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

Average Error: 20.6 → 17.8

Time: 11.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999999024225:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r751589 = 2.0;
        double r751590 = x;
        double r751591 = sqrt(r751590);
        double r751592 = r751589 * r751591;
        double r751593 = y;
        double r751594 = z;
        double r751595 = t;
        double r751596 = r751594 * r751595;
        double r751597 = 3.0;
        double r751598 = r751596 / r751597;
        double r751599 = r751593 - r751598;
        double r751600 = cos(r751599);
        double r751601 = r751592 * r751600;
        double r751602 = a;
        double r751603 = b;
        double r751604 = r751603 * r751597;
        double r751605 = r751602 / r751604;
        double r751606 = r751601 - r751605;
        return r751606;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r751607 = y;
        double r751608 = z;
        double r751609 = t;
        double r751610 = r751608 * r751609;
        double r751611 = 3.0;
        double r751612 = r751610 / r751611;
        double r751613 = r751607 - r751612;
        double r751614 = cos(r751613);
        double r751615 = 0.9999999999990242;
        bool r751616 = r751614 <= r751615;
        double r751617 = 2.0;
        double r751618 = x;
        double r751619 = sqrt(r751618);
        double r751620 = r751617 * r751619;
        double r751621 = cos(r751607);
        double r751622 = cos(r751612);
        double r751623 = 3.0;
        double r751624 = pow(r751622, r751623);
        double r751625 = cbrt(r751624);
        double r751626 = r751621 * r751625;
        double r751627 = r751620 * r751626;
        double r751628 = sin(r751607);
        double r751629 = 0.3333333333333333;
        double r751630 = r751609 * r751608;
        double r751631 = r751629 * r751630;
        double r751632 = sin(r751631);
        double r751633 = cbrt(r751632);
        double r751634 = r751633 * r751633;
        double r751635 = r751634 * r751633;
        double r751636 = r751628 * r751635;
        double r751637 = r751620 * r751636;
        double r751638 = r751627 + r751637;
        double r751639 = a;
        double r751640 = b;
        double r751641 = r751640 * r751611;
        double r751642 = r751639 / r751641;
        double r751643 = r751638 - r751642;
        double r751644 = 1.0;
        double r751645 = 0.5;
        double r751646 = 2.0;
        double r751647 = pow(r751607, r751646);
        double r751648 = r751645 * r751647;
        double r751649 = r751644 - r751648;
        double r751650 = r751620 * r751649;
        double r751651 = r751650 - r751642;
        double r751652 = r751616 ? r751643 : r751651;
        return r751652;
}

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	18.3
Herbie	17.8

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999990242

   1. Initial program 19.4

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

   3. Applied cos-diff 18.8

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

   4. Applied distribute-lft-in 18.8

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

   5. Taylor expanded around inf 18.9

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right) - \frac{a}{b \cdot 3}\]
   6. Using strategy rm

   7. Applied add-cbrt-cube 18.9

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{z \cdot t}{3}\right) \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \cos \left(\frac{z \cdot t}{3}\right)}}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3}\]

   8. Simplified 18.9

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \sqrt[3]{\color{blue}{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 18.9

       \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)}\right)\right) - \frac{a}{b \cdot 3}\]

   if 0.9999999999990242 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.7

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

   2. Taylor expanded around 0 15.9

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999999024225:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e+129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))