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

Average Error: 5.9 → 6.2

Time: 12.9s

Precision: 64

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.6574930852975223 \cdot 10^{154} \lor \neg \left(x \le 7.5134315393536342 \cdot 10^{-290}\right):\\ \;\;\;\;\mathsf{fma}\left(t, {\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r780934 = x;
        double r780935 = 18.0;
        double r780936 = r780934 * r780935;
        double r780937 = y;
        double r780938 = r780936 * r780937;
        double r780939 = z;
        double r780940 = r780938 * r780939;
        double r780941 = t;
        double r780942 = r780940 * r780941;
        double r780943 = a;
        double r780944 = 4.0;
        double r780945 = r780943 * r780944;
        double r780946 = r780945 * r780941;
        double r780947 = r780942 - r780946;
        double r780948 = b;
        double r780949 = c;
        double r780950 = r780948 * r780949;
        double r780951 = r780947 + r780950;
        double r780952 = r780934 * r780944;
        double r780953 = i;
        double r780954 = r780952 * r780953;
        double r780955 = r780951 - r780954;
        double r780956 = j;
        double r780957 = 27.0;
        double r780958 = r780956 * r780957;
        double r780959 = k;
        double r780960 = r780958 * r780959;
        double r780961 = r780955 - r780960;
        return r780961;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r780962 = x;
        double r780963 = -1.6574930852975223e+154;
        bool r780964 = r780962 <= r780963;
        double r780965 = 7.513431539353634e-290;
        bool r780966 = r780962 <= r780965;
        double r780967 = !r780966;
        bool r780968 = r780964 || r780967;
        double r780969 = t;
        double r780970 = 18.0;
        double r780971 = z;
        double r780972 = y;
        double r780973 = r780971 * r780972;
        double r780974 = r780962 * r780973;
        double r780975 = r780970 * r780974;
        double r780976 = 1.0;
        double r780977 = pow(r780975, r780976);
        double r780978 = a;
        double r780979 = 4.0;
        double r780980 = r780978 * r780979;
        double r780981 = r780977 - r780980;
        double r780982 = b;
        double r780983 = c;
        double r780984 = r780982 * r780983;
        double r780985 = i;
        double r780986 = r780979 * r780985;
        double r780987 = j;
        double r780988 = 27.0;
        double r780989 = r780987 * r780988;
        double r780990 = k;
        double r780991 = r780989 * r780990;
        double r780992 = fma(r780962, r780986, r780991);
        double r780993 = r780984 - r780992;
        double r780994 = fma(r780969, r780981, r780993);
        double r780995 = r780962 * r780970;
        double r780996 = r780995 * r780972;
        double r780997 = r780971 * r780969;
        double r780998 = r780996 * r780997;
        double r780999 = r780980 * r780969;
        double r781000 = r780998 - r780999;
        double r781001 = r781000 + r780984;
        double r781002 = r780962 * r780979;
        double r781003 = r781002 * r780985;
        double r781004 = r781001 - r781003;
        double r781005 = r781004 - r780991;
        double r781006 = r780968 ? r780994 : r781005;
        return r781006;
}

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

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Target

Original	5.9
Target	1.7
Herbie	6.2

\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -1.6574930852975223e+154 or 7.513431539353634e-290 < x

   1. Initial program 7.9

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   2. Simplified 7.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
   3. Using strategy rm

   4. Applied pow1 7.9

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot \color{blue}{{z}^{1}} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

   5. Applied pow1 7.9

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot \color{blue}{{y}^{1}}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

   6. Applied pow1 7.9

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot \color{blue}{{18}^{1}}\right) \cdot {y}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

   7. Applied pow1 7.9

      \[\leadsto \mathsf{fma}\left(t, \left(\left(\color{blue}{{x}^{1}} \cdot {18}^{1}\right) \cdot {y}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

   8. Applied pow-prod-down 7.9

      \[\leadsto \mathsf{fma}\left(t, \left(\color{blue}{{\left(x \cdot 18\right)}^{1}} \cdot {y}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

   9. Applied pow-prod-down 7.9

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{{\left(\left(x \cdot 18\right) \cdot y\right)}^{1}} \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

   10. Applied pow-prod-down 7.9

       \[\leadsto \mathsf{fma}\left(t, \color{blue}{{\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right)}^{1}} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

   11. Simplified 7.1

       \[\leadsto \mathsf{fma}\left(t, {\color{blue}{\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

   if -1.6574930852975223e+154 < x < 7.513431539353634e-290

   1. Initial program 3.5

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
   2. Using strategy rm

   3. Applied associate-*l* 5.1

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6574930852975223 \cdot 10^{154} \lor \neg \left(x \le 7.5134315393536342 \cdot 10^{-290}\right):\\ \;\;\;\;\mathsf{fma}\left(t, {\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))