Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, A

Average Error: 3.7 → 0.4

Time: 19.5s

Precision: 64

Math

\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t = -\infty \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le 4.296785106576348656228515128564970887864 \cdot 10^{306}\right):\\ \;\;\;\;\left(x \cdot 2 - \left(\left(z \cdot 9\right) \cdot t\right) \cdot y\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot b\right) \cdot 27\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r572959 = x;
        double r572960 = 2.0;
        double r572961 = r572959 * r572960;
        double r572962 = y;
        double r572963 = 9.0;
        double r572964 = r572962 * r572963;
        double r572965 = z;
        double r572966 = r572964 * r572965;
        double r572967 = t;
        double r572968 = r572966 * r572967;
        double r572969 = r572961 - r572968;
        double r572970 = a;
        double r572971 = 27.0;
        double r572972 = r572970 * r572971;
        double r572973 = b;
        double r572974 = r572972 * r572973;
        double r572975 = r572969 + r572974;
        return r572975;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r572976 = y;
        double r572977 = 9.0;
        double r572978 = r572976 * r572977;
        double r572979 = z;
        double r572980 = r572978 * r572979;
        double r572981 = t;
        double r572982 = r572980 * r572981;
        double r572983 = -inf.0;
        bool r572984 = r572982 <= r572983;
        double r572985 = 4.2967851065763487e+306;
        bool r572986 = r572982 <= r572985;
        double r572987 = !r572986;
        bool r572988 = r572984 || r572987;
        double r572989 = x;
        double r572990 = 2.0;
        double r572991 = r572989 * r572990;
        double r572992 = r572979 * r572977;
        double r572993 = r572992 * r572981;
        double r572994 = r572993 * r572976;
        double r572995 = r572991 - r572994;
        double r572996 = a;
        double r572997 = 27.0;
        double r572998 = r572996 * r572997;
        double r572999 = b;
        double r573000 = r572998 * r572999;
        double r573001 = r572995 + r573000;
        double r573002 = r572991 - r572982;
        double r573003 = r572996 * r572999;
        double r573004 = r573003 * r572997;
        double r573005 = r573002 + r573004;
        double r573006 = r572988 ? r573001 : r573005;
        return r573006;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	3.7
Target	2.6
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811188954625810696587370427881 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* (* (* y 9.0) z) t) < -inf.0 or 4.2967851065763487e+306 < (* (* (* y 9.0) z) t)

   1. Initial program 63.4

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 62.6

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot \left(9 \cdot z\right)\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   4. Simplified 62.6

      \[\leadsto \left(x \cdot 2 - \left(y \cdot \color{blue}{\left(z \cdot 9\right)}\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   5. Using strategy rm

   6. Applied associate-*l* 0.8

      \[\leadsto \left(x \cdot 2 - \color{blue}{y \cdot \left(\left(z \cdot 9\right) \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

   7. Simplified 0.8

      \[\leadsto \left(x \cdot 2 - y \cdot \color{blue}{\left(\left(9 \cdot z\right) \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

   if -inf.0 < (* (* (* y 9.0) z) t) < 4.2967851065763487e+306

   1. Initial program 0.5

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   2. Taylor expanded around 0 0.4

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{27 \cdot \left(a \cdot b\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t = -\infty \lor \neg \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le 4.296785106576348656228515128564970887864 \cdot 10^{306}\right):\\ \;\;\;\;\left(x \cdot 2 - \left(\left(z \cdot 9\right) \cdot t\right) \cdot y\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot b\right) \cdot 27\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))