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

Average Error: 20.8 → 18.0

Time: 10.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999845195289002731:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \left(2 \cdot \log \left(\sqrt[3]{e^{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r812717 = 2.0;
        double r812718 = x;
        double r812719 = sqrt(r812718);
        double r812720 = r812717 * r812719;
        double r812721 = y;
        double r812722 = z;
        double r812723 = t;
        double r812724 = r812722 * r812723;
        double r812725 = 3.0;
        double r812726 = r812724 / r812725;
        double r812727 = r812721 - r812726;
        double r812728 = cos(r812727);
        double r812729 = r812720 * r812728;
        double r812730 = a;
        double r812731 = b;
        double r812732 = r812731 * r812725;
        double r812733 = r812730 / r812732;
        double r812734 = r812729 - r812733;
        return r812734;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r812735 = y;
        double r812736 = z;
        double r812737 = t;
        double r812738 = r812736 * r812737;
        double r812739 = 3.0;
        double r812740 = r812738 / r812739;
        double r812741 = r812735 - r812740;
        double r812742 = cos(r812741);
        double r812743 = 0.9998451952890027;
        bool r812744 = r812742 <= r812743;
        double r812745 = 2.0;
        double r812746 = x;
        double r812747 = sqrt(r812746);
        double r812748 = r812745 * r812747;
        double r812749 = cos(r812735);
        double r812750 = cos(r812740);
        double r812751 = r812749 * r812750;
        double r812752 = 2.0;
        double r812753 = sin(r812735);
        double r812754 = 0.3333333333333333;
        double r812755 = r812737 * r812736;
        double r812756 = r812754 * r812755;
        double r812757 = sin(r812756);
        double r812758 = r812753 * r812757;
        double r812759 = exp(r812758);
        double r812760 = cbrt(r812759);
        double r812761 = log(r812760);
        double r812762 = r812752 * r812761;
        double r812763 = r812762 + r812761;
        double r812764 = r812751 + r812763;
        double r812765 = r812748 * r812764;
        double r812766 = a;
        double r812767 = b;
        double r812768 = r812767 * r812739;
        double r812769 = r812766 / r812768;
        double r812770 = r812765 - r812769;
        double r812771 = 1.0;
        double r812772 = 0.5;
        double r812773 = pow(r812735, r812752);
        double r812774 = r812772 * r812773;
        double r812775 = r812771 - r812774;
        double r812776 = r812748 * r812775;
        double r812777 = r812776 - r812769;
        double r812778 = r812744 ? r812770 : r812777;
        return r812778;
}

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	20.8
Target	18.9
Herbie	18.0

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9998451952890027

   1. Initial program 20.3

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

   3. Applied cos-diff 19.5

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

   4. Taylor expanded around inf 19.6

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

   6. Applied add-log-exp 19.6

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

   8. Applied add-cube-cbrt 19.6

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

   9. Applied log-prod 19.6

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

   10. Simplified 19.6

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

   if 0.9998451952890027 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.6

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

   2. Taylor expanded around 0 15.5

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999845195289002731:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \left(2 \cdot \log \left(\sqrt[3]{e^{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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))))