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

Average Error: 5.1 → 4.2

Time: 32.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.5364366178162196 \cdot 10^{-144}:\\ \;\;\;\;\left(\left(c \cdot b - \left(j \cdot k\right) \cdot 27.0\right) - \left(i \cdot x\right) \cdot 4.0\right) + t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot z\right) \cdot y - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le 6.983384062178601 \cdot 10^{-99}:\\ \;\;\;\;\left(-t \cdot \left(a \cdot 4.0\right)\right) + \left(\left(c \cdot b - \left(j \cdot k\right) \cdot 27.0\right) - \left(i \cdot x\right) \cdot 4.0\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(\left(y \cdot z\right) \cdot 18.0\right) \cdot x - a \cdot 4.0\right) + \left(\left(c \cdot b - \left(j \cdot k\right) \cdot 27.0\right) - \left(i \cdot x\right) \cdot 4.0\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r5826517 = x;
        double r5826518 = 18.0;
        double r5826519 = r5826517 * r5826518;
        double r5826520 = y;
        double r5826521 = r5826519 * r5826520;
        double r5826522 = z;
        double r5826523 = r5826521 * r5826522;
        double r5826524 = t;
        double r5826525 = r5826523 * r5826524;
        double r5826526 = a;
        double r5826527 = 4.0;
        double r5826528 = r5826526 * r5826527;
        double r5826529 = r5826528 * r5826524;
        double r5826530 = r5826525 - r5826529;
        double r5826531 = b;
        double r5826532 = c;
        double r5826533 = r5826531 * r5826532;
        double r5826534 = r5826530 + r5826533;
        double r5826535 = r5826517 * r5826527;
        double r5826536 = i;
        double r5826537 = r5826535 * r5826536;
        double r5826538 = r5826534 - r5826537;
        double r5826539 = j;
        double r5826540 = 27.0;
        double r5826541 = r5826539 * r5826540;
        double r5826542 = k;
        double r5826543 = r5826541 * r5826542;
        double r5826544 = r5826538 - r5826543;
        return r5826544;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r5826545 = t;
        double r5826546 = -4.5364366178162196e-144;
        bool r5826547 = r5826545 <= r5826546;
        double r5826548 = c;
        double r5826549 = b;
        double r5826550 = r5826548 * r5826549;
        double r5826551 = j;
        double r5826552 = k;
        double r5826553 = r5826551 * r5826552;
        double r5826554 = 27.0;
        double r5826555 = r5826553 * r5826554;
        double r5826556 = r5826550 - r5826555;
        double r5826557 = i;
        double r5826558 = x;
        double r5826559 = r5826557 * r5826558;
        double r5826560 = 4.0;
        double r5826561 = r5826559 * r5826560;
        double r5826562 = r5826556 - r5826561;
        double r5826563 = 18.0;
        double r5826564 = r5826558 * r5826563;
        double r5826565 = z;
        double r5826566 = r5826564 * r5826565;
        double r5826567 = y;
        double r5826568 = r5826566 * r5826567;
        double r5826569 = a;
        double r5826570 = r5826569 * r5826560;
        double r5826571 = r5826568 - r5826570;
        double r5826572 = r5826545 * r5826571;
        double r5826573 = r5826562 + r5826572;
        double r5826574 = 6.983384062178601e-99;
        bool r5826575 = r5826545 <= r5826574;
        double r5826576 = r5826545 * r5826570;
        double r5826577 = -r5826576;
        double r5826578 = r5826577 + r5826562;
        double r5826579 = r5826567 * r5826565;
        double r5826580 = r5826579 * r5826563;
        double r5826581 = r5826580 * r5826558;
        double r5826582 = r5826581 - r5826570;
        double r5826583 = r5826545 * r5826582;
        double r5826584 = r5826583 + r5826562;
        double r5826585 = r5826575 ? r5826578 : r5826584;
        double r5826586 = r5826547 ? r5826573 : r5826585;
        return r5826586;
}

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 3 regimes

2. if t < -4.5364366178162196e-144

   1. Initial program 2.9

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

   2. Simplified 3.8

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

   3. Taylor expanded around inf 3.7

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

   5. Applied *-un-lft-identity 3.7

      \[\leadsto \left(\left(c \cdot b - 27.0 \cdot \left(j \cdot k\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) + \left(\left(x \cdot 18.0\right) \cdot \left(y \cdot z\right) - a \cdot 4.0\right) \cdot \color{blue}{\left(1 \cdot t\right)}\]

   6. Applied associate-*r* 3.7

      \[\leadsto \left(\left(c \cdot b - 27.0 \cdot \left(j \cdot k\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) + \color{blue}{\left(\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot z\right) - a \cdot 4.0\right) \cdot 1\right) \cdot t}\]

   7. Simplified 3.7

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

   9. Applied *-un-lft-identity 3.7

      \[\leadsto \left(\left(c \cdot b - 27.0 \cdot \left(j \cdot k\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) + \left(x \cdot \left(\left(y \cdot z\right) \cdot 18.0\right) - 4.0 \cdot a\right) \cdot \color{blue}{\left(1 \cdot t\right)}\]

   10. Applied associate-*r* 3.7

       \[\leadsto \left(\left(c \cdot b - 27.0 \cdot \left(j \cdot k\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) + \color{blue}{\left(\left(x \cdot \left(\left(y \cdot z\right) \cdot 18.0\right) - 4.0 \cdot a\right) \cdot 1\right) \cdot t}\]

   11. Simplified 3.1

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

   if -4.5364366178162196e-144 < t < 6.983384062178601e-99

   1. Initial program 8.7

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

   2. Simplified 9.5

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

   3. Taylor expanded around inf 9.4

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

   5. Applied *-un-lft-identity 9.4

      \[\leadsto \left(\left(c \cdot b - 27.0 \cdot \left(j \cdot k\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) + \left(\left(x \cdot 18.0\right) \cdot \left(y \cdot z\right) - a \cdot 4.0\right) \cdot \color{blue}{\left(1 \cdot t\right)}\]

   6. Applied associate-*r* 9.4

      \[\leadsto \left(\left(c \cdot b - 27.0 \cdot \left(j \cdot k\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) + \color{blue}{\left(\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot z\right) - a \cdot 4.0\right) \cdot 1\right) \cdot t}\]

   7. Simplified 9.5

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

   8. Taylor expanded around 0 5.8

      \[\leadsto \left(\left(c \cdot b - 27.0 \cdot \left(j \cdot k\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) + \left(\color{blue}{0} - 4.0 \cdot a\right) \cdot t\]

   if 6.983384062178601e-99 < t

   1. Initial program 2.4

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

   2. Simplified 3.2

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

   3. Taylor expanded around inf 3.2

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

   5. Applied *-un-lft-identity 3.2

      \[\leadsto \left(\left(c \cdot b - 27.0 \cdot \left(j \cdot k\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) + \left(\left(x \cdot 18.0\right) \cdot \left(y \cdot z\right) - a \cdot 4.0\right) \cdot \color{blue}{\left(1 \cdot t\right)}\]

   6. Applied associate-*r* 3.2

      \[\leadsto \left(\left(c \cdot b - 27.0 \cdot \left(j \cdot k\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) + \color{blue}{\left(\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot z\right) - a \cdot 4.0\right) \cdot 1\right) \cdot t}\]

   7. Simplified 3.2

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

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.5364366178162196 \cdot 10^{-144}:\\ \;\;\;\;\left(\left(c \cdot b - \left(j \cdot k\right) \cdot 27.0\right) - \left(i \cdot x\right) \cdot 4.0\right) + t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot z\right) \cdot y - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le 6.983384062178601 \cdot 10^{-99}:\\ \;\;\;\;\left(-t \cdot \left(a \cdot 4.0\right)\right) + \left(\left(c \cdot b - \left(j \cdot k\right) \cdot 27.0\right) - \left(i \cdot x\right) \cdot 4.0\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(\left(y \cdot z\right) \cdot 18.0\right) \cdot x - a \cdot 4.0\right) + \left(\left(c \cdot b - \left(j \cdot k\right) \cdot 27.0\right) - \left(i \cdot x\right) \cdot 4.0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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)))