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

Average Error: 20.3 → 17.8

Time: 16.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99990718372146614:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r716605 = 2.0;
        double r716606 = x;
        double r716607 = sqrt(r716606);
        double r716608 = r716605 * r716607;
        double r716609 = y;
        double r716610 = z;
        double r716611 = t;
        double r716612 = r716610 * r716611;
        double r716613 = 3.0;
        double r716614 = r716612 / r716613;
        double r716615 = r716609 - r716614;
        double r716616 = cos(r716615);
        double r716617 = r716608 * r716616;
        double r716618 = a;
        double r716619 = b;
        double r716620 = r716619 * r716613;
        double r716621 = r716618 / r716620;
        double r716622 = r716617 - r716621;
        return r716622;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r716623 = y;
        double r716624 = z;
        double r716625 = t;
        double r716626 = r716624 * r716625;
        double r716627 = 3.0;
        double r716628 = r716626 / r716627;
        double r716629 = r716623 - r716628;
        double r716630 = cos(r716629);
        double r716631 = 0.9999071837214661;
        bool r716632 = r716630 <= r716631;
        double r716633 = 2.0;
        double r716634 = x;
        double r716635 = sqrt(r716634);
        double r716636 = r716633 * r716635;
        double r716637 = cos(r716623);
        double r716638 = 0.3333333333333333;
        double r716639 = r716625 * r716624;
        double r716640 = r716638 * r716639;
        double r716641 = cos(r716640);
        double r716642 = r716637 * r716641;
        double r716643 = r716636 * r716642;
        double r716644 = sin(r716640);
        double r716645 = sin(r716623);
        double r716646 = r716644 * r716645;
        double r716647 = r716636 * r716646;
        double r716648 = cbrt(r716647);
        double r716649 = r716648 * r716648;
        double r716650 = r716649 * r716648;
        double r716651 = r716643 + r716650;
        double r716652 = a;
        double r716653 = b;
        double r716654 = r716653 * r716627;
        double r716655 = r716652 / r716654;
        double r716656 = r716651 - r716655;
        double r716657 = 1.0;
        double r716658 = 0.5;
        double r716659 = 2.0;
        double r716660 = pow(r716623, r716659);
        double r716661 = r716658 * r716660;
        double r716662 = r716657 - r716661;
        double r716663 = r716636 * r716662;
        double r716664 = r716663 - r716655;
        double r716665 = r716632 ? r716656 : r716664;
        return r716665;
}

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	20.3
Target	18.4
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))) < 0.9999071837214661

   1. Initial program 19.9

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

   3. Applied cos-diff 19.4

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

   4. Applied distribute-lft-in 19.4

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

   5. Taylor expanded around inf 19.4

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

   6. Taylor expanded around inf 19.3

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

   8. Applied add-cube-cbrt 19.3

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

   if 0.9999071837214661 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 20.9

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

   2. Taylor expanded around 0 15.2

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99990718372146614:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))))