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

Average Error: 5.5 → 4.6

Time: 7.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.82739249994089036 \cdot 10^{-158} \lor \neg \left(t \le 4.1450304107497156 \cdot 10^{-137}\right):\\ \;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{\left(j \cdot 27\right) \cdot k} \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\left(j \cdot 27\right) \cdot k}} \cdot \sqrt[3]{\sqrt[3]{\left(j \cdot 27\right) \cdot k}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(j \cdot 27\right) \cdot k}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{\left(j \cdot 27\right) \cdot k} \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right) \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r866945 = x;
        double r866946 = 18.0;
        double r866947 = r866945 * r866946;
        double r866948 = y;
        double r866949 = r866947 * r866948;
        double r866950 = z;
        double r866951 = r866949 * r866950;
        double r866952 = t;
        double r866953 = r866951 * r866952;
        double r866954 = a;
        double r866955 = 4.0;
        double r866956 = r866954 * r866955;
        double r866957 = r866956 * r866952;
        double r866958 = r866953 - r866957;
        double r866959 = b;
        double r866960 = c;
        double r866961 = r866959 * r866960;
        double r866962 = r866958 + r866961;
        double r866963 = r866945 * r866955;
        double r866964 = i;
        double r866965 = r866963 * r866964;
        double r866966 = r866962 - r866965;
        double r866967 = j;
        double r866968 = 27.0;
        double r866969 = r866967 * r866968;
        double r866970 = k;
        double r866971 = r866969 * r866970;
        double r866972 = r866966 - r866971;
        return r866972;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r866973 = t;
        double r866974 = -2.8273924999408904e-158;
        bool r866975 = r866973 <= r866974;
        double r866976 = 4.1450304107497156e-137;
        bool r866977 = r866973 <= r866976;
        double r866978 = !r866977;
        bool r866979 = r866975 || r866978;
        double r866980 = x;
        double r866981 = 18.0;
        double r866982 = r866980 * r866981;
        double r866983 = y;
        double r866984 = r866982 * r866983;
        double r866985 = z;
        double r866986 = r866984 * r866985;
        double r866987 = a;
        double r866988 = 4.0;
        double r866989 = r866987 * r866988;
        double r866990 = r866986 - r866989;
        double r866991 = r866973 * r866990;
        double r866992 = b;
        double r866993 = c;
        double r866994 = r866992 * r866993;
        double r866995 = r866980 * r866988;
        double r866996 = i;
        double r866997 = r866995 * r866996;
        double r866998 = j;
        double r866999 = 27.0;
        double r867000 = r866998 * r866999;
        double r867001 = k;
        double r867002 = r867000 * r867001;
        double r867003 = cbrt(r867002);
        double r867004 = r867003 * r867003;
        double r867005 = cbrt(r867003);
        double r867006 = r867005 * r867005;
        double r867007 = r867006 * r867005;
        double r867008 = r867004 * r867007;
        double r867009 = r866997 + r867008;
        double r867010 = r866994 - r867009;
        double r867011 = r866991 + r867010;
        double r867012 = 0.0;
        double r867013 = r867012 - r866989;
        double r867014 = r866973 * r867013;
        double r867015 = r867004 * r867003;
        double r867016 = r866997 + r867015;
        double r867017 = r866994 - r867016;
        double r867018 = r867014 + r867017;
        double r867019 = r866979 ? r867011 : r867018;
        return r867019;
}

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	1.3
Herbie	4.6

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

Derivation

1. Split input into 2 regimes

2. if t < -2.8273924999408904e-158 or 4.1450304107497156e-137 < t

   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. Simplified 3.5

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

   4. Applied add-cube-cbrt 3.7

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

   6. Applied add-cube-cbrt 3.8

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

   if -2.8273924999408904e-158 < t < 4.1450304107497156e-137

   1. Initial program 9.5

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

   2. Simplified 9.5

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

   4. Applied add-cube-cbrt 9.8

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

   5. Taylor expanded around 0 6.3

      \[\leadsto t \cdot \left(\color{blue}{0} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{\left(j \cdot 27\right) \cdot k} \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right) \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.82739249994089036 \cdot 10^{-158} \lor \neg \left(t \le 4.1450304107497156 \cdot 10^{-137}\right):\\ \;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{\left(j \cdot 27\right) \cdot k} \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\left(j \cdot 27\right) \cdot k}} \cdot \sqrt[3]{\sqrt[3]{\left(j \cdot 27\right) \cdot k}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(j \cdot 27\right) \cdot k}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{\left(j \cdot 27\right) \cdot k} \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right) \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))