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

Average Error: 20.6 → 17.8

Time: 19.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le \frac{1125898768688837}{1125899906842624}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right)} \cdot \sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right)}\right) \cdot \sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right)}\right) + \sin \left(\frac{6004799503160661}{18014398509481984} \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r684190 = 2.0;
        double r684191 = x;
        double r684192 = sqrt(r684191);
        double r684193 = r684190 * r684192;
        double r684194 = y;
        double r684195 = z;
        double r684196 = t;
        double r684197 = r684195 * r684196;
        double r684198 = 3.0;
        double r684199 = r684197 / r684198;
        double r684200 = r684194 - r684199;
        double r684201 = cos(r684200);
        double r684202 = r684193 * r684201;
        double r684203 = a;
        double r684204 = b;
        double r684205 = r684204 * r684198;
        double r684206 = r684203 / r684205;
        double r684207 = r684202 - r684206;
        return r684207;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r684208 = y;
        double r684209 = z;
        double r684210 = t;
        double r684211 = r684209 * r684210;
        double r684212 = 3.0;
        double r684213 = r684211 / r684212;
        double r684214 = r684208 - r684213;
        double r684215 = cos(r684214);
        double r684216 = 1125898768688837.0;
        double r684217 = 1125899906842624.0;
        double r684218 = r684216 / r684217;
        bool r684219 = r684215 <= r684218;
        double r684220 = 2.0;
        double r684221 = x;
        double r684222 = sqrt(r684221);
        double r684223 = r684220 * r684222;
        double r684224 = cos(r684208);
        double r684225 = cbrt(r684213);
        double r684226 = r684225 * r684225;
        double r684227 = r684226 * r684225;
        double r684228 = cos(r684227);
        double r684229 = cbrt(r684228);
        double r684230 = r684229 * r684229;
        double r684231 = r684230 * r684229;
        double r684232 = r684224 * r684231;
        double r684233 = 6004799503160661.0;
        double r684234 = 18014398509481984.0;
        double r684235 = r684233 / r684234;
        double r684236 = r684210 * r684209;
        double r684237 = r684235 * r684236;
        double r684238 = sin(r684237);
        double r684239 = sin(r684208);
        double r684240 = r684238 * r684239;
        double r684241 = r684232 + r684240;
        double r684242 = r684223 * r684241;
        double r684243 = a;
        double r684244 = b;
        double r684245 = r684244 * r684212;
        double r684246 = r684243 / r684245;
        double r684247 = r684242 - r684246;
        double r684248 = 1.0;
        double r684249 = 0.5;
        double r684250 = 2.0;
        double r684251 = pow(r684208, r684250);
        double r684252 = r684249 * r684251;
        double r684253 = r684248 - r684252;
        double r684254 = r684223 * r684253;
        double r684255 = r684254 - r684246;
        double r684256 = r684219 ? r684247 : r684255;
        return r684256;
}

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	17.8

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999989891163681

   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 add-cube-cbrt 20.0

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

   5. Applied cos-diff 19.2

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

   6. Simplified 19.2

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

   7. Taylor expanded around inf 19.2

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

   8. Simplified 19.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) + \color{blue}{\sin \left(\frac{6004799503160661}{18014398509481984} \cdot \left(t \cdot z\right)\right)} \cdot \sin y\right) - \frac{a}{b \cdot 3}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 19.2

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right)} \cdot \sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right)}\right) \cdot \sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right)}\right)} + \sin \left(\frac{6004799503160661}{18014398509481984} \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\]

   if 0.9999989891163681 < (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. Taylor expanded around 0 15.3

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le \frac{1125898768688837}{1125899906842624}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right)} \cdot \sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right)}\right) \cdot \sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right)}\right) + \sin \left(\frac{6004799503160661}{18014398509481984} \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.333333333333333315 z) t)))) (/ (/ a 3) b)) (if (< z 3.51629061355598715e106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.333333333333333315 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

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