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

Average Error: 20.7 → 17.9

Time: 13.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99999999987750376:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r1027173 = 2.0;
        double r1027174 = x;
        double r1027175 = sqrt(r1027174);
        double r1027176 = r1027173 * r1027175;
        double r1027177 = y;
        double r1027178 = z;
        double r1027179 = t;
        double r1027180 = r1027178 * r1027179;
        double r1027181 = 3.0;
        double r1027182 = r1027180 / r1027181;
        double r1027183 = r1027177 - r1027182;
        double r1027184 = cos(r1027183);
        double r1027185 = r1027176 * r1027184;
        double r1027186 = a;
        double r1027187 = b;
        double r1027188 = r1027187 * r1027181;
        double r1027189 = r1027186 / r1027188;
        double r1027190 = r1027185 - r1027189;
        return r1027190;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r1027191 = y;
        double r1027192 = z;
        double r1027193 = t;
        double r1027194 = r1027192 * r1027193;
        double r1027195 = 3.0;
        double r1027196 = r1027194 / r1027195;
        double r1027197 = r1027191 - r1027196;
        double r1027198 = cos(r1027197);
        double r1027199 = 0.9999999998775038;
        bool r1027200 = r1027198 <= r1027199;
        double r1027201 = 2.0;
        double r1027202 = x;
        double r1027203 = sqrt(r1027202);
        double r1027204 = r1027201 * r1027203;
        double r1027205 = cos(r1027191);
        double r1027206 = 0.3333333333333333;
        double r1027207 = r1027193 * r1027192;
        double r1027208 = r1027206 * r1027207;
        double r1027209 = cos(r1027208);
        double r1027210 = r1027205 * r1027209;
        double r1027211 = sin(r1027191);
        double r1027212 = sin(r1027208);
        double r1027213 = r1027211 * r1027212;
        double r1027214 = r1027210 + r1027213;
        double r1027215 = r1027204 * r1027214;
        double r1027216 = a;
        double r1027217 = b;
        double r1027218 = r1027216 / r1027217;
        double r1027219 = r1027218 / r1027195;
        double r1027220 = r1027215 - r1027219;
        double r1027221 = 1.0;
        double r1027222 = 0.5;
        double r1027223 = 2.0;
        double r1027224 = pow(r1027191, r1027223);
        double r1027225 = r1027222 * r1027224;
        double r1027226 = r1027221 - r1027225;
        double r1027227 = r1027204 * r1027226;
        double r1027228 = r1027217 * r1027195;
        double r1027229 = r1027216 / r1027228;
        double r1027230 = r1027227 - r1027229;
        double r1027231 = r1027200 ? r1027220 : r1027230;
        return r1027231;
}

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	20.7
Target	18.7
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))) < 0.9999999998775038

   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 19.6

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

   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. Taylor expanded around inf 19.6

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

   7. Applied associate-/r* 19.6

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

   if 0.9999999998775038 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.5

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

   2. Taylor expanded around 0 15.0

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99999999987750376:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))))