quadp (p42, positive)

Average Error: 33.5 → 13.2

Time: 40.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 7.579106596092839 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 3.188391854507535 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{1}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \cdot \left(-4 \cdot c\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{b + b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r2386950 = b;
        double r2386951 = -r2386950;
        double r2386952 = r2386950 * r2386950;
        double r2386953 = 4.0;
        double r2386954 = a;
        double r2386955 = c;
        double r2386956 = r2386954 * r2386955;
        double r2386957 = r2386953 * r2386956;
        double r2386958 = r2386952 - r2386957;
        double r2386959 = sqrt(r2386958);
        double r2386960 = r2386951 + r2386959;
        double r2386961 = 2.0;
        double r2386962 = r2386961 * r2386954;
        double r2386963 = r2386960 / r2386962;
        return r2386963;
}

↓

double f(double a, double b, double c) {
        double r2386964 = b;
        double r2386965 = 7.579106596092839e-190;
        bool r2386966 = r2386964 <= r2386965;
        double r2386967 = a;
        double r2386968 = c;
        double r2386969 = r2386967 * r2386968;
        double r2386970 = -4.0;
        double r2386971 = r2386964 * r2386964;
        double r2386972 = fma(r2386969, r2386970, r2386971);
        double r2386973 = sqrt(r2386972);
        double r2386974 = r2386973 / r2386967;
        double r2386975 = r2386964 / r2386967;
        double r2386976 = r2386974 - r2386975;
        double r2386977 = 2.0;
        double r2386978 = r2386976 / r2386977;
        double r2386979 = 3.188391854507535e+149;
        bool r2386980 = r2386964 <= r2386979;
        double r2386981 = 1.0;
        double r2386982 = r2386964 + r2386973;
        double r2386983 = r2386981 / r2386982;
        double r2386984 = r2386970 * r2386968;
        double r2386985 = r2386983 * r2386984;
        double r2386986 = r2386985 / r2386977;
        double r2386987 = r2386964 + r2386964;
        double r2386988 = r2386984 / r2386987;
        double r2386989 = r2386988 / r2386977;
        double r2386990 = r2386980 ? r2386986 : r2386989;
        double r2386991 = r2386966 ? r2386978 : r2386990;
        return r2386991;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.5
Target	21.2
Herbie	13.2

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

Derivation

1. Split input into 3 regimes

2. if b < 7.579106596092839e-190

   1. Initial program 20.7

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

   2. Simplified 20.6

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

   4. Applied div-sub 20.7

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

   if 7.579106596092839e-190 < b < 3.188391854507535e+149

   1. Initial program 38.6

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

   2. Simplified 38.6

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

   4. Applied clear-num 38.7

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

   6. Applied flip-- 38.8

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

   7. Applied associate-/r/ 38.8

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

   8. Applied associate-/r* 38.8

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

   9. Simplified 13.6

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

   10. Taylor expanded around inf 6.5

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

   12. Applied div-inv 6.7

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

   if 3.188391854507535e+149 < b

   1. Initial program 62.3

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

   2. Simplified 62.3

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

   4. Applied clear-num 62.3

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

   6. Applied flip-- 62.4

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

   7. Applied associate-/r/ 62.4

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

   8. Applied associate-/r* 62.4

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

   9. Simplified 39.6

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

   10. Taylor expanded around inf 39.4

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

   11. Taylor expanded around 0 1.8

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 7.579106596092839 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 3.188391854507535 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{1}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \cdot \left(-4 \cdot c\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot c}{b + b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +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)))