Average Error: 43.7 → 0.2
Time: 10.7s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c \cdot \left(4 \cdot a\right)}{a} \cdot \frac{1}{2}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{c \cdot \left(4 \cdot a\right)}{a} \cdot \frac{1}{2}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}
double f(double a, double b, double c) {
        double r43934 = b;
        double r43935 = -r43934;
        double r43936 = r43934 * r43934;
        double r43937 = 4.0;
        double r43938 = a;
        double r43939 = r43937 * r43938;
        double r43940 = c;
        double r43941 = r43939 * r43940;
        double r43942 = r43936 - r43941;
        double r43943 = sqrt(r43942);
        double r43944 = r43935 + r43943;
        double r43945 = 2.0;
        double r43946 = r43945 * r43938;
        double r43947 = r43944 / r43946;
        return r43947;
}
↓
double f(double a, double b, double c) {
        double r43948 = c;
        double r43949 = 4.0;
        double r43950 = a;
        double r43951 = r43949 * r43950;
        double r43952 = r43948 * r43951;
        double r43953 = r43952 / r43950;
        double r43954 = 1.0;
        double r43955 = 2.0;
        double r43956 = r43954 / r43955;
        double r43957 = r43953 * r43956;
        double r43958 = b;
        double r43959 = -r43958;
        double r43960 = r43958 * r43958;
        double r43961 = r43951 * r43948;
        double r43962 = r43960 - r43961;
        double r43963 = sqrt(r43962);
        double r43964 = r43959 - r43963;
        double r43965 = r43957 / r43964;
        return r43965;
}

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 43.7

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

  3. Applied flip-+43.7

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

  4. Simplified0.4

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

  6. Applied clear-num0.5

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

  7. Simplified0.5

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

  9. Applied add-sqr-sqrt0.5

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

  10. Applied associate-/l*0.5

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

  11. Simplified0.3

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

  13. Applied associate-/r/0.5

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

  14. Applied associate-/r*0.3

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

  15. Simplified0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 
(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)))