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

Average Error: 20.8 → 18.0

Time: 11.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999845195289002731:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) - \left(\sqrt[3]{\sin y \cdot \sin \left(-0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(-0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(-0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r1286132 = 2.0;
        double r1286133 = x;
        double r1286134 = sqrt(r1286133);
        double r1286135 = r1286132 * r1286134;
        double r1286136 = y;
        double r1286137 = z;
        double r1286138 = t;
        double r1286139 = r1286137 * r1286138;
        double r1286140 = 3.0;
        double r1286141 = r1286139 / r1286140;
        double r1286142 = r1286136 - r1286141;
        double r1286143 = cos(r1286142);
        double r1286144 = r1286135 * r1286143;
        double r1286145 = a;
        double r1286146 = b;
        double r1286147 = r1286146 * r1286140;
        double r1286148 = r1286145 / r1286147;
        double r1286149 = r1286144 - r1286148;
        return r1286149;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r1286150 = y;
        double r1286151 = z;
        double r1286152 = t;
        double r1286153 = r1286151 * r1286152;
        double r1286154 = 3.0;
        double r1286155 = r1286153 / r1286154;
        double r1286156 = r1286150 - r1286155;
        double r1286157 = cos(r1286156);
        double r1286158 = 0.9998451952890027;
        bool r1286159 = r1286157 <= r1286158;
        double r1286160 = 2.0;
        double r1286161 = x;
        double r1286162 = sqrt(r1286161);
        double r1286163 = r1286160 * r1286162;
        double r1286164 = cos(r1286150);
        double r1286165 = cos(r1286155);
        double r1286166 = r1286164 * r1286165;
        double r1286167 = sin(r1286150);
        double r1286168 = 0.3333333333333333;
        double r1286169 = r1286152 * r1286151;
        double r1286170 = r1286168 * r1286169;
        double r1286171 = -r1286170;
        double r1286172 = sin(r1286171);
        double r1286173 = r1286167 * r1286172;
        double r1286174 = cbrt(r1286173);
        double r1286175 = r1286174 * r1286174;
        double r1286176 = r1286175 * r1286174;
        double r1286177 = r1286166 - r1286176;
        double r1286178 = r1286163 * r1286177;
        double r1286179 = a;
        double r1286180 = b;
        double r1286181 = r1286180 * r1286154;
        double r1286182 = r1286179 / r1286181;
        double r1286183 = r1286178 - r1286182;
        double r1286184 = 1.0;
        double r1286185 = 0.5;
        double r1286186 = 2.0;
        double r1286187 = pow(r1286150, r1286186);
        double r1286188 = r1286185 * r1286187;
        double r1286189 = r1286184 - r1286188;
        double r1286190 = r1286163 * r1286189;
        double r1286191 = r1286190 - r1286182;
        double r1286192 = r1286159 ? r1286183 : r1286191;
        return r1286192;
}

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	18.9
Herbie	18.0

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9998451952890027

   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 sub-neg 20.3

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

   4. Applied cos-sum 19.5

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

   5. Simplified 19.5

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

   6. Taylor expanded around inf 19.6

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

   8. Applied add-cube-cbrt 19.6

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

   if 0.9998451952890027 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.6

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

   2. Taylor expanded around 0 15.5

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999845195289002731:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) - \left(\sqrt[3]{\sin y \cdot \sin \left(-0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin y \cdot \sin \left(-0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin y \cdot \sin \left(-0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))))