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

Average Error: 21.1 → 18.7

Time: 24.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -1.428478097956767831303888965158296736042 \cdot 10^{226} \lor \neg \left(z \cdot t \le 7.274902941706250528826933509874936047753 \cdot 10^{294}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos y \cdot \mathsf{log1p}\left(\sqrt[3]{{\left(\mathsf{expm1}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right)}^{3}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r524114 = 2.0;
        double r524115 = x;
        double r524116 = sqrt(r524115);
        double r524117 = r524114 * r524116;
        double r524118 = y;
        double r524119 = z;
        double r524120 = t;
        double r524121 = r524119 * r524120;
        double r524122 = 3.0;
        double r524123 = r524121 / r524122;
        double r524124 = r524118 - r524123;
        double r524125 = cos(r524124);
        double r524126 = r524117 * r524125;
        double r524127 = a;
        double r524128 = b;
        double r524129 = r524128 * r524122;
        double r524130 = r524127 / r524129;
        double r524131 = r524126 - r524130;
        return r524131;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r524132 = z;
        double r524133 = t;
        double r524134 = r524132 * r524133;
        double r524135 = -1.4284780979567678e+226;
        bool r524136 = r524134 <= r524135;
        double r524137 = 7.27490294170625e+294;
        bool r524138 = r524134 <= r524137;
        double r524139 = !r524138;
        bool r524140 = r524136 || r524139;
        double r524141 = 2.0;
        double r524142 = x;
        double r524143 = sqrt(r524142);
        double r524144 = r524141 * r524143;
        double r524145 = y;
        double r524146 = 2.0;
        double r524147 = pow(r524145, r524146);
        double r524148 = -0.5;
        double r524149 = 1.0;
        double r524150 = fma(r524147, r524148, r524149);
        double r524151 = r524144 * r524150;
        double r524152 = a;
        double r524153 = b;
        double r524154 = 3.0;
        double r524155 = r524153 * r524154;
        double r524156 = r524152 / r524155;
        double r524157 = r524151 - r524156;
        double r524158 = cos(r524145);
        double r524159 = 0.3333333333333333;
        double r524160 = r524133 * r524132;
        double r524161 = r524159 * r524160;
        double r524162 = cos(r524161);
        double r524163 = expm1(r524162);
        double r524164 = 3.0;
        double r524165 = pow(r524163, r524164);
        double r524166 = cbrt(r524165);
        double r524167 = log1p(r524166);
        double r524168 = r524158 * r524167;
        double r524169 = r524168 * r524144;
        double r524170 = sin(r524145);
        double r524171 = r524134 / r524154;
        double r524172 = sin(r524171);
        double r524173 = r524170 * r524172;
        double r524174 = r524144 * r524173;
        double r524175 = r524169 + r524174;
        double r524176 = r524175 - r524156;
        double r524177 = r524140 ? r524157 : r524176;
        return r524177;
}

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	19.0
Herbie	18.7

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

Derivation

1. Split input into 2 regimes

2. if (* z t) < -1.4284780979567678e+226 or 7.27490294170625e+294 < (* z t)

   1. Initial program 57.2

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

   2. Taylor expanded around 0 45.8

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

   3. Simplified 45.8

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

   if -1.4284780979567678e+226 < (* z t) < 7.27490294170625e+294

   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 13.2

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

   4. Applied distribute-lft-in 13.2

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

   5. Simplified 13.2

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

   6. Taylor expanded around inf 13.2

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

   8. Applied log1p-expm1-u 13.2

      \[\leadsto \left(\left(\cos y \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right)}\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   9. Using strategy rm

   10. Applied add-cbrt-cube 13.2

       \[\leadsto \left(\left(\cos y \cdot \mathsf{log1p}\left(\color{blue}{\sqrt[3]{\left(\mathsf{expm1}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right) \cdot \mathsf{expm1}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   11. Simplified 13.2

       \[\leadsto \left(\left(\cos y \cdot \mathsf{log1p}\left(\sqrt[3]{\color{blue}{{\left(\mathsf{expm1}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right)}^{3}}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]
3. Recombined 2 regimes into one program.

4. Final simplification 18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -1.428478097956767831303888965158296736042 \cdot 10^{226} \lor \neg \left(z \cdot t \le 7.274902941706250528826933509874936047753 \cdot 10^{294}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos y \cdot \mathsf{log1p}\left(\sqrt[3]{{\left(\mathsf{expm1}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right)}^{3}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +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 -1.379333748723514e129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.333333333333333315 z) t)))) (/ (/ a 3) b)) (if (< z 3.51629061355598715e106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.333333333333333315 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))