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

Average Error: 5.1 → 1.6

Time: 53.9s

Precision: 64

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 1.070131500677526 \cdot 10^{+283}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27.0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r19801383 = x;
        double r19801384 = 18.0;
        double r19801385 = r19801383 * r19801384;
        double r19801386 = y;
        double r19801387 = r19801385 * r19801386;
        double r19801388 = z;
        double r19801389 = r19801387 * r19801388;
        double r19801390 = t;
        double r19801391 = r19801389 * r19801390;
        double r19801392 = a;
        double r19801393 = 4.0;
        double r19801394 = r19801392 * r19801393;
        double r19801395 = r19801394 * r19801390;
        double r19801396 = r19801391 - r19801395;
        double r19801397 = b;
        double r19801398 = c;
        double r19801399 = r19801397 * r19801398;
        double r19801400 = r19801396 + r19801399;
        double r19801401 = r19801383 * r19801393;
        double r19801402 = i;
        double r19801403 = r19801401 * r19801402;
        double r19801404 = r19801400 - r19801403;
        double r19801405 = j;
        double r19801406 = 27.0;
        double r19801407 = r19801405 * r19801406;
        double r19801408 = k;
        double r19801409 = r19801407 * r19801408;
        double r19801410 = r19801404 - r19801409;
        return r19801410;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r19801411 = t;
        double r19801412 = x;
        double r19801413 = 18.0;
        double r19801414 = r19801412 * r19801413;
        double r19801415 = y;
        double r19801416 = r19801414 * r19801415;
        double r19801417 = z;
        double r19801418 = r19801416 * r19801417;
        double r19801419 = r19801411 * r19801418;
        double r19801420 = a;
        double r19801421 = 4.0;
        double r19801422 = r19801420 * r19801421;
        double r19801423 = r19801422 * r19801411;
        double r19801424 = r19801419 - r19801423;
        double r19801425 = c;
        double r19801426 = b;
        double r19801427 = r19801425 * r19801426;
        double r19801428 = r19801424 + r19801427;
        double r19801429 = r19801412 * r19801421;
        double r19801430 = i;
        double r19801431 = r19801429 * r19801430;
        double r19801432 = r19801428 - r19801431;
        double r19801433 = -inf.0;
        bool r19801434 = r19801432 <= r19801433;
        double r19801435 = r19801417 * r19801412;
        double r19801436 = r19801435 * r19801411;
        double r19801437 = r19801415 * r19801436;
        double r19801438 = r19801437 * r19801413;
        double r19801439 = r19801438 - r19801423;
        double r19801440 = r19801427 + r19801439;
        double r19801441 = r19801440 - r19801431;
        double r19801442 = j;
        double r19801443 = 27.0;
        double r19801444 = k;
        double r19801445 = r19801443 * r19801444;
        double r19801446 = r19801442 * r19801445;
        double r19801447 = r19801441 - r19801446;
        double r19801448 = 1.070131500677526e+283;
        bool r19801449 = r19801432 <= r19801448;
        double r19801450 = r19801442 * r19801444;
        double r19801451 = r19801450 * r19801443;
        double r19801452 = r19801432 - r19801451;
        double r19801453 = r19801449 ? r19801452 : r19801447;
        double r19801454 = r19801434 ? r19801447 : r19801453;
        return r19801454;
}

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

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 1.070131500677526e+283 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 41.7

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

   2. Taylor expanded around inf 27.4

      \[\leadsto \left(\left(\left(\color{blue}{18.0 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
   3. Using strategy rm

   4. Applied associate-*r* 27.5

      \[\leadsto \left(\left(\left(18.0 \cdot \left(t \cdot \color{blue}{\left(\left(x \cdot z\right) \cdot y\right)}\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
   5. Using strategy rm

   6. Applied associate-*l* 27.5

      \[\leadsto \left(\left(\left(18.0 \cdot \left(t \cdot \left(\left(x \cdot z\right) \cdot y\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{j \cdot \left(27.0 \cdot k\right)}\]
   7. Using strategy rm

   8. Applied associate-*r* 11.6

      \[\leadsto \left(\left(\left(18.0 \cdot \color{blue}{\left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\]

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 1.070131500677526e+283

   1. Initial program 0.3

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

   2. Taylor expanded around 0 0.2

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{27.0 \cdot \left(j \cdot k\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 1.070131500677526 \cdot 10^{+283}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27.0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))