Average Error: 52.6 → 0.4
Time: 7.6s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r40062 = b;
        double r40063 = -r40062;
        double r40064 = r40062 * r40062;
        double r40065 = 4.0;
        double r40066 = a;
        double r40067 = r40065 * r40066;
        double r40068 = c;
        double r40069 = r40067 * r40068;
        double r40070 = r40064 - r40069;
        double r40071 = sqrt(r40070);
        double r40072 = r40063 + r40071;
        double r40073 = 2.0;
        double r40074 = r40073 * r40066;
        double r40075 = r40072 / r40074;
        return r40075;
}
↓
double f(double a, double b, double c) {
        double r40076 = 0.0;
        double r40077 = 4.0;
        double r40078 = a;
        double r40079 = c;
        double r40080 = r40078 * r40079;
        double r40081 = r40077 * r40080;
        double r40082 = r40076 + r40081;
        double r40083 = b;
        double r40084 = -r40083;
        double r40085 = 6.0;
        double r40086 = pow(r40083, r40085);
        double r40087 = r40077 * r40078;
        double r40088 = r40087 * r40079;
        double r40089 = 3.0;
        double r40090 = pow(r40088, r40089);
        double r40091 = r40086 - r40090;
        double r40092 = fma(r40083, r40083, r40088);
        double r40093 = r40080 * r40092;
        double r40094 = 4.0;
        double r40095 = pow(r40083, r40094);
        double r40096 = fma(r40077, r40093, r40095);
        double r40097 = r40091 / r40096;
        double r40098 = sqrt(r40097);
        double r40099 = r40084 - r40098;
        double r40100 = r40082 / r40099;
        double r40101 = 2.0;
        double r40102 = r40101 * r40078;
        double r40103 = r40100 / r40102;
        return r40103;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.6

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

  3. Applied flip-+52.6

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

  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 flip3--0.4

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

  7. Simplified0.4

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{\color{blue}{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{2 \cdot a}\]

  8. Simplified0.4

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\color{blue}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}}}}{2 \cdot a}\]

  9. Final simplification0.4

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}}}{2 \cdot a}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))