quadp (p42, positive)

Average Error: 34.6 → 16.3

Time: 34.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 2.49445624012960862396084940365110205816 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 3.224491050532555179035846228386712352959 \cdot 10^{112}:\\ \;\;\;\;\frac{\left(\frac{c}{\sqrt[3]{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}} \cdot \left(\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}} \cdot \sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}\right)}} \cdot \frac{a \cdot 4}{a}\right) \cdot \frac{-1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} + b} \cdot \sqrt[3]{b + \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot \left(-4\right), a, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot \left(-4\right), a, b \cdot b\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot \left(-4\right), a, b \cdot b\right)}}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot 4, -c, 0\right)}{b \cdot 2}}{a}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r97930 = b;
        double r97931 = -r97930;
        double r97932 = r97930 * r97930;
        double r97933 = 4.0;
        double r97934 = a;
        double r97935 = c;
        double r97936 = r97934 * r97935;
        double r97937 = r97933 * r97936;
        double r97938 = r97932 - r97937;
        double r97939 = sqrt(r97938);
        double r97940 = r97931 + r97939;
        double r97941 = 2.0;
        double r97942 = r97941 * r97934;
        double r97943 = r97940 / r97942;
        return r97943;
}

↓

double f(double a, double b, double c) {
        double r97944 = b;
        double r97945 = 2.4944562401296086e-289;
        bool r97946 = r97944 <= r97945;
        double r97947 = a;
        double r97948 = c;
        double r97949 = -r97948;
        double r97950 = r97947 * r97949;
        double r97951 = 4.0;
        double r97952 = r97944 * r97944;
        double r97953 = fma(r97950, r97951, r97952);
        double r97954 = sqrt(r97953);
        double r97955 = r97954 - r97944;
        double r97956 = r97955 / r97947;
        double r97957 = 2.0;
        double r97958 = r97956 / r97957;
        double r97959 = 3.224491050532555e+112;
        bool r97960 = r97944 <= r97959;
        double r97961 = fma(r97951, r97950, r97952);
        double r97962 = sqrt(r97961);
        double r97963 = r97944 + r97962;
        double r97964 = cbrt(r97963);
        double r97965 = r97964 * r97964;
        double r97966 = r97964 * r97965;
        double r97967 = cbrt(r97966);
        double r97968 = r97948 / r97967;
        double r97969 = r97947 * r97951;
        double r97970 = r97969 / r97947;
        double r97971 = r97968 * r97970;
        double r97972 = -1.0;
        double r97973 = r97954 + r97944;
        double r97974 = cbrt(r97973);
        double r97975 = -r97951;
        double r97976 = r97948 * r97975;
        double r97977 = fma(r97976, r97947, r97952);
        double r97978 = sqrt(r97977);
        double r97979 = cbrt(r97978);
        double r97980 = r97979 * r97979;
        double r97981 = r97980 * r97979;
        double r97982 = r97944 + r97981;
        double r97983 = cbrt(r97982);
        double r97984 = r97974 * r97983;
        double r97985 = r97972 / r97984;
        double r97986 = r97971 * r97985;
        double r97987 = r97986 / r97957;
        double r97988 = 0.0;
        double r97989 = fma(r97969, r97949, r97988);
        double r97990 = 2.0;
        double r97991 = r97944 * r97990;
        double r97992 = r97989 / r97991;
        double r97993 = r97992 / r97947;
        double r97994 = r97993 / r97957;
        double r97995 = r97960 ? r97987 : r97994;
        double r97996 = r97946 ? r97958 : r97995;
        return r97996;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.6
Target	21.0
Herbie	16.3

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

Derivation

1. Split input into 3 regimes

2. if b < 2.4944562401296086e-289

   1. Initial program 22.5

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

   2. Simplified 22.5

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

   4. Applied *-un-lft-identity 22.5

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

   5. Applied *-un-lft-identity 22.5

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

   6. Applied times-frac 22.5

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

   7. Simplified 22.5

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

   8. Simplified 22.5

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

   if 2.4944562401296086e-289 < b < 3.224491050532555e+112

   1. Initial program 33.8

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

   2. Simplified 33.8

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

   4. Applied flip-- 33.8

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

   5. Simplified 15.8

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

   6. Simplified 15.8

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

   8. Applied *-un-lft-identity 15.8

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

   9. Applied add-cube-cbrt 16.5

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

   10. Applied *-un-lft-identity 16.5

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

   11. Applied times-frac 16.5

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

   12. Applied times-frac 15.8

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

   13. Simplified 15.8

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

   14. Simplified 9.1

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

   16. Applied add-cube-cbrt 9.1

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

   17. Simplified 9.1

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

   18. Simplified 9.1

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

   20. Applied add-cube-cbrt 9.1

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

   21. Simplified 9.2

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

   22. Simplified 9.2

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

   if 3.224491050532555e+112 < b

   1. Initial program 60.8

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

   2. Simplified 60.8

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

   4. Applied flip-- 60.8

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

   5. Simplified 32.6

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

   6. Simplified 32.6

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

   7. Taylor expanded around 0 13.2

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

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.49445624012960862396084940365110205816 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 3.224491050532555179035846228386712352959 \cdot 10^{112}:\\ \;\;\;\;\frac{\left(\frac{c}{\sqrt[3]{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}} \cdot \left(\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}} \cdot \sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}\right)}} \cdot \frac{a \cdot 4}{a}\right) \cdot \frac{-1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} + b} \cdot \sqrt[3]{b + \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot \left(-4\right), a, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot \left(-4\right), a, b \cdot b\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot \left(-4\right), a, b \cdot b\right)}}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot 4, -c, 0\right)}{b \cdot 2}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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)))