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

Average Error: 20.6 → 17.9

Time: 31.7s

Precision: 64

Math

\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9999647789220716953551004735345486551523:\\ \;\;\;\;\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \cos y + \sqrt[3]{{\left(\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)}^{3}}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r588511 = 2.0;
        double r588512 = x;
        double r588513 = sqrt(r588512);
        double r588514 = r588511 * r588513;
        double r588515 = y;
        double r588516 = z;
        double r588517 = t;
        double r588518 = r588516 * r588517;
        double r588519 = 3.0;
        double r588520 = r588518 / r588519;
        double r588521 = r588515 - r588520;
        double r588522 = cos(r588521);
        double r588523 = r588514 * r588522;
        double r588524 = a;
        double r588525 = b;
        double r588526 = r588525 * r588519;
        double r588527 = r588524 / r588526;
        double r588528 = r588523 - r588527;
        return r588528;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r588529 = y;
        double r588530 = t;
        double r588531 = z;
        double r588532 = r588530 * r588531;
        double r588533 = 3.0;
        double r588534 = r588532 / r588533;
        double r588535 = r588529 - r588534;
        double r588536 = cos(r588535);
        double r588537 = 0.9999647789220717;
        bool r588538 = r588536 <= r588537;
        double r588539 = 0.3333333333333333;
        double r588540 = r588539 * r588532;
        double r588541 = cos(r588540);
        double r588542 = cos(r588529);
        double r588543 = r588541 * r588542;
        double r588544 = sin(r588540);
        double r588545 = sin(r588529);
        double r588546 = r588544 * r588545;
        double r588547 = 3.0;
        double r588548 = pow(r588546, r588547);
        double r588549 = cbrt(r588548);
        double r588550 = r588543 + r588549;
        double r588551 = x;
        double r588552 = sqrt(r588551);
        double r588553 = 2.0;
        double r588554 = r588552 * r588553;
        double r588555 = r588550 * r588554;
        double r588556 = a;
        double r588557 = b;
        double r588558 = r588556 / r588557;
        double r588559 = r588558 / r588533;
        double r588560 = r588555 - r588559;
        double r588561 = 1.0;
        double r588562 = 0.5;
        double r588563 = r588529 * r588529;
        double r588564 = r588562 * r588563;
        double r588565 = r588561 - r588564;
        double r588566 = r588565 * r588554;
        double r588567 = r588566 - r588559;
        double r588568 = r588538 ? r588560 : r588567;
        return r588568;
}

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

Target

Original	20.6
Target	18.6
Herbie	17.9

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999647789220717

   1. Initial program 20.0

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]

   2. Simplified 20.0

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - z \cdot \frac{t}{3}\right) - \frac{\frac{a}{b}}{3}}\]
   3. Using strategy rm

   4. Applied cos-diff 19.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) + \sin y \cdot \sin \left(z \cdot \frac{t}{3}\right)\right)} - \frac{\frac{a}{b}}{3}\]

   5. Simplified 19.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\cos \left(z \cdot \frac{t}{3}\right) \cdot \cos y} + \sin y \cdot \sin \left(z \cdot \frac{t}{3}\right)\right) - \frac{\frac{a}{b}}{3}\]

   6. Taylor expanded around inf 19.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \cos y + \sin y \cdot \sin \left(z \cdot \frac{t}{3}\right)\right) - \frac{\frac{a}{b}}{3}\]

   7. Simplified 19.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right)} \cdot \cos y + \sin y \cdot \sin \left(z \cdot \frac{t}{3}\right)\right) - \frac{\frac{a}{b}}{3}\]

   8. Taylor expanded around inf 19.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \cos y + \sin y \cdot \color{blue}{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) - \frac{\frac{a}{b}}{3}\]

   9. Simplified 19.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \cos y + \sin y \cdot \color{blue}{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right)}\right) - \frac{\frac{a}{b}}{3}\]
   10. Using strategy rm

   11. Applied add-cbrt-cube 19.3

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \cos y + \sin y \cdot \color{blue}{\sqrt[3]{\left(\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right) \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right)\right) \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right)}}\right) - \frac{\frac{a}{b}}{3}\]

   12. Applied add-cbrt-cube 19.3

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \cos y + \color{blue}{\sqrt[3]{\left(\sin y \cdot \sin y\right) \cdot \sin y}} \cdot \sqrt[3]{\left(\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right) \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right)\right) \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right)}\right) - \frac{\frac{a}{b}}{3}\]

   13. Applied cbrt-unprod 19.3

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \cos y + \color{blue}{\sqrt[3]{\left(\left(\sin y \cdot \sin y\right) \cdot \sin y\right) \cdot \left(\left(\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right) \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right)\right) \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(z \cdot t\right)\right)\right)}}\right) - \frac{\frac{a}{b}}{3}\]

   14. Simplified 19.3

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \cos y + \sqrt[3]{\color{blue}{{\left(\sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \sin y\right)}^{3}}}\right) - \frac{\frac{a}{b}}{3}\]

   if 0.9999647789220717 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.8

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]

   2. Simplified 21.7

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - z \cdot \frac{t}{3}\right) - \frac{\frac{a}{b}}{3}}\]

   3. Taylor expanded around 0 15.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{\frac{a}{b}}{3}\]

   4. Simplified 15.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right)} - \frac{\frac{a}{b}}{3}\]
3. Recombined 2 regimes into one program.

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9999647789220716953551004735345486551523:\\ \;\;\;\;\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \cos y + \sqrt[3]{{\left(\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)}^{3}}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))