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

Average Error: 20.7 → 17.8

Time: 29.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 8.139896335094742174168253426990729793648 \cdot 10^{149}:\\ \;\;\;\;\left(\left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r436045 = 2.0;
        double r436046 = x;
        double r436047 = sqrt(r436046);
        double r436048 = r436045 * r436047;
        double r436049 = y;
        double r436050 = z;
        double r436051 = t;
        double r436052 = r436050 * r436051;
        double r436053 = 3.0;
        double r436054 = r436052 / r436053;
        double r436055 = r436049 - r436054;
        double r436056 = cos(r436055);
        double r436057 = r436048 * r436056;
        double r436058 = a;
        double r436059 = b;
        double r436060 = r436059 * r436053;
        double r436061 = r436058 / r436060;
        double r436062 = r436057 - r436061;
        return r436062;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r436063 = 2.0;
        double r436064 = x;
        double r436065 = sqrt(r436064);
        double r436066 = r436063 * r436065;
        double r436067 = y;
        double r436068 = z;
        double r436069 = t;
        double r436070 = r436068 * r436069;
        double r436071 = 3.0;
        double r436072 = r436070 / r436071;
        double r436073 = r436067 - r436072;
        double r436074 = cos(r436073);
        double r436075 = r436066 * r436074;
        double r436076 = 8.139896335094742e+149;
        bool r436077 = r436075 <= r436076;
        double r436078 = cos(r436067);
        double r436079 = 0.3333333333333333;
        double r436080 = r436069 * r436068;
        double r436081 = r436079 * r436080;
        double r436082 = cos(r436081);
        double r436083 = cbrt(r436082);
        double r436084 = cos(r436072);
        double r436085 = cbrt(r436084);
        double r436086 = r436083 * r436085;
        double r436087 = r436086 * r436085;
        double r436088 = r436078 * r436087;
        double r436089 = r436088 * r436066;
        double r436090 = sin(r436067);
        double r436091 = sin(r436072);
        double r436092 = r436090 * r436091;
        double r436093 = r436092 * r436066;
        double r436094 = r436089 + r436093;
        double r436095 = a;
        double r436096 = b;
        double r436097 = r436095 / r436096;
        double r436098 = r436097 / r436071;
        double r436099 = r436094 - r436098;
        double r436100 = 2.0;
        double r436101 = pow(r436067, r436100);
        double r436102 = -0.5;
        double r436103 = 1.0;
        double r436104 = fma(r436101, r436102, r436103);
        double r436105 = r436066 * r436104;
        double r436106 = r436096 * r436071;
        double r436107 = r436095 / r436106;
        double r436108 = r436105 - r436107;
        double r436109 = r436077 ? r436099 : r436108;
        return r436109;
}

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	20.7
Target	18.7
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)))) < 8.139896335094742e+149

   1. Initial program 14.6

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

   3. Applied cos-diff 14.0

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

   4. Applied distribute-lft-in 14.0

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

   5. Simplified 14.0

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

   6. Simplified 14.0

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

   8. Applied associate-/r* 14.1

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

   10. Applied add-cube-cbrt 14.1

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

   11. Taylor expanded around inf 14.1

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

   if 8.139896335094742e+149 < (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))

   1. Initial program 60.9

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

   2. Taylor expanded around 0 42.3

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

   3. Simplified 42.3

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 8.139896335094742174168253426990729793648 \cdot 10^{149}:\\ \;\;\;\;\left(\left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left({y}^{2}, \frac{-1}{2}, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(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))))