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

Average Error: 5.6 → 1.4

Time: 28.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le -6.642944391539530496197547352303770744813 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(y \cdot \left(\left(t \cdot x\right) \cdot z\right)\right) \cdot 18 - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 1.953937894704433840283649332874256587502 \cdot 10^{253}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(y \cdot \left(\left(t \cdot x\right) \cdot z\right)\right) \cdot 18 - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r28240989 = x;
        double r28240990 = 18.0;
        double r28240991 = r28240989 * r28240990;
        double r28240992 = y;
        double r28240993 = r28240991 * r28240992;
        double r28240994 = z;
        double r28240995 = r28240993 * r28240994;
        double r28240996 = t;
        double r28240997 = r28240995 * r28240996;
        double r28240998 = a;
        double r28240999 = 4.0;
        double r28241000 = r28240998 * r28240999;
        double r28241001 = r28241000 * r28240996;
        double r28241002 = r28240997 - r28241001;
        double r28241003 = b;
        double r28241004 = c;
        double r28241005 = r28241003 * r28241004;
        double r28241006 = r28241002 + r28241005;
        double r28241007 = r28240989 * r28240999;
        double r28241008 = i;
        double r28241009 = r28241007 * r28241008;
        double r28241010 = r28241006 - r28241009;
        double r28241011 = j;
        double r28241012 = 27.0;
        double r28241013 = r28241011 * r28241012;
        double r28241014 = k;
        double r28241015 = r28241013 * r28241014;
        double r28241016 = r28241010 - r28241015;
        return r28241016;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r28241017 = t;
        double r28241018 = x;
        double r28241019 = 18.0;
        double r28241020 = r28241018 * r28241019;
        double r28241021 = y;
        double r28241022 = r28241020 * r28241021;
        double r28241023 = z;
        double r28241024 = r28241022 * r28241023;
        double r28241025 = r28241017 * r28241024;
        double r28241026 = a;
        double r28241027 = 4.0;
        double r28241028 = r28241026 * r28241027;
        double r28241029 = r28241028 * r28241017;
        double r28241030 = r28241025 - r28241029;
        double r28241031 = c;
        double r28241032 = b;
        double r28241033 = r28241031 * r28241032;
        double r28241034 = r28241030 + r28241033;
        double r28241035 = r28241018 * r28241027;
        double r28241036 = i;
        double r28241037 = r28241035 * r28241036;
        double r28241038 = r28241034 - r28241037;
        double r28241039 = -6.6429443915395305e+305;
        bool r28241040 = r28241038 <= r28241039;
        double r28241041 = r28241017 * r28241018;
        double r28241042 = r28241041 * r28241023;
        double r28241043 = r28241021 * r28241042;
        double r28241044 = r28241043 * r28241019;
        double r28241045 = r28241018 * r28241036;
        double r28241046 = fma(r28241017, r28241026, r28241045);
        double r28241047 = 27.0;
        double r28241048 = k;
        double r28241049 = j;
        double r28241050 = r28241048 * r28241049;
        double r28241051 = r28241047 * r28241050;
        double r28241052 = fma(r28241027, r28241046, r28241051);
        double r28241053 = r28241044 - r28241052;
        double r28241054 = fma(r28241032, r28241031, r28241053);
        double r28241055 = 1.953937894704434e+253;
        bool r28241056 = r28241038 <= r28241055;
        double r28241057 = r28241049 * r28241047;
        double r28241058 = r28241057 * r28241048;
        double r28241059 = r28241038 - r28241058;
        double r28241060 = r28241056 ? r28241059 : r28241054;
        double r28241061 = r28241040 ? r28241054 : r28241060;
        return r28241061;
}

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	5.6
Target	1.5
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -6.6429443915395305e+305 or 1.953937894704434e+253 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 32.5

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

   2. Simplified 9.4

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

   4. Applied associate-*r* 7.0

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

   6. Applied associate-*r* 7.0

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

   if -6.6429443915395305e+305 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 1.953937894704434e+253

   1. Initial program 0.4

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le -6.642944391539530496197547352303770744813 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(y \cdot \left(\left(t \cdot x\right) \cdot z\right)\right) \cdot 18 - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 1.953937894704433840283649332874256587502 \cdot 10^{253}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(y \cdot \left(\left(t \cdot x\right) \cdot z\right)\right) \cdot 18 - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))