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

Average Error: 20.8 → 18.8

Time: 12.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -5.40746193454883711 \cdot 10^{253} \lor \neg \left(z \cdot t \le 3.85908544518920173 \cdot 10^{291}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\left(\sqrt[3]{z \cdot t} \cdot \sqrt[3]{\frac{1}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3}}}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r743172 = 2.0;
        double r743173 = x;
        double r743174 = sqrt(r743173);
        double r743175 = r743172 * r743174;
        double r743176 = y;
        double r743177 = z;
        double r743178 = t;
        double r743179 = r743177 * r743178;
        double r743180 = 3.0;
        double r743181 = r743179 / r743180;
        double r743182 = r743176 - r743181;
        double r743183 = cos(r743182);
        double r743184 = r743175 * r743183;
        double r743185 = a;
        double r743186 = b;
        double r743187 = r743186 * r743180;
        double r743188 = r743185 / r743187;
        double r743189 = r743184 - r743188;
        return r743189;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r743190 = z;
        double r743191 = t;
        double r743192 = r743190 * r743191;
        double r743193 = -5.407461934548837e+253;
        bool r743194 = r743192 <= r743193;
        double r743195 = 3.859085445189202e+291;
        bool r743196 = r743192 <= r743195;
        double r743197 = !r743196;
        bool r743198 = r743194 || r743197;
        double r743199 = 2.0;
        double r743200 = x;
        double r743201 = sqrt(r743200);
        double r743202 = r743199 * r743201;
        double r743203 = 1.0;
        double r743204 = 0.5;
        double r743205 = y;
        double r743206 = 2.0;
        double r743207 = pow(r743205, r743206);
        double r743208 = r743204 * r743207;
        double r743209 = r743203 - r743208;
        double r743210 = r743202 * r743209;
        double r743211 = a;
        double r743212 = b;
        double r743213 = 3.0;
        double r743214 = r743212 * r743213;
        double r743215 = r743211 / r743214;
        double r743216 = r743210 - r743215;
        double r743217 = cbrt(r743192);
        double r743218 = r743203 / r743213;
        double r743219 = cbrt(r743218);
        double r743220 = r743217 * r743219;
        double r743221 = r743192 / r743213;
        double r743222 = cbrt(r743221);
        double r743223 = r743220 * r743222;
        double r743224 = r743222 * r743222;
        double r743225 = cbrt(r743224);
        double r743226 = cbrt(r743222);
        double r743227 = r743225 * r743226;
        double r743228 = r743223 * r743227;
        double r743229 = r743205 - r743228;
        double r743230 = cos(r743229);
        double r743231 = r743202 * r743230;
        double r743232 = r743231 - r743215;
        double r743233 = r743198 ? r743216 : r743232;
        return r743233;
}

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	18.8

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

Derivation

1. Split input into 2 regimes

2. if (* z t) < -5.407461934548837e+253 or 3.859085445189202e+291 < (* z t)

   1. Initial program 58.7

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

   2. Taylor expanded around 0 45.1

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

   if -5.407461934548837e+253 < (* z t) < 3.859085445189202e+291

   1. Initial program 14.1

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

   3. Applied add-cube-cbrt 14.1

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

   5. Applied div-inv 14.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\color{blue}{\left(z \cdot t\right) \cdot \frac{1}{3}}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) - \frac{a}{b \cdot 3}\]

   6. Applied cbrt-prod 14.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\color{blue}{\left(\sqrt[3]{z \cdot t} \cdot \sqrt[3]{\frac{1}{3}}\right)} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) - \frac{a}{b \cdot 3}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 14.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\left(\sqrt[3]{z \cdot t} \cdot \sqrt[3]{\frac{1}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}}}\right) - \frac{a}{b \cdot 3}\]

   9. Applied cbrt-prod 14.1

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

4. Final simplification 18.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -5.40746193454883711 \cdot 10^{253} \lor \neg \left(z \cdot t \le 3.85908544518920173 \cdot 10^{291}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\left(\sqrt[3]{z \cdot t} \cdot \sqrt[3]{\frac{1}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3}}}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(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))))