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

Average Error: 20.5 → 17.6

Time: 21.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999999864552790995730902068316936:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\frac{z \cdot t}{3}\right) \cdot \cos y - \sin y \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{z}{\frac{3}{t}}\right)}\right)}^{6}} \cdot \left(-\sqrt[3]{\sin \left(\frac{z}{\frac{3}{t}}\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r620966 = 2.0;
        double r620967 = x;
        double r620968 = sqrt(r620967);
        double r620969 = r620966 * r620968;
        double r620970 = y;
        double r620971 = z;
        double r620972 = t;
        double r620973 = r620971 * r620972;
        double r620974 = 3.0;
        double r620975 = r620973 / r620974;
        double r620976 = r620970 - r620975;
        double r620977 = cos(r620976);
        double r620978 = r620969 * r620977;
        double r620979 = a;
        double r620980 = b;
        double r620981 = r620980 * r620974;
        double r620982 = r620979 / r620981;
        double r620983 = r620978 - r620982;
        return r620983;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r620984 = y;
        double r620985 = z;
        double r620986 = t;
        double r620987 = r620985 * r620986;
        double r620988 = 3.0;
        double r620989 = r620987 / r620988;
        double r620990 = r620984 - r620989;
        double r620991 = cos(r620990);
        double r620992 = 0.9999999999999865;
        bool r620993 = r620991 <= r620992;
        double r620994 = 2.0;
        double r620995 = x;
        double r620996 = sqrt(r620995);
        double r620997 = r620994 * r620996;
        double r620998 = cos(r620989);
        double r620999 = cos(r620984);
        double r621000 = r620998 * r620999;
        double r621001 = sin(r620984);
        double r621002 = r620988 / r620986;
        double r621003 = r620985 / r621002;
        double r621004 = sin(r621003);
        double r621005 = cbrt(r621004);
        double r621006 = 6.0;
        double r621007 = pow(r621005, r621006);
        double r621008 = cbrt(r621007);
        double r621009 = -r621005;
        double r621010 = r621008 * r621009;
        double r621011 = r621001 * r621010;
        double r621012 = r621000 - r621011;
        double r621013 = r620997 * r621012;
        double r621014 = a;
        double r621015 = b;
        double r621016 = r621015 * r620988;
        double r621017 = r621014 / r621016;
        double r621018 = r621013 - r621017;
        double r621019 = 1.0;
        double r621020 = 0.5;
        double r621021 = 2.0;
        double r621022 = pow(r620984, r621021);
        double r621023 = r621020 * r621022;
        double r621024 = r621019 - r621023;
        double r621025 = r620997 * r621024;
        double r621026 = r621025 - r621017;
        double r621027 = r620993 ? r621018 : r621026;
        return r621027;
}

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	20.5
Target	18.5
Herbie	17.6

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999999865

   1. Initial program 19.6

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

   3. Applied sub-neg 19.6

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

   4. Applied cos-sum 18.9

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

   5. Simplified 18.9

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

   7. Applied add-cbrt-cube 18.9

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

   8. Simplified 18.9

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

   10. Applied add-cube-cbrt 18.9

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

   11. Applied unpow-prod-down 18.9

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

   12. Applied cbrt-prod 18.9

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

   13. Simplified 18.9

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\frac{z \cdot t}{3}\right) \cdot \cos y - \sin y \cdot \left(\color{blue}{\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{z}{\frac{3}{t}}\right)}\right)}^{6}}} \cdot \sqrt[3]{{\left(\sqrt[3]{-\sin \left(\frac{z}{\frac{3}{t}}\right)}\right)}^{3}}\right)\right) - \frac{a}{b \cdot 3}\]

   14. Simplified 18.9

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\frac{z \cdot t}{3}\right) \cdot \cos y - \sin y \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{z}{\frac{3}{t}}\right)}\right)}^{6}} \cdot \color{blue}{\left(-\sqrt[3]{\sin \left(\frac{z}{\frac{3}{t}}\right)}\right)}\right)\right) - \frac{a}{b \cdot 3}\]

   if 0.9999999999999865 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.0

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

   2. Taylor expanded around 0 15.5

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999999864552790995730902068316936:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\frac{z \cdot t}{3}\right) \cdot \cos y - \sin y \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sin \left(\frac{z}{\frac{3}{t}}\right)}\right)}^{6}} \cdot \left(-\sqrt[3]{\sin \left(\frac{z}{\frac{3}{t}}\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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))))