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

Average Error: 20.3 → 17.8

Time: 20.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -8.150590203871196948341785027487202501958 \cdot 10^{263}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \le -7.209482062165938737652173765899207053792 \cdot 10^{58}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \le 6.844977881112784128232038768992631634214 \cdot 10^{294}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \log \left(e^{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r624962 = 2.0;
        double r624963 = x;
        double r624964 = sqrt(r624963);
        double r624965 = r624962 * r624964;
        double r624966 = y;
        double r624967 = z;
        double r624968 = t;
        double r624969 = r624967 * r624968;
        double r624970 = 3.0;
        double r624971 = r624969 / r624970;
        double r624972 = r624966 - r624971;
        double r624973 = cos(r624972);
        double r624974 = r624965 * r624973;
        double r624975 = a;
        double r624976 = b;
        double r624977 = r624976 * r624970;
        double r624978 = r624975 / r624977;
        double r624979 = r624974 - r624978;
        return r624979;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r624980 = z;
        double r624981 = t;
        double r624982 = r624980 * r624981;
        double r624983 = -8.150590203871197e+263;
        bool r624984 = r624982 <= r624983;
        double r624985 = 2.0;
        double r624986 = x;
        double r624987 = sqrt(r624986);
        double r624988 = r624985 * r624987;
        double r624989 = 1.0;
        double r624990 = 0.5;
        double r624991 = y;
        double r624992 = 2.0;
        double r624993 = pow(r624991, r624992);
        double r624994 = r624990 * r624993;
        double r624995 = r624989 - r624994;
        double r624996 = r624988 * r624995;
        double r624997 = a;
        double r624998 = b;
        double r624999 = 3.0;
        double r625000 = r624998 * r624999;
        double r625001 = r624997 / r625000;
        double r625002 = r624996 - r625001;
        double r625003 = -7.209482062165939e+58;
        bool r625004 = r624982 <= r625003;
        double r625005 = r624982 / r624999;
        double r625006 = cbrt(r625005);
        double r625007 = r625006 * r625006;
        double r625008 = r625007 * r625006;
        double r625009 = r624991 - r625008;
        double r625010 = cos(r625009);
        double r625011 = r624988 * r625010;
        double r625012 = r625011 - r625001;
        double r625013 = 6.844977881112784e+294;
        bool r625014 = r624982 <= r625013;
        double r625015 = cos(r624991);
        double r625016 = 0.3333333333333333;
        double r625017 = r624981 * r624980;
        double r625018 = r625016 * r625017;
        double r625019 = cos(r625018);
        double r625020 = exp(r625019);
        double r625021 = log(r625020);
        double r625022 = r625015 * r625021;
        double r625023 = r624988 * r625022;
        double r625024 = sin(r624991);
        double r625025 = sin(r625018);
        double r625026 = r625024 * r625025;
        double r625027 = r624988 * r625026;
        double r625028 = r625023 + r625027;
        double r625029 = r625028 - r625001;
        double r625030 = r625014 ? r625029 : r625002;
        double r625031 = r625004 ? r625012 : r625030;
        double r625032 = r624984 ? r625002 : r625031;
        return r625032;
}

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	18.3
Herbie	17.8

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

Derivation

1. Split input into 3 regimes

2. if (* z t) < -8.150590203871197e+263 or 6.844977881112784e+294 < (* z t)

   1. Initial program 59.2

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

   2. Taylor expanded around 0 45.3

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

   if -8.150590203871197e+263 < (* z t) < -7.209482062165939e+58

   1. Initial program 33.0

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

   3. Applied add-cube-cbrt 33.0

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

   if -7.209482062165939e+58 < (* z t) < 6.844977881112784e+294

   1. Initial program 10.2

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

   3. Applied cos-diff 9.6

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

   4. Applied distribute-lft-in 9.6

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

   5. Taylor expanded around inf 9.7

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

   6. Taylor expanded around inf 9.6

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

   8. Applied add-log-exp 9.6

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -8.150590203871196948341785027487202501958 \cdot 10^{263}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \le -7.209482062165938737652173765899207053792 \cdot 10^{58}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \le 6.844977881112784128232038768992631634214 \cdot 10^{294}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \log \left(e^{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(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))))