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

Average Error: 21.1 → 18.3

Time: 9.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999999932276395497865451034158468:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r756165 = 2.0;
        double r756166 = x;
        double r756167 = sqrt(r756166);
        double r756168 = r756165 * r756167;
        double r756169 = y;
        double r756170 = z;
        double r756171 = t;
        double r756172 = r756170 * r756171;
        double r756173 = 3.0;
        double r756174 = r756172 / r756173;
        double r756175 = r756169 - r756174;
        double r756176 = cos(r756175);
        double r756177 = r756168 * r756176;
        double r756178 = a;
        double r756179 = b;
        double r756180 = r756179 * r756173;
        double r756181 = r756178 / r756180;
        double r756182 = r756177 - r756181;
        return r756182;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r756183 = y;
        double r756184 = z;
        double r756185 = t;
        double r756186 = r756184 * r756185;
        double r756187 = 3.0;
        double r756188 = r756186 / r756187;
        double r756189 = r756183 - r756188;
        double r756190 = cos(r756189);
        double r756191 = 0.9999999999999932;
        bool r756192 = r756190 <= r756191;
        double r756193 = 2.0;
        double r756194 = x;
        double r756195 = sqrt(r756194);
        double r756196 = r756193 * r756195;
        double r756197 = cos(r756183);
        double r756198 = 0.3333333333333333;
        double r756199 = r756185 * r756184;
        double r756200 = r756198 * r756199;
        double r756201 = cos(r756200);
        double r756202 = r756197 * r756201;
        double r756203 = r756196 * r756202;
        double r756204 = sin(r756183);
        double r756205 = sin(r756188);
        double r756206 = r756204 * r756205;
        double r756207 = r756196 * r756206;
        double r756208 = cbrt(r756207);
        double r756209 = r756208 * r756208;
        double r756210 = sin(r756200);
        double r756211 = r756204 * r756210;
        double r756212 = r756196 * r756211;
        double r756213 = cbrt(r756212);
        double r756214 = r756209 * r756213;
        double r756215 = r756203 + r756214;
        double r756216 = a;
        double r756217 = b;
        double r756218 = r756217 * r756187;
        double r756219 = r756216 / r756218;
        double r756220 = r756215 - r756219;
        double r756221 = 1.0;
        double r756222 = 0.5;
        double r756223 = 2.0;
        double r756224 = pow(r756183, r756223);
        double r756225 = r756222 * r756224;
        double r756226 = r756221 - r756225;
        double r756227 = r756196 * r756226;
        double r756228 = r756227 - r756219;
        double r756229 = r756192 ? r756220 : r756228;
        return r756229;
}

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	21.1
Target	19.1
Herbie	18.3

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999999932

   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. Applied distribute-lft-in 19.6

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

   6. Applied add-cube-cbrt 19.6

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

   7. Taylor expanded around inf 19.6

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

   8. Taylor expanded around inf 19.6

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

   if 0.9999999999999932 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.5

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

   2. Taylor expanded around 0 15.8

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

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999999932276395497865451034158468:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))