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

Average Error: 5.8 → 2.0

Time: 22.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.433898367772061 \cdot 10^{-33}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18.0 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;t \le 7.338288124293042 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(z \cdot \left(x \cdot t\right)\right) \cdot y\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18.0 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r37518377 = x;
        double r37518378 = 18.0;
        double r37518379 = r37518377 * r37518378;
        double r37518380 = y;
        double r37518381 = r37518379 * r37518380;
        double r37518382 = z;
        double r37518383 = r37518381 * r37518382;
        double r37518384 = t;
        double r37518385 = r37518383 * r37518384;
        double r37518386 = a;
        double r37518387 = 4.0;
        double r37518388 = r37518386 * r37518387;
        double r37518389 = r37518388 * r37518384;
        double r37518390 = r37518385 - r37518389;
        double r37518391 = b;
        double r37518392 = c;
        double r37518393 = r37518391 * r37518392;
        double r37518394 = r37518390 + r37518393;
        double r37518395 = r37518377 * r37518387;
        double r37518396 = i;
        double r37518397 = r37518395 * r37518396;
        double r37518398 = r37518394 - r37518397;
        double r37518399 = j;
        double r37518400 = 27.0;
        double r37518401 = r37518399 * r37518400;
        double r37518402 = k;
        double r37518403 = r37518401 * r37518402;
        double r37518404 = r37518398 - r37518403;
        return r37518404;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r37518405 = t;
        double r37518406 = -5.433898367772061e-33;
        bool r37518407 = r37518405 <= r37518406;
        double r37518408 = b;
        double r37518409 = c;
        double r37518410 = r37518408 * r37518409;
        double r37518411 = 18.0;
        double r37518412 = z;
        double r37518413 = y;
        double r37518414 = r37518412 * r37518413;
        double r37518415 = x;
        double r37518416 = r37518414 * r37518415;
        double r37518417 = r37518405 * r37518416;
        double r37518418 = r37518411 * r37518417;
        double r37518419 = a;
        double r37518420 = 4.0;
        double r37518421 = r37518419 * r37518420;
        double r37518422 = r37518421 * r37518405;
        double r37518423 = r37518418 - r37518422;
        double r37518424 = r37518410 + r37518423;
        double r37518425 = r37518415 * r37518420;
        double r37518426 = i;
        double r37518427 = r37518425 * r37518426;
        double r37518428 = r37518424 - r37518427;
        double r37518429 = 27.0;
        double r37518430 = j;
        double r37518431 = k;
        double r37518432 = r37518430 * r37518431;
        double r37518433 = r37518429 * r37518432;
        double r37518434 = r37518428 - r37518433;
        double r37518435 = 7.338288124293042e+80;
        bool r37518436 = r37518405 <= r37518435;
        double r37518437 = r37518415 * r37518405;
        double r37518438 = r37518412 * r37518437;
        double r37518439 = r37518438 * r37518413;
        double r37518440 = r37518439 * r37518411;
        double r37518441 = r37518440 - r37518422;
        double r37518442 = r37518410 + r37518441;
        double r37518443 = r37518442 - r37518427;
        double r37518444 = r37518429 * r37518430;
        double r37518445 = r37518444 * r37518431;
        double r37518446 = r37518443 - r37518445;
        double r37518447 = r37518436 ? r37518446 : r37518434;
        double r37518448 = r37518407 ? r37518434 : r37518447;
        return r37518448;
}

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	5.8
Target	1.7
Herbie	2.0

\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.68027943805222:\\ \;\;\;\;\left(\left(18.0 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) + \left(c \cdot b - 27.0 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -5.433898367772061e-33 or 7.338288124293042e+80 < t

   1. Initial program 2.2

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

   2. Taylor expanded around inf 2.1

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

   3. Taylor expanded around 0 2.0

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

   if -5.433898367772061e-33 < t < 7.338288124293042e+80

   1. Initial program 7.6

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

   2. Taylor expanded around inf 8.5

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

   4. Applied associate-*r* 5.9

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

   6. Applied associate-*r* 2.0

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.433898367772061 \cdot 10^{-33}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18.0 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;t \le 7.338288124293042 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(z \cdot \left(x \cdot t\right)\right) \cdot y\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18.0 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))