Average Error: 43.4 → 0.2
Time: 5.8s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{2} \cdot \frac{\frac{c \cdot 4}{1}}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{2} \cdot \frac{\frac{c \cdot 4}{1}}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}
double f(double a, double b, double c) {
        double r32881 = b;
        double r32882 = -r32881;
        double r32883 = r32881 * r32881;
        double r32884 = 4.0;
        double r32885 = a;
        double r32886 = r32884 * r32885;
        double r32887 = c;
        double r32888 = r32886 * r32887;
        double r32889 = r32883 - r32888;
        double r32890 = sqrt(r32889);
        double r32891 = r32882 + r32890;
        double r32892 = 2.0;
        double r32893 = r32892 * r32885;
        double r32894 = r32891 / r32893;
        return r32894;
}
↓
double f(double a, double b, double c) {
        double r32895 = 1.0;
        double r32896 = 2.0;
        double r32897 = r32895 / r32896;
        double r32898 = c;
        double r32899 = 4.0;
        double r32900 = r32898 * r32899;
        double r32901 = r32900 / r32895;
        double r32902 = b;
        double r32903 = sqrt(r32902);
        double r32904 = -r32903;
        double r32905 = r32902 * r32902;
        double r32906 = a;
        double r32907 = r32899 * r32906;
        double r32908 = r32907 * r32898;
        double r32909 = r32905 - r32908;
        double r32910 = sqrt(r32909);
        double r32911 = -r32910;
        double r32912 = fma(r32903, r32904, r32911);
        double r32913 = r32901 / r32912;
        double r32914 = r32897 * r32913;
        return r32914;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.4

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

  3. Applied flip-+43.4

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

  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 *-un-lft-identity0.4

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

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

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

  8. Applied times-frac0.4

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

  9. Applied times-frac0.4

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  13. Applied associate-/r*0.2

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

  14. Simplified0.2

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

  16. Applied add-sqr-sqrt0.3

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

  17. Applied distribute-rgt-neg-in0.3

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

  18. Applied fma-neg0.2

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

  19. Final simplification0.2

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

Reproduce

herbie shell --seed 2020056 +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)))