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

Average Error: 20.0 → 17.3

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9999896787288287125505803487612865865231:\\ \;\;\;\;\frac{\left(\sqrt{x} \cdot 2\right) \cdot \left({\left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right)\right)}^{3} + {\left(\sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \sin y\right)}^{3}\right)}{\left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right)\right) \cdot \left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right)\right) + \left(\left(\sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \sin y\right) \cdot \left(\sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \sin y\right) - \left(\sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \sin y\right) \cdot \left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right)\right)\right)} - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \left(1 + \frac{-1}{2} \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r35780101 = 2.0;
        double r35780102 = x;
        double r35780103 = sqrt(r35780102);
        double r35780104 = r35780101 * r35780103;
        double r35780105 = y;
        double r35780106 = z;
        double r35780107 = t;
        double r35780108 = r35780106 * r35780107;
        double r35780109 = 3.0;
        double r35780110 = r35780108 / r35780109;
        double r35780111 = r35780105 - r35780110;
        double r35780112 = cos(r35780111);
        double r35780113 = r35780104 * r35780112;
        double r35780114 = a;
        double r35780115 = b;
        double r35780116 = r35780115 * r35780109;
        double r35780117 = r35780114 / r35780116;
        double r35780118 = r35780113 - r35780117;
        return r35780118;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r35780119 = y;
        double r35780120 = t;
        double r35780121 = z;
        double r35780122 = r35780120 * r35780121;
        double r35780123 = 3.0;
        double r35780124 = r35780122 / r35780123;
        double r35780125 = r35780119 - r35780124;
        double r35780126 = cos(r35780125);
        double r35780127 = 0.9999896787288287;
        bool r35780128 = r35780126 <= r35780127;
        double r35780129 = x;
        double r35780130 = sqrt(r35780129);
        double r35780131 = 2.0;
        double r35780132 = r35780130 * r35780131;
        double r35780133 = cos(r35780119);
        double r35780134 = 0.3333333333333333;
        double r35780135 = r35780122 * r35780134;
        double r35780136 = cos(r35780135);
        double r35780137 = r35780133 * r35780136;
        double r35780138 = 3.0;
        double r35780139 = pow(r35780137, r35780138);
        double r35780140 = sin(r35780135);
        double r35780141 = sin(r35780119);
        double r35780142 = r35780140 * r35780141;
        double r35780143 = pow(r35780142, r35780138);
        double r35780144 = r35780139 + r35780143;
        double r35780145 = r35780132 * r35780144;
        double r35780146 = r35780137 * r35780137;
        double r35780147 = r35780142 * r35780142;
        double r35780148 = r35780142 * r35780137;
        double r35780149 = r35780147 - r35780148;
        double r35780150 = r35780146 + r35780149;
        double r35780151 = r35780145 / r35780150;
        double r35780152 = a;
        double r35780153 = b;
        double r35780154 = r35780153 * r35780123;
        double r35780155 = r35780152 / r35780154;
        double r35780156 = r35780151 - r35780155;
        double r35780157 = 1.0;
        double r35780158 = -0.5;
        double r35780159 = r35780119 * r35780119;
        double r35780160 = r35780158 * r35780159;
        double r35780161 = r35780157 + r35780160;
        double r35780162 = r35780132 * r35780161;
        double r35780163 = r35780162 - r35780155;
        double r35780164 = r35780128 ? r35780156 : r35780163;
        return r35780164;
}

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	20.0
Target	18.0
Herbie	17.3

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999896787288287

   1. Initial program 19.2

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

   3. Applied cos-diff 18.6

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

   4. Taylor expanded around inf 18.6

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

   5. Taylor expanded around inf 18.6

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

   7. Applied flip3-+ 18.6

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

   8. Applied associate-*r/ 18.6

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

   if 0.9999896787288287 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.2

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

   2. Taylor expanded around 0 15.1

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

   3. Simplified 15.1

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

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9999896787288287125505803487612865865231:\\ \;\;\;\;\frac{\left(\sqrt{x} \cdot 2\right) \cdot \left({\left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right)\right)}^{3} + {\left(\sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \sin y\right)}^{3}\right)}{\left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right)\right) \cdot \left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right)\right) + \left(\left(\sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \sin y\right) \cdot \left(\sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \sin y\right) - \left(\sin \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right) \cdot \sin y\right) \cdot \left(\cos y \cdot \cos \left(\left(t \cdot z\right) \cdot 0.3333333333333333148296162562473909929395\right)\right)\right)} - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \left(1 + \frac{-1}{2} \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))))