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

Average Error: 5.9 → 1.8

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 1.408347514255849598590216064760682864053 \cdot 10^{307}\right):\\ \;\;\;\;\left(\left(\left(x \cdot \left(\left(18 \cdot \left(y \cdot z\right)\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r740952 = x;
        double r740953 = 18.0;
        double r740954 = r740952 * r740953;
        double r740955 = y;
        double r740956 = r740954 * r740955;
        double r740957 = z;
        double r740958 = r740956 * r740957;
        double r740959 = t;
        double r740960 = r740958 * r740959;
        double r740961 = a;
        double r740962 = 4.0;
        double r740963 = r740961 * r740962;
        double r740964 = r740963 * r740959;
        double r740965 = r740960 - r740964;
        double r740966 = b;
        double r740967 = c;
        double r740968 = r740966 * r740967;
        double r740969 = r740965 + r740968;
        double r740970 = r740952 * r740962;
        double r740971 = i;
        double r740972 = r740970 * r740971;
        double r740973 = r740969 - r740972;
        double r740974 = j;
        double r740975 = 27.0;
        double r740976 = r740974 * r740975;
        double r740977 = k;
        double r740978 = r740976 * r740977;
        double r740979 = r740973 - r740978;
        return r740979;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r740980 = x;
        double r740981 = 18.0;
        double r740982 = r740980 * r740981;
        double r740983 = y;
        double r740984 = r740982 * r740983;
        double r740985 = z;
        double r740986 = r740984 * r740985;
        double r740987 = t;
        double r740988 = r740986 * r740987;
        double r740989 = a;
        double r740990 = 4.0;
        double r740991 = r740989 * r740990;
        double r740992 = r740991 * r740987;
        double r740993 = r740988 - r740992;
        double r740994 = b;
        double r740995 = c;
        double r740996 = r740994 * r740995;
        double r740997 = r740993 + r740996;
        double r740998 = r740980 * r740990;
        double r740999 = i;
        double r741000 = r740998 * r740999;
        double r741001 = r740997 - r741000;
        double r741002 = -inf.0;
        bool r741003 = r741001 <= r741002;
        double r741004 = 1.4083475142558496e+307;
        bool r741005 = r741001 <= r741004;
        double r741006 = !r741005;
        bool r741007 = r741003 || r741006;
        double r741008 = r740983 * r740985;
        double r741009 = r740981 * r741008;
        double r741010 = r741009 * r740987;
        double r741011 = r740980 * r741010;
        double r741012 = r741011 - r740992;
        double r741013 = r741012 + r740996;
        double r741014 = r741013 - r741000;
        double r741015 = j;
        double r741016 = 27.0;
        double r741017 = k;
        double r741018 = r741016 * r741017;
        double r741019 = r741015 * r741018;
        double r741020 = r741014 - r741019;
        double r741021 = r741001 - r741019;
        double r741022 = r741007 ? r741020 : r741021;
        return r741022;
}

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	5.9
Target	1.7
Herbie	1.8

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

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)) < -inf.0 or 1.4083475142558496e+307 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 63.0

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

   3. Applied associate-*l* 40.5

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

   5. Applied associate-*l* 40.2

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

   7. Applied associate-*l* 40.3

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

   9. Applied associate-*l* 16.8

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

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 1.4083475142558496e+307

   1. Initial program 0.3

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

   3. Applied associate-*l* 0.3

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 1.408347514255849598590216064760682864053 \cdot 10^{307}\right):\\ \;\;\;\;\left(\left(\left(x \cdot \left(\left(18 \cdot \left(y \cdot z\right)\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))