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

Average Error: 5.4 → 3.7

Time: 15.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.15846637414394352 \cdot 10^{131} \lor \neg \left(z \le 3.14022389051985691 \cdot 10^{43}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, {\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r868983 = x;
        double r868984 = 18.0;
        double r868985 = r868983 * r868984;
        double r868986 = y;
        double r868987 = r868985 * r868986;
        double r868988 = z;
        double r868989 = r868987 * r868988;
        double r868990 = t;
        double r868991 = r868989 * r868990;
        double r868992 = a;
        double r868993 = 4.0;
        double r868994 = r868992 * r868993;
        double r868995 = r868994 * r868990;
        double r868996 = r868991 - r868995;
        double r868997 = b;
        double r868998 = c;
        double r868999 = r868997 * r868998;
        double r869000 = r868996 + r868999;
        double r869001 = r868983 * r868993;
        double r869002 = i;
        double r869003 = r869001 * r869002;
        double r869004 = r869000 - r869003;
        double r869005 = j;
        double r869006 = 27.0;
        double r869007 = r869005 * r869006;
        double r869008 = k;
        double r869009 = r869007 * r869008;
        double r869010 = r869004 - r869009;
        return r869010;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r869011 = z;
        double r869012 = -4.1584663741439435e+131;
        bool r869013 = r869011 <= r869012;
        double r869014 = 3.140223890519857e+43;
        bool r869015 = r869011 <= r869014;
        double r869016 = !r869015;
        bool r869017 = r869013 || r869016;
        double r869018 = t;
        double r869019 = x;
        double r869020 = 18.0;
        double r869021 = r869019 * r869020;
        double r869022 = y;
        double r869023 = r869021 * r869022;
        double r869024 = r869023 * r869011;
        double r869025 = a;
        double r869026 = 4.0;
        double r869027 = r869025 * r869026;
        double r869028 = r869024 - r869027;
        double r869029 = b;
        double r869030 = c;
        double r869031 = r869029 * r869030;
        double r869032 = i;
        double r869033 = r869026 * r869032;
        double r869034 = j;
        double r869035 = 27.0;
        double r869036 = k;
        double r869037 = r869035 * r869036;
        double r869038 = r869034 * r869037;
        double r869039 = fma(r869019, r869033, r869038);
        double r869040 = r869031 - r869039;
        double r869041 = fma(r869018, r869028, r869040);
        double r869042 = r869011 * r869022;
        double r869043 = r869019 * r869042;
        double r869044 = r869020 * r869043;
        double r869045 = 1.0;
        double r869046 = pow(r869044, r869045);
        double r869047 = r869046 - r869027;
        double r869048 = r869034 * r869035;
        double r869049 = r869048 * r869036;
        double r869050 = fma(r869019, r869033, r869049);
        double r869051 = r869031 - r869050;
        double r869052 = fma(r869018, r869047, r869051);
        double r869053 = r869017 ? r869041 : r869052;
        return r869053;
}

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	5.4
Target	1.7
Herbie	3.7

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

Derivation

1. Split input into 2 regimes

2. if z < -4.1584663741439435e+131 or 3.140223890519857e+43 < z

   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 associate-*l* 8.0

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

   if -4.1584663741439435e+131 < z < 3.140223890519857e+43

   1. Initial program 4.4

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

   2. Simplified 4.4

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

   4. Applied pow1 4.4

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

   5. Applied pow1 4.4

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

   6. Applied pow1 4.4

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

   7. Applied pow1 4.4

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

   8. Applied pow-prod-down 4.4

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

   9. Applied pow-prod-down 4.4

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

   10. Applied pow-prod-down 4.4

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

   11. Simplified 1.9

       \[\leadsto \mathsf{fma}\left(t, {\color{blue}{\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.15846637414394352 \cdot 10^{131} \lor \neg \left(z \le 3.14022389051985691 \cdot 10^{43}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, {\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))