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

Average Error: 20.3 → 17.7

Time: 38.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999884525819578984240365571167785674334:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r137211101 = 2.0;
        double r137211102 = x;
        double r137211103 = sqrt(r137211102);
        double r137211104 = r137211101 * r137211103;
        double r137211105 = y;
        double r137211106 = z;
        double r137211107 = t;
        double r137211108 = r137211106 * r137211107;
        double r137211109 = 3.0;
        double r137211110 = r137211108 / r137211109;
        double r137211111 = r137211105 - r137211110;
        double r137211112 = cos(r137211111);
        double r137211113 = r137211104 * r137211112;
        double r137211114 = a;
        double r137211115 = b;
        double r137211116 = r137211115 * r137211109;
        double r137211117 = r137211114 / r137211116;
        double r137211118 = r137211113 - r137211117;
        return r137211118;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r137211119 = y;
        double r137211120 = z;
        double r137211121 = t;
        double r137211122 = r137211120 * r137211121;
        double r137211123 = 3.0;
        double r137211124 = r137211122 / r137211123;
        double r137211125 = r137211119 - r137211124;
        double r137211126 = cos(r137211125);
        double r137211127 = 0.9999884525819579;
        bool r137211128 = r137211126 <= r137211127;
        double r137211129 = 2.0;
        double r137211130 = x;
        double r137211131 = sqrt(r137211130);
        double r137211132 = r137211129 * r137211131;
        double r137211133 = cos(r137211119);
        double r137211134 = 0.3333333333333333;
        double r137211135 = r137211121 * r137211120;
        double r137211136 = r137211134 * r137211135;
        double r137211137 = cos(r137211136);
        double r137211138 = r137211133 * r137211137;
        double r137211139 = sin(r137211119);
        double r137211140 = sin(r137211136);
        double r137211141 = r137211139 * r137211140;
        double r137211142 = r137211138 + r137211141;
        double r137211143 = r137211132 * r137211142;
        double r137211144 = a;
        double r137211145 = b;
        double r137211146 = r137211144 / r137211145;
        double r137211147 = r137211146 / r137211123;
        double r137211148 = r137211143 - r137211147;
        double r137211149 = 1.0;
        double r137211150 = 0.5;
        double r137211151 = r137211119 * r137211119;
        double r137211152 = r137211150 * r137211151;
        double r137211153 = r137211149 - r137211152;
        double r137211154 = r137211132 * r137211153;
        double r137211155 = r137211145 * r137211123;
        double r137211156 = r137211144 / r137211155;
        double r137211157 = r137211154 - r137211156;
        double r137211158 = r137211128 ? r137211148 : r137211157;
        return r137211158;
}

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	20.3
Target	18.4
Herbie	17.7

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999884525819579

   1. Initial program 19.9

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

   2. Taylor expanded around inf 19.9

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

   4. Applied cos-diff 19.3

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

   6. Applied associate-/r* 19.3

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

   if 0.9999884525819579 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 20.8

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

   2. Taylor expanded around inf 20.8

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

   3. Taylor expanded around 0 14.9

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

   4. Simplified 14.9

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

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999884525819578984240365571167785674334:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(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))))