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

Average Error: 20.9 → 18.6

Time: 12.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -1.094007139762150192715724278337881239255 \cdot 10^{305} \lor \neg \left(z \cdot t \le 3.189636586473062064553892727344787175045 \cdot 10^{234}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \sqrt[3]{{\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}^{3}}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r723275 = 2.0;
        double r723276 = x;
        double r723277 = sqrt(r723276);
        double r723278 = r723275 * r723277;
        double r723279 = y;
        double r723280 = z;
        double r723281 = t;
        double r723282 = r723280 * r723281;
        double r723283 = 3.0;
        double r723284 = r723282 / r723283;
        double r723285 = r723279 - r723284;
        double r723286 = cos(r723285);
        double r723287 = r723278 * r723286;
        double r723288 = a;
        double r723289 = b;
        double r723290 = r723289 * r723283;
        double r723291 = r723288 / r723290;
        double r723292 = r723287 - r723291;
        return r723292;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r723293 = z;
        double r723294 = t;
        double r723295 = r723293 * r723294;
        double r723296 = -1.0940071397621502e+305;
        bool r723297 = r723295 <= r723296;
        double r723298 = 3.189636586473062e+234;
        bool r723299 = r723295 <= r723298;
        double r723300 = !r723299;
        bool r723301 = r723297 || r723300;
        double r723302 = 2.0;
        double r723303 = x;
        double r723304 = sqrt(r723303);
        double r723305 = r723302 * r723304;
        double r723306 = 1.0;
        double r723307 = 0.5;
        double r723308 = y;
        double r723309 = 2.0;
        double r723310 = pow(r723308, r723309);
        double r723311 = r723307 * r723310;
        double r723312 = r723306 - r723311;
        double r723313 = r723305 * r723312;
        double r723314 = a;
        double r723315 = b;
        double r723316 = 3.0;
        double r723317 = r723315 * r723316;
        double r723318 = r723314 / r723317;
        double r723319 = r723313 - r723318;
        double r723320 = cos(r723308);
        double r723321 = 0.3333333333333333;
        double r723322 = r723294 * r723293;
        double r723323 = r723321 * r723322;
        double r723324 = cos(r723323);
        double r723325 = 3.0;
        double r723326 = pow(r723324, r723325);
        double r723327 = cbrt(r723326);
        double r723328 = r723320 * r723327;
        double r723329 = r723305 * r723328;
        double r723330 = sin(r723308);
        double r723331 = sin(r723323);
        double r723332 = r723330 * r723331;
        double r723333 = r723305 * r723332;
        double r723334 = r723329 + r723333;
        double r723335 = r723334 - r723318;
        double r723336 = r723301 ? r723319 : r723335;
        return r723336;
}

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	20.9
Target	18.9
Herbie	18.6

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

Derivation

1. Split input into 2 regimes

2. if (* z t) < -1.0940071397621502e+305 or 3.189636586473062e+234 < (* z t)

   1. Initial program 56.8

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

   2. Taylor expanded around 0 45.4

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

   if -1.0940071397621502e+305 < (* z t) < 3.189636586473062e+234

   1. Initial program 13.8

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

   3. Applied cos-diff 13.3

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

   4. Applied distribute-lft-in 13.3

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

   5. Taylor expanded around inf 13.3

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

   6. Taylor expanded around inf 13.3

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

   8. Applied add-cbrt-cube 13.3

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

   9. Simplified 13.3

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

4. Final simplification 18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -1.094007139762150192715724278337881239255 \cdot 10^{305} \lor \neg \left(z \cdot t \le 3.189636586473062064553892727344787175045 \cdot 10^{234}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \sqrt[3]{{\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}^{3}}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(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))))