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

Average Error: 20.8 → 17.8

Time: 10.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999999942379:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r746189 = 2.0;
        double r746190 = x;
        double r746191 = sqrt(r746190);
        double r746192 = r746189 * r746191;
        double r746193 = y;
        double r746194 = z;
        double r746195 = t;
        double r746196 = r746194 * r746195;
        double r746197 = 3.0;
        double r746198 = r746196 / r746197;
        double r746199 = r746193 - r746198;
        double r746200 = cos(r746199);
        double r746201 = r746192 * r746200;
        double r746202 = a;
        double r746203 = b;
        double r746204 = r746203 * r746197;
        double r746205 = r746202 / r746204;
        double r746206 = r746201 - r746205;
        return r746206;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r746207 = y;
        double r746208 = z;
        double r746209 = t;
        double r746210 = r746208 * r746209;
        double r746211 = 3.0;
        double r746212 = r746210 / r746211;
        double r746213 = r746207 - r746212;
        double r746214 = cos(r746213);
        double r746215 = 0.9999999999999424;
        bool r746216 = r746214 <= r746215;
        double r746217 = 2.0;
        double r746218 = x;
        double r746219 = sqrt(r746218);
        double r746220 = r746217 * r746219;
        double r746221 = cos(r746207);
        double r746222 = 0.3333333333333333;
        double r746223 = r746209 * r746208;
        double r746224 = r746222 * r746223;
        double r746225 = cos(r746224);
        double r746226 = r746221 * r746225;
        double r746227 = r746220 * r746226;
        double r746228 = sin(r746207);
        double r746229 = sin(r746224);
        double r746230 = cbrt(r746229);
        double r746231 = r746230 * r746230;
        double r746232 = r746231 * r746230;
        double r746233 = r746228 * r746232;
        double r746234 = r746220 * r746233;
        double r746235 = r746227 + r746234;
        double r746236 = a;
        double r746237 = b;
        double r746238 = r746237 * r746211;
        double r746239 = r746236 / r746238;
        double r746240 = r746235 - r746239;
        double r746241 = 1.0;
        double r746242 = 0.5;
        double r746243 = 2.0;
        double r746244 = pow(r746207, r746243);
        double r746245 = r746242 * r746244;
        double r746246 = r746241 - r746245;
        double r746247 = r746220 * r746246;
        double r746248 = r746247 - r746239;
        double r746249 = r746216 ? r746240 : r746248;
        return r746249;
}

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	18.7
Herbie	17.8

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999999424

   1. Initial program 19.6

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

   3. Applied cos-diff 18.8

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

   4. Applied distribute-lft-in 18.8

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

   5. Taylor expanded around inf 18.8

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

   6. Taylor expanded around inf 18.8

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

   8. Applied add-cube-cbrt 18.8

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

   if 0.9999999999999424 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.8

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

   2. Taylor expanded around 0 16.0

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999999942379:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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))))