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

Average Error: 20.2 → 17.4

Time: 33.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 3.7752219740464399 \cdot 10^{301}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \log \left(e^{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) + \sin y \cdot \left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r468984 = 2.0;
        double r468985 = x;
        double r468986 = sqrt(r468985);
        double r468987 = r468984 * r468986;
        double r468988 = y;
        double r468989 = z;
        double r468990 = t;
        double r468991 = r468989 * r468990;
        double r468992 = 3.0;
        double r468993 = r468991 / r468992;
        double r468994 = r468988 - r468993;
        double r468995 = cos(r468994);
        double r468996 = r468987 * r468995;
        double r468997 = a;
        double r468998 = b;
        double r468999 = r468998 * r468992;
        double r469000 = r468997 / r468999;
        double r469001 = r468996 - r469000;
        return r469001;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r469002 = z;
        double r469003 = t;
        double r469004 = r469002 * r469003;
        double r469005 = -inf.0;
        bool r469006 = r469004 <= r469005;
        double r469007 = 3.77522197404644e+301;
        bool r469008 = r469004 <= r469007;
        double r469009 = !r469008;
        bool r469010 = r469006 || r469009;
        double r469011 = 2.0;
        double r469012 = x;
        double r469013 = sqrt(r469012);
        double r469014 = r469011 * r469013;
        double r469015 = -0.5;
        double r469016 = y;
        double r469017 = 2.0;
        double r469018 = pow(r469016, r469017);
        double r469019 = 1.0;
        double r469020 = fma(r469015, r469018, r469019);
        double r469021 = r469014 * r469020;
        double r469022 = a;
        double r469023 = b;
        double r469024 = 3.0;
        double r469025 = r469023 * r469024;
        double r469026 = r469022 / r469025;
        double r469027 = r469021 - r469026;
        double r469028 = cos(r469016);
        double r469029 = 0.3333333333333333;
        double r469030 = r469003 * r469002;
        double r469031 = r469029 * r469030;
        double r469032 = cos(r469031);
        double r469033 = exp(r469032);
        double r469034 = log(r469033);
        double r469035 = r469028 * r469034;
        double r469036 = sin(r469016);
        double r469037 = r469004 / r469024;
        double r469038 = sin(r469037);
        double r469039 = cbrt(r469038);
        double r469040 = r469039 * r469039;
        double r469041 = r469040 * r469039;
        double r469042 = r469036 * r469041;
        double r469043 = r469035 + r469042;
        double r469044 = r469014 * r469043;
        double r469045 = r469044 - r469026;
        double r469046 = r469010 ? r469027 : r469045;
        return r469046;
}

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	20.2
Target	18.2
Herbie	17.4

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

Derivation

1. Split input into 2 regimes

2. if (* z t) < -inf.0 or 3.77522197404644e+301 < (* z t)

   1. Initial program 63.3

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

   2. Taylor expanded around 0 44.5

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

   3. Simplified 44.5

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

   if -inf.0 < (* z t) < 3.77522197404644e+301

   1. Initial program 14.1

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

   3. Applied cos-diff 13.5

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

   4. Taylor expanded around inf 13.6

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

   6. Applied add-cube-cbrt 13.6

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

   8. Applied add-log-exp 13.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\log \left(e^{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)} + \sin y \cdot \left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\]
3. Recombined 2 regimes into one program.

4. Final simplification 17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 3.7752219740464399 \cdot 10^{301}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \log \left(e^{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) + \sin y \cdot \left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(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))))