quadp (p42, positive)

Average Error: 34.0 → 13.5

Time: 42.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 3.456075129616807 \cdot 10^{-304}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b\right)}{2}\\ \mathbf{elif}\;b \le 2.6656023684116586 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{\frac{-4 \cdot c}{\frac{a}{a}}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-4 \cdot c}{\frac{a}{a}}}{b \cdot 2}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1910947 = b;
        double r1910948 = -r1910947;
        double r1910949 = r1910947 * r1910947;
        double r1910950 = 4.0;
        double r1910951 = a;
        double r1910952 = c;
        double r1910953 = r1910951 * r1910952;
        double r1910954 = r1910950 * r1910953;
        double r1910955 = r1910949 - r1910954;
        double r1910956 = sqrt(r1910955);
        double r1910957 = r1910948 + r1910956;
        double r1910958 = 2.0;
        double r1910959 = r1910958 * r1910951;
        double r1910960 = r1910957 / r1910959;
        return r1910960;
}

↓

double f(double a, double b, double c) {
        double r1910961 = b;
        double r1910962 = 3.456075129616807e-304;
        bool r1910963 = r1910961 <= r1910962;
        double r1910964 = 1.0;
        double r1910965 = a;
        double r1910966 = r1910964 / r1910965;
        double r1910967 = c;
        double r1910968 = r1910965 * r1910967;
        double r1910969 = -4.0;
        double r1910970 = r1910961 * r1910961;
        double r1910971 = fma(r1910968, r1910969, r1910970);
        double r1910972 = sqrt(r1910971);
        double r1910973 = r1910972 - r1910961;
        double r1910974 = r1910966 * r1910973;
        double r1910975 = 2.0;
        double r1910976 = r1910974 / r1910975;
        double r1910977 = 2.6656023684116586e+55;
        bool r1910978 = r1910961 <= r1910977;
        double r1910979 = r1910969 * r1910967;
        double r1910980 = r1910965 / r1910965;
        double r1910981 = r1910979 / r1910980;
        double r1910982 = fma(r1910979, r1910965, r1910970);
        double r1910983 = sqrt(r1910982);
        double r1910984 = r1910983 + r1910961;
        double r1910985 = r1910981 / r1910984;
        double r1910986 = r1910985 / r1910975;
        double r1910987 = r1910961 * r1910975;
        double r1910988 = r1910981 / r1910987;
        double r1910989 = r1910988 / r1910975;
        double r1910990 = r1910978 ? r1910986 : r1910989;
        double r1910991 = r1910963 ? r1910976 : r1910990;
        return r1910991;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.0
Target	20.9
Herbie	13.5

\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < 3.456075129616807e-304

   1. Initial program 21.5

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

   2. Simplified 21.4

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

   4. Applied div-inv 21.6

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

   if 3.456075129616807e-304 < b < 2.6656023684116586e+55

   1. Initial program 29.8

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

   2. Simplified 29.8

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

   4. Applied flip-- 29.9

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

   5. Simplified 17.1

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

   7. Applied *-un-lft-identity 17.1

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

   8. Applied *-un-lft-identity 17.1

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

   9. Applied *-un-lft-identity 17.1

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

   10. Applied times-frac 17.1

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

   11. Applied times-frac 17.1

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

   12. Simplified 17.1

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

   13. Simplified 9.6

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

   if 2.6656023684116586e+55 < b

   1. Initial program 56.9

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

   2. Simplified 56.9

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

   4. Applied flip-- 57.0

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

   5. Simplified 28.4

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

   7. Applied *-un-lft-identity 28.4

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

   8. Applied *-un-lft-identity 28.4

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

   9. Applied *-un-lft-identity 28.4

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

   10. Applied times-frac 28.4

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

   11. Applied times-frac 28.4

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

   12. Simplified 28.4

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

   13. Simplified 25.6

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

   15. Applied add-cube-cbrt 25.7

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

   16. Applied sqrt-prod 25.7

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

   17. Applied fma-def 25.7

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

   18. Taylor expanded around 0 4.5

       \[\leadsto \frac{1 \cdot \frac{\frac{-4 \cdot c}{\frac{a}{a}}}{\color{blue}{2 \cdot b}}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3.456075129616807 \cdot 10^{-304}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b\right)}{2}\\ \mathbf{elif}\;b \le 2.6656023684116586 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{\frac{-4 \cdot c}{\frac{a}{a}}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-4 \cdot c}{\frac{a}{a}}}{b \cdot 2}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))