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

Average Error: 5.5 → 1.7

Time: 25.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.15848203748868583 \cdot 10^{-15} \lor \neg \left(x \le 4.75949776591493624 \cdot 10^{-71}\right):\\ \;\;\;\;\left(\left(\left(x \cdot \left(\left(18 \cdot y\right) \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sqrt[3]{t} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r753975 = x;
        double r753976 = 18.0;
        double r753977 = r753975 * r753976;
        double r753978 = y;
        double r753979 = r753977 * r753978;
        double r753980 = z;
        double r753981 = r753979 * r753980;
        double r753982 = t;
        double r753983 = r753981 * r753982;
        double r753984 = a;
        double r753985 = 4.0;
        double r753986 = r753984 * r753985;
        double r753987 = r753986 * r753982;
        double r753988 = r753983 - r753987;
        double r753989 = b;
        double r753990 = c;
        double r753991 = r753989 * r753990;
        double r753992 = r753988 + r753991;
        double r753993 = r753975 * r753985;
        double r753994 = i;
        double r753995 = r753993 * r753994;
        double r753996 = r753992 - r753995;
        double r753997 = j;
        double r753998 = 27.0;
        double r753999 = r753997 * r753998;
        double r754000 = k;
        double r754001 = r753999 * r754000;
        double r754002 = r753996 - r754001;
        return r754002;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r754003 = x;
        double r754004 = -1.1584820374886858e-15;
        bool r754005 = r754003 <= r754004;
        double r754006 = 4.759497765914936e-71;
        bool r754007 = r754003 <= r754006;
        double r754008 = !r754007;
        bool r754009 = r754005 || r754008;
        double r754010 = 18.0;
        double r754011 = y;
        double r754012 = r754010 * r754011;
        double r754013 = z;
        double r754014 = t;
        double r754015 = r754013 * r754014;
        double r754016 = r754012 * r754015;
        double r754017 = r754003 * r754016;
        double r754018 = a;
        double r754019 = 4.0;
        double r754020 = r754018 * r754019;
        double r754021 = r754020 * r754014;
        double r754022 = r754017 - r754021;
        double r754023 = b;
        double r754024 = c;
        double r754025 = r754023 * r754024;
        double r754026 = r754022 + r754025;
        double r754027 = r754003 * r754019;
        double r754028 = i;
        double r754029 = r754027 * r754028;
        double r754030 = r754026 - r754029;
        double r754031 = j;
        double r754032 = 27.0;
        double r754033 = r754031 * r754032;
        double r754034 = k;
        double r754035 = r754033 * r754034;
        double r754036 = r754030 - r754035;
        double r754037 = r754003 * r754010;
        double r754038 = r754037 * r754011;
        double r754039 = r754038 * r754013;
        double r754040 = cbrt(r754014);
        double r754041 = r754040 * r754040;
        double r754042 = r754039 * r754041;
        double r754043 = r754042 * r754040;
        double r754044 = r754043 - r754021;
        double r754045 = r754044 + r754025;
        double r754046 = r754045 - r754029;
        double r754047 = r754046 - r754035;
        double r754048 = r754009 ? r754036 : r754047;
        return r754048;
}

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	5.5
Target	1.7
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if x < -1.1584820374886858e-15 or 4.759497765914936e-71 < x

   1. Initial program 10.4

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

   3. Applied associate-*l* 7.5

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

   5. Applied associate-*l* 7.5

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

   7. Applied associate-*l* 1.9

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

   if -1.1584820374886858e-15 < x < 4.759497765914936e-71

   1. Initial program 1.5

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

   3. Applied add-cube-cbrt 1.6

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

   4. Applied associate-*r* 1.6

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.15848203748868583 \cdot 10^{-15} \lor \neg \left(x \le 4.75949776591493624 \cdot 10^{-71}\right):\\ \;\;\;\;\left(\left(\left(x \cdot \left(\left(18 \cdot y\right) \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sqrt[3]{t} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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)))