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

Average Error: 20.6 → 17.8

Time: 33.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 1.965264242388164234247715384116746023602 \cdot 10^{152}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\sqrt[3]{\frac{1}{3}} \cdot \left({\left(\sqrt[3]{-1}\right)}^{2} \cdot \left(z \cdot \left(t \cdot {\left(\sqrt[3]{0.3333333333333333148296162562473909929395}\right)}^{2}\right)\right)\right)\right) - \sin y \cdot \sin \left(-\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r568002 = 2.0;
        double r568003 = x;
        double r568004 = sqrt(r568003);
        double r568005 = r568002 * r568004;
        double r568006 = y;
        double r568007 = z;
        double r568008 = t;
        double r568009 = r568007 * r568008;
        double r568010 = 3.0;
        double r568011 = r568009 / r568010;
        double r568012 = r568006 - r568011;
        double r568013 = cos(r568012);
        double r568014 = r568005 * r568013;
        double r568015 = a;
        double r568016 = b;
        double r568017 = r568016 * r568010;
        double r568018 = r568015 / r568017;
        double r568019 = r568014 - r568018;
        return r568019;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r568020 = 2.0;
        double r568021 = x;
        double r568022 = sqrt(r568021);
        double r568023 = r568020 * r568022;
        double r568024 = y;
        double r568025 = z;
        double r568026 = t;
        double r568027 = r568025 * r568026;
        double r568028 = 3.0;
        double r568029 = r568027 / r568028;
        double r568030 = r568024 - r568029;
        double r568031 = cos(r568030);
        double r568032 = r568023 * r568031;
        double r568033 = 1.9652642423881642e+152;
        bool r568034 = r568032 <= r568033;
        double r568035 = cos(r568024);
        double r568036 = 1.0;
        double r568037 = r568036 / r568028;
        double r568038 = cbrt(r568037);
        double r568039 = -1.0;
        double r568040 = cbrt(r568039);
        double r568041 = 2.0;
        double r568042 = pow(r568040, r568041);
        double r568043 = 0.3333333333333333;
        double r568044 = cbrt(r568043);
        double r568045 = pow(r568044, r568041);
        double r568046 = r568026 * r568045;
        double r568047 = r568025 * r568046;
        double r568048 = r568042 * r568047;
        double r568049 = r568038 * r568048;
        double r568050 = cos(r568049);
        double r568051 = r568035 * r568050;
        double r568052 = sin(r568024);
        double r568053 = cbrt(r568029);
        double r568054 = r568053 * r568053;
        double r568055 = sqrt(r568028);
        double r568056 = r568025 / r568055;
        double r568057 = cbrt(r568056);
        double r568058 = r568026 / r568055;
        double r568059 = cbrt(r568058);
        double r568060 = r568057 * r568059;
        double r568061 = r568054 * r568060;
        double r568062 = -r568061;
        double r568063 = sin(r568062);
        double r568064 = r568052 * r568063;
        double r568065 = r568051 - r568064;
        double r568066 = r568023 * r568065;
        double r568067 = a;
        double r568068 = b;
        double r568069 = r568068 * r568028;
        double r568070 = r568067 / r568069;
        double r568071 = r568066 - r568070;
        double r568072 = 0.5;
        double r568073 = pow(r568024, r568041);
        double r568074 = r568072 * r568073;
        double r568075 = r568036 - r568074;
        double r568076 = r568023 * r568075;
        double r568077 = r568076 - r568070;
        double r568078 = r568034 ? r568071 : r568077;
        return r568078;
}

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	20.6
Target	18.6
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 2 regimes

2. if (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) < 1.9652642423881642e+152

   1. Initial program 14.5

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

   3. Applied add-cube-cbrt 14.5

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

   5. Applied add-sqr-sqrt 14.4

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

   6. Applied times-frac 14.5

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

   7. Applied cbrt-prod 14.5

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

   9. Applied sub-neg 14.5

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

   10. Applied cos-sum 13.9

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

   11. Simplified 13.9

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

   12. Taylor expanded around -inf 13.9

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(\left({\left(\sqrt[3]{-1}\right)}^{2} \cdot \left(z \cdot \left(t \cdot {\left(\sqrt[3]{0.3333333333333333148296162562473909929395}\right)}^{2}\right)\right)\right) \cdot {\left(\frac{1}{{\left(\sqrt{3}\right)}^{2}}\right)}^{\frac{1}{3}}\right)} - \sin y \cdot \sin \left(-\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   13. Simplified 13.9

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(\sqrt[3]{\frac{1}{3}} \cdot \left({\left(\sqrt[3]{-1}\right)}^{2} \cdot \left(z \cdot \left(t \cdot {\left(\sqrt[3]{0.3333333333333333148296162562473909929395}\right)}^{2}\right)\right)\right)\right)} - \sin y \cdot \sin \left(-\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   if 1.9652642423881642e+152 < (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))

   1. Initial program 62.5

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

   2. Taylor expanded around 0 44.3

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 1.965264242388164234247715384116746023602 \cdot 10^{152}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\sqrt[3]{\frac{1}{3}} \cdot \left({\left(\sqrt[3]{-1}\right)}^{2} \cdot \left(z \cdot \left(t \cdot {\left(\sqrt[3]{0.3333333333333333148296162562473909929395}\right)}^{2}\right)\right)\right)\right) - \sin y \cdot \sin \left(-\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt{3}}} \cdot \sqrt[3]{\frac{t}{\sqrt{3}}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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))))