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

Average Error: 5.4 → 3.9

Time: 17.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.6487252448166965 \cdot 10^{-103} \lor \neg \left(z \le 2.96658959257465003 \cdot 10^{-73}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, {\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r1054979 = x;
        double r1054980 = 18.0;
        double r1054981 = r1054979 * r1054980;
        double r1054982 = y;
        double r1054983 = r1054981 * r1054982;
        double r1054984 = z;
        double r1054985 = r1054983 * r1054984;
        double r1054986 = t;
        double r1054987 = r1054985 * r1054986;
        double r1054988 = a;
        double r1054989 = 4.0;
        double r1054990 = r1054988 * r1054989;
        double r1054991 = r1054990 * r1054986;
        double r1054992 = r1054987 - r1054991;
        double r1054993 = b;
        double r1054994 = c;
        double r1054995 = r1054993 * r1054994;
        double r1054996 = r1054992 + r1054995;
        double r1054997 = r1054979 * r1054989;
        double r1054998 = i;
        double r1054999 = r1054997 * r1054998;
        double r1055000 = r1054996 - r1054999;
        double r1055001 = j;
        double r1055002 = 27.0;
        double r1055003 = r1055001 * r1055002;
        double r1055004 = k;
        double r1055005 = r1055003 * r1055004;
        double r1055006 = r1055000 - r1055005;
        return r1055006;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r1055007 = z;
        double r1055008 = -3.6487252448166965e-103;
        bool r1055009 = r1055007 <= r1055008;
        double r1055010 = 2.96658959257465e-73;
        bool r1055011 = r1055007 <= r1055010;
        double r1055012 = !r1055011;
        bool r1055013 = r1055009 || r1055012;
        double r1055014 = t;
        double r1055015 = x;
        double r1055016 = 18.0;
        double r1055017 = r1055015 * r1055016;
        double r1055018 = y;
        double r1055019 = r1055017 * r1055018;
        double r1055020 = r1055019 * r1055007;
        double r1055021 = a;
        double r1055022 = 4.0;
        double r1055023 = r1055021 * r1055022;
        double r1055024 = r1055020 - r1055023;
        double r1055025 = b;
        double r1055026 = c;
        double r1055027 = r1055025 * r1055026;
        double r1055028 = i;
        double r1055029 = r1055022 * r1055028;
        double r1055030 = j;
        double r1055031 = 27.0;
        double r1055032 = k;
        double r1055033 = r1055031 * r1055032;
        double r1055034 = r1055030 * r1055033;
        double r1055035 = fma(r1055015, r1055029, r1055034);
        double r1055036 = r1055027 - r1055035;
        double r1055037 = fma(r1055014, r1055024, r1055036);
        double r1055038 = r1055007 * r1055018;
        double r1055039 = r1055015 * r1055038;
        double r1055040 = r1055016 * r1055039;
        double r1055041 = 1.0;
        double r1055042 = pow(r1055040, r1055041);
        double r1055043 = r1055042 - r1055023;
        double r1055044 = r1055030 * r1055031;
        double r1055045 = cbrt(r1055032);
        double r1055046 = r1055045 * r1055045;
        double r1055047 = r1055044 * r1055046;
        double r1055048 = r1055047 * r1055045;
        double r1055049 = fma(r1055015, r1055029, r1055048);
        double r1055050 = r1055027 - r1055049;
        double r1055051 = fma(r1055014, r1055043, r1055050);
        double r1055052 = r1055013 ? r1055037 : r1055051;
        return r1055052;
}

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	1.6
Herbie	3.9

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

Derivation

1. Split input into 2 regimes

2. if z < -3.6487252448166965e-103 or 2.96658959257465e-73 < z

   1. Initial program 5.7

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

   2. Simplified 5.7

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

   4. Applied associate-*l* 5.7

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

   if -3.6487252448166965e-103 < z < 2.96658959257465e-73

   1. Initial program 4.9

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

   2. Simplified 5.0

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

   4. Applied pow1 5.0

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

   5. Applied pow1 5.0

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

   6. Applied pow1 5.0

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

   7. Applied pow1 5.0

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

   8. Applied pow-prod-down 5.0

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

   9. Applied pow-prod-down 5.0

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

   10. Applied pow-prod-down 5.0

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

   11. Simplified 1.0

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

   13. Applied add-cube-cbrt 1.2

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

   14. Applied associate-*r* 1.2

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

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.6487252448166965 \cdot 10^{-103} \lor \neg \left(z \le 2.96658959257465003 \cdot 10^{-73}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, {\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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)))