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

Average Error: 21.0 → 18.5

Time: 29.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9997361592973614818902206025086343288422:\\ \;\;\;\;\left(\left(\sqrt{x} \cdot 2\right) \cdot \left(\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) \cdot \cos y\right) + \left(\sqrt{x} \cdot 2\right) \cdot \left(\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r37072189 = 2.0;
        double r37072190 = x;
        double r37072191 = sqrt(r37072190);
        double r37072192 = r37072189 * r37072191;
        double r37072193 = y;
        double r37072194 = z;
        double r37072195 = t;
        double r37072196 = r37072194 * r37072195;
        double r37072197 = 3.0;
        double r37072198 = r37072196 / r37072197;
        double r37072199 = r37072193 - r37072198;
        double r37072200 = cos(r37072199);
        double r37072201 = r37072192 * r37072200;
        double r37072202 = a;
        double r37072203 = b;
        double r37072204 = r37072203 * r37072197;
        double r37072205 = r37072202 / r37072204;
        double r37072206 = r37072201 - r37072205;
        return r37072206;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r37072207 = y;
        double r37072208 = t;
        double r37072209 = z;
        double r37072210 = r37072208 * r37072209;
        double r37072211 = 3.0;
        double r37072212 = r37072210 / r37072211;
        double r37072213 = r37072207 - r37072212;
        double r37072214 = cos(r37072213);
        double r37072215 = 0.9997361592973615;
        bool r37072216 = r37072214 <= r37072215;
        double r37072217 = x;
        double r37072218 = sqrt(r37072217);
        double r37072219 = 2.0;
        double r37072220 = r37072218 * r37072219;
        double r37072221 = 0.3333333333333333;
        double r37072222 = r37072221 * r37072210;
        double r37072223 = cos(r37072222);
        double r37072224 = cbrt(r37072223);
        double r37072225 = r37072224 * r37072224;
        double r37072226 = r37072224 * r37072225;
        double r37072227 = cos(r37072207);
        double r37072228 = r37072226 * r37072227;
        double r37072229 = r37072220 * r37072228;
        double r37072230 = sin(r37072222);
        double r37072231 = sin(r37072207);
        double r37072232 = r37072230 * r37072231;
        double r37072233 = r37072220 * r37072232;
        double r37072234 = r37072229 + r37072233;
        double r37072235 = a;
        double r37072236 = b;
        double r37072237 = r37072236 * r37072211;
        double r37072238 = r37072235 / r37072237;
        double r37072239 = r37072234 - r37072238;
        double r37072240 = 1.0;
        double r37072241 = 0.5;
        double r37072242 = r37072207 * r37072207;
        double r37072243 = r37072241 * r37072242;
        double r37072244 = r37072240 - r37072243;
        double r37072245 = r37072244 * r37072220;
        double r37072246 = r37072245 - r37072238;
        double r37072247 = r37072216 ? r37072239 : r37072246;
        return r37072247;
}

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	21.0
Target	18.9
Herbie	18.5

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9997361592973615

   1. Initial program 20.6

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

   3. Applied cos-diff 20.0

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

   4. Applied distribute-lft-in 20.0

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

   5. Taylor expanded around inf 20.0

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

   6. Simplified 20.0

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

   8. Applied add-cube-cbrt 20.0

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

   9. Taylor expanded around inf 20.0

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

   10. Simplified 20.0

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

   if 0.9997361592973615 < (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 16.1

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

   3. Simplified 16.1

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

4. Final simplification 18.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9997361592973614818902206025086343288422:\\ \;\;\;\;\left(\left(\sqrt{x} \cdot 2\right) \cdot \left(\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) \cdot \cos y\right) + \left(\sqrt{x} \cdot 2\right) \cdot \left(\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))