36075 = r36071 * r36074;
        double r36076 = r36070 + r36075;
        double r36077 = r36076 / r36072;
        double r36078 = 1.0;
        double r36079 = b;
        double r36080 = -r36079;
        double r36081 = r36079 * r36079;
        double r36082 = r36071 * r36072;
        double r36083 = r36082 * r36073;
        double r36084 = r36081 - r36083;
        double r36085 = sqrt(r36084);
        double r36086 = r36080 - r36085;
        double r36087 = r36078 / r36086;
        double r36088 = 2.0;
        double r36089 = r36087 / r36088;
        double r36090 = r36077 * r36089;
        return r36090;
}

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 43.8

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

  3. Applied flip-+43.8

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

  4. Simplified0.4

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

  6. Applied flip-+0.5

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

  7. Applied associate-/l/0.5

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

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

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

  10. Applied times-frac0.5

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

  11. Applied times-frac0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(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)))