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

Average Error: 5.6 → 5.6

Time: 1.5m

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r604372 = x;
        double r604373 = 18.0;
        double r604374 = r604372 * r604373;
        double r604375 = y;
        double r604376 = r604374 * r604375;
        double r604377 = z;
        double r604378 = r604376 * r604377;
        double r604379 = t;
        double r604380 = r604378 * r604379;
        double r604381 = a;
        double r604382 = 4.0;
        double r604383 = r604381 * r604382;
        double r604384 = r604383 * r604379;
        double r604385 = r604380 - r604384;
        double r604386 = b;
        double r604387 = c;
        double r604388 = r604386 * r604387;
        double r604389 = r604385 + r604388;
        double r604390 = r604372 * r604382;
        double r604391 = i;
        double r604392 = r604390 * r604391;
        double r604393 = r604389 - r604392;
        double r604394 = j;
        double r604395 = 27.0;
        double r604396 = r604394 * r604395;
        double r604397 = k;
        double r604398 = r604396 * r604397;
        double r604399 = r604393 - r604398;
        return r604399;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r604400 = t;
        double r604401 = x;
        double r604402 = 18.0;
        double r604403 = r604401 * r604402;
        double r604404 = y;
        double r604405 = r604403 * r604404;
        double r604406 = z;
        double r604407 = r604405 * r604406;
        double r604408 = a;
        double r604409 = 4.0;
        double r604410 = r604408 * r604409;
        double r604411 = r604407 - r604410;
        double r604412 = r604400 * r604411;
        double r604413 = b;
        double r604414 = c;
        double r604415 = r604413 * r604414;
        double r604416 = r604401 * r604409;
        double r604417 = i;
        double r604418 = r604416 * r604417;
        double r604419 = r604415 - r604418;
        double r604420 = r604412 + r604419;
        double r604421 = j;
        double r604422 = 27.0;
        double r604423 = k;
        double r604424 = r604422 * r604423;
        double r604425 = r604421 * r604424;
        double r604426 = r604420 - r604425;
        return r604426;
}

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	5.6

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

Derivation

1. Split input into 3 regimes

2. if t < -4.3830139062668545e+58

   1. Initial program 1.4

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

   3. Applied associate-*l* 1.3

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

   if -4.3830139062668545e+58 < t < 3.6462011735036985e+44

   1. Initial program 7.2

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

   3. Applied associate-*l* 4.4

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

   if 3.6462011735036985e+44 < t

   1. Initial program 1.8

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

   3. Applied associate-*l* 2.0

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

4. Final simplification 5.6

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

Reproduce

herbie shell --seed 2019303 
(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 165.680279438052224) (+ (- (* (* 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)))