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

Average Error: 20.5 → 17.9

Time: 9.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999000948875516:\\ \;\;\;\;\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right) \cdot \cos y\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right) \cdot \cos y\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \sin y\right) \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r800194 = 2.0;
        double r800195 = x;
        double r800196 = sqrt(r800195);
        double r800197 = r800194 * r800196;
        double r800198 = y;
        double r800199 = z;
        double r800200 = t;
        double r800201 = r800199 * r800200;
        double r800202 = 3.0;
        double r800203 = r800201 / r800202;
        double r800204 = r800198 - r800203;
        double r800205 = cos(r800204);
        double r800206 = r800197 * r800205;
        double r800207 = a;
        double r800208 = b;
        double r800209 = r800208 * r800202;
        double r800210 = r800207 / r800209;
        double r800211 = r800206 - r800210;
        return r800211;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r800212 = y;
        double r800213 = z;
        double r800214 = t;
        double r800215 = r800213 * r800214;
        double r800216 = 3.0;
        double r800217 = r800215 / r800216;
        double r800218 = r800212 - r800217;
        double r800219 = cos(r800218);
        double r800220 = 0.9999000948875516;
        bool r800221 = r800219 <= r800220;
        double r800222 = 2.0;
        double r800223 = x;
        double r800224 = sqrt(r800223);
        double r800225 = r800222 * r800224;
        double r800226 = cos(r800217);
        double r800227 = exp(r800226);
        double r800228 = sqrt(r800227);
        double r800229 = log(r800228);
        double r800230 = cos(r800212);
        double r800231 = r800229 * r800230;
        double r800232 = r800225 * r800231;
        double r800233 = r800232 + r800232;
        double r800234 = sin(r800212);
        double r800235 = r800225 * r800234;
        double r800236 = sin(r800217);
        double r800237 = r800235 * r800236;
        double r800238 = r800233 + r800237;
        double r800239 = a;
        double r800240 = b;
        double r800241 = r800240 * r800216;
        double r800242 = r800239 / r800241;
        double r800243 = r800238 - r800242;
        double r800244 = 1.0;
        double r800245 = 0.5;
        double r800246 = 2.0;
        double r800247 = pow(r800212, r800246);
        double r800248 = r800245 * r800247;
        double r800249 = r800244 - r800248;
        double r800250 = r800225 * r800249;
        double r800251 = r800250 - r800242;
        double r800252 = r800221 ? r800243 : r800251;
        return r800252;
}

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	20.5
Target	18.5
Herbie	17.9

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999000948875516

   1. Initial program 20.1

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
   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. Applied distribute-lft-in 19.5

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

   6. Applied associate-*r* 19.5

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

   8. Applied add-log-exp 19.5

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

   10. Applied add-sqr-sqrt 19.5

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

   11. Applied log-prod 19.5

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

   12. Applied distribute-lft-in 19.5

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

   13. Applied distribute-lft-in 19.5

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

   14. Simplified 19.5

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

   15. Simplified 19.5

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

   if 0.9999000948875516 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.1

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

   2. Taylor expanded around 0 15.1

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999000948875516:\\ \;\;\;\;\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right) \cdot \cos y\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right) \cdot \cos y\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \sin y\right) \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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))))