Cubic critical, narrow range

Average Error: 28.8 → 16.1

Time: 30.9s

Precision: 64

\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le 315.4848387613182580935244914144277572632:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r134949 = b;
        double r134950 = -r134949;
        double r134951 = r134949 * r134949;
        double r134952 = 3.0;
        double r134953 = a;
        double r134954 = r134952 * r134953;
        double r134955 = c;
        double r134956 = r134954 * r134955;
        double r134957 = r134951 - r134956;
        double r134958 = sqrt(r134957);
        double r134959 = r134950 + r134958;
        double r134960 = r134959 / r134954;
        return r134960;
}

↓

double f(double a, double b, double c) {
        double r134961 = b;
        double r134962 = 315.48483876131826;
        bool r134963 = r134961 <= r134962;
        double r134964 = r134961 * r134961;
        double r134965 = a;
        double r134966 = c;
        double r134967 = r134965 * r134966;
        double r134968 = 3.0;
        double r134969 = r134967 * r134968;
        double r134970 = r134964 - r134969;
        double r134971 = r134970 - r134964;
        double r134972 = sqrt(r134970);
        double r134973 = r134961 + r134972;
        double r134974 = r134971 / r134973;
        double r134975 = r134974 / r134965;
        double r134976 = r134975 / r134968;
        double r134977 = -0.5;
        double r134978 = r134966 * r134977;
        double r134979 = r134978 / r134961;
        double r134980 = r134963 ? r134976 : r134979;
        return r134980;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 315.48483876131826

   1. Initial program 16.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Simplified 16.4

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

   4. Applied flip-- 16.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}}{a}}{3}\]

   5. Simplified 15.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right) - b \cdot b}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{a}}{3}\]

   6. Simplified 15.4

      \[\leadsto \frac{\frac{\frac{\left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right) - b \cdot b}{\color{blue}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}}{a}}{3}\]

   if 315.48483876131826 < b

   1. Initial program 36.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Simplified 36.0

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{a}}{3}}\]

   3. Taylor expanded around inf 16.6

      \[\leadsto \frac{\frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{a}}{3}\]

   4. Simplified 16.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1.5}{\frac{b}{a \cdot c}}}}{a}}{3}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 16.6

      \[\leadsto \frac{\frac{\frac{-1.5}{\frac{b}{a \cdot c}}}{a}}{\color{blue}{1 \cdot 3}}\]

   7. Applied *-un-lft-identity 16.6

      \[\leadsto \frac{\frac{\frac{-1.5}{\frac{b}{a \cdot c}}}{\color{blue}{1 \cdot a}}}{1 \cdot 3}\]

   8. Applied *-un-lft-identity 16.6

      \[\leadsto \frac{\frac{\frac{-1.5}{\frac{\color{blue}{1 \cdot b}}{a \cdot c}}}{1 \cdot a}}{1 \cdot 3}\]

   9. Applied times-frac 16.6

      \[\leadsto \frac{\frac{\frac{-1.5}{\color{blue}{\frac{1}{a} \cdot \frac{b}{c}}}}{1 \cdot a}}{1 \cdot 3}\]

   10. Applied *-un-lft-identity 16.6

       \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot -1.5}}{\frac{1}{a} \cdot \frac{b}{c}}}{1 \cdot a}}{1 \cdot 3}\]

   11. Applied times-frac 16.6

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{1}{a}} \cdot \frac{-1.5}{\frac{b}{c}}}}{1 \cdot a}}{1 \cdot 3}\]

   12. Applied times-frac 16.6

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\frac{1}{a}}}{1} \cdot \frac{\frac{-1.5}{\frac{b}{c}}}{a}}}{1 \cdot 3}\]

   13. Applied times-frac 16.6

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\frac{1}{a}}}{1}}{1} \cdot \frac{\frac{\frac{-1.5}{\frac{b}{c}}}{a}}{3}}\]

   14. Simplified 16.6

       \[\leadsto \color{blue}{a} \cdot \frac{\frac{\frac{-1.5}{\frac{b}{c}}}{a}}{3}\]

   15. Taylor expanded around 0 16.5

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]

   16. Simplified 16.5

       \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 315.4848387613182580935244914144277572632:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))