quadp (p42, positive)

Average Error: 34.2 → 15.8

Time: 30.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 3.32986367031880887531689935783430471444 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 5.806955844129337209678827027234678404075 \cdot 10^{152}:\\ \;\;\;\;\frac{\frac{-1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} + b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} + b}} \cdot \left(\frac{c}{\sqrt[3]{b + \sqrt{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}} \cdot \frac{a \cdot 4}{a}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot 4, -c, 0\right)}{b \cdot 2}}{a}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r69937 = b;
        double r69938 = -r69937;
        double r69939 = r69937 * r69937;
        double r69940 = 4.0;
        double r69941 = a;
        double r69942 = c;
        double r69943 = r69941 * r69942;
        double r69944 = r69940 * r69943;
        double r69945 = r69939 - r69944;
        double r69946 = sqrt(r69945);
        double r69947 = r69938 + r69946;
        double r69948 = 2.0;
        double r69949 = r69948 * r69941;
        double r69950 = r69947 / r69949;
        return r69950;
}

↓

double f(double a, double b, double c) {
        double r69951 = b;
        double r69952 = 3.329863670318809e-219;
        bool r69953 = r69951 <= r69952;
        double r69954 = a;
        double r69955 = c;
        double r69956 = -r69955;
        double r69957 = r69954 * r69956;
        double r69958 = 4.0;
        double r69959 = r69951 * r69951;
        double r69960 = fma(r69957, r69958, r69959);
        double r69961 = sqrt(r69960);
        double r69962 = r69961 / r69954;
        double r69963 = r69951 / r69954;
        double r69964 = r69962 - r69963;
        double r69965 = 2.0;
        double r69966 = r69964 / r69965;
        double r69967 = 5.806955844129337e+152;
        bool r69968 = r69951 <= r69967;
        double r69969 = -1.0;
        double r69970 = r69961 + r69951;
        double r69971 = cbrt(r69970);
        double r69972 = r69971 * r69971;
        double r69973 = r69969 / r69972;
        double r69974 = fma(r69958, r69957, r69959);
        double r69975 = sqrt(r69974);
        double r69976 = sqrt(r69975);
        double r69977 = r69976 * r69976;
        double r69978 = r69951 + r69977;
        double r69979 = cbrt(r69978);
        double r69980 = r69955 / r69979;
        double r69981 = r69954 * r69958;
        double r69982 = r69981 / r69954;
        double r69983 = r69980 * r69982;
        double r69984 = r69973 * r69983;
        double r69985 = r69984 / r69965;
        double r69986 = 0.0;
        double r69987 = fma(r69981, r69956, r69986);
        double r69988 = 2.0;
        double r69989 = r69951 * r69988;
        double r69990 = r69987 / r69989;
        double r69991 = r69990 / r69954;
        double r69992 = r69991 / r69965;
        double r69993 = r69968 ? r69985 : r69992;
        double r69994 = r69953 ? r69966 : r69993;
        return r69994;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	21.0
Herbie	15.8

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < 3.329863670318809e-219

   1. Initial program 21.2

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

   2. Simplified 21.2

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

   4. Applied div-sub 21.2

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

   5. Simplified 21.2

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

   if 3.329863670318809e-219 < b < 5.806955844129337e+152

   1. Initial program 38.0

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

   2. Simplified 38.0

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

   4. Applied flip-- 38.0

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

   5. Simplified 15.6

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

   6. Simplified 15.6

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

   8. Applied *-un-lft-identity 15.6

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

   9. Applied add-cube-cbrt 16.2

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

   10. Applied *-un-lft-identity 16.2

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

   11. Applied times-frac 16.3

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

   12. Applied times-frac 15.1

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

   13. Simplified 15.1

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

   14. Simplified 7.9

       \[\leadsto \frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(\left(-c\right) \cdot a, 4, b \cdot b\right)}} \cdot \sqrt[3]{b + \sqrt{\mathsf{fma}\left(\left(-c\right) \cdot a, 4, b \cdot b\right)}}} \cdot \color{blue}{\left(\frac{a \cdot 4}{a} \cdot \frac{-c}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(\left(-c\right) \cdot a, 4, b \cdot b\right)}}}\right)}}{2}\]
   15. Using strategy rm

   16. Applied add-sqr-sqrt 7.9

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

   17. Applied sqrt-prod 7.9

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

   18. Simplified 7.9

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

   19. Simplified 7.9

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

   if 5.806955844129337e+152 < b

   1. Initial program 63.8

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

   2. Simplified 63.8

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

   4. Applied flip-- 63.8

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

   5. Simplified 38.4

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

   6. Simplified 38.4

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

   7. Taylor expanded around 0 14.3

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot 4, -c, 0\right)}{\color{blue}{2 \cdot b}}}{a}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3.32986367031880887531689935783430471444 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 5.806955844129337209678827027234678404075 \cdot 10^{152}:\\ \;\;\;\;\frac{\frac{-1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} + b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} + b}} \cdot \left(\frac{c}{\sqrt[3]{b + \sqrt{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}} \cdot \frac{a \cdot 4}{a}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot 4, -c, 0\right)}{b \cdot 2}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))