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

Average Accuracy: 67.3% → 73.4%

Time: 44.6s

Precision: binary64

Cost: 47112

Math

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

↓

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{{\cos y}^{2} \cdot \left(x \cdot 4\right)} - t_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+180}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{{\left(\sqrt[3]{\frac{\frac{-3}{z}}{t}}\right)}^{3}}\right) - \sin y \cdot \sin \left(t \cdot \frac{z}{-3}\right)\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot \left(2 \cdot \sqrt{x}\right) + \frac{\frac{a}{b}}{-3}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))))
   (if (<= (* z t) -2e+197)
     (- (sqrt (* (pow (cos y) 2.0) (* x 4.0))) t_1)
     (if (<= (* z t) 1e+180)
       (-
        (*
         2.0
         (*
          (sqrt x)
          (-
           (* (cos y) (cos (/ 1.0 (pow (cbrt (/ (/ -3.0 z) t)) 3.0))))
           (* (sin y) (sin (* t (/ z -3.0)))))))
        t_1)
       (+ (* (cos y) (* 2.0 (sqrt x))) (/ (/ a b) -3.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double tmp;
	if ((z * t) <= -2e+197) {
		tmp = sqrt((pow(cos(y), 2.0) * (x * 4.0))) - t_1;
	} else if ((z * t) <= 1e+180) {
		tmp = (2.0 * (sqrt(x) * ((cos(y) * cos((1.0 / pow(cbrt(((-3.0 / z) / t)), 3.0)))) - (sin(y) * sin((t * (z / -3.0))))))) - t_1;
	} else {
		tmp = (cos(y) * (2.0 * sqrt(x))) + ((a / b) / -3.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double tmp;
	if ((z * t) <= -2e+197) {
		tmp = Math.sqrt((Math.pow(Math.cos(y), 2.0) * (x * 4.0))) - t_1;
	} else if ((z * t) <= 1e+180) {
		tmp = (2.0 * (Math.sqrt(x) * ((Math.cos(y) * Math.cos((1.0 / Math.pow(Math.cbrt(((-3.0 / z) / t)), 3.0)))) - (Math.sin(y) * Math.sin((t * (z / -3.0))))))) - t_1;
	} else {
		tmp = (Math.cos(y) * (2.0 * Math.sqrt(x))) + ((a / b) / -3.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	tmp = 0.0
	if (Float64(z * t) <= -2e+197)
		tmp = Float64(sqrt(Float64((cos(y) ^ 2.0) * Float64(x * 4.0))) - t_1);
	elseif (Float64(z * t) <= 1e+180)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(cos(y) * cos(Float64(1.0 / (cbrt(Float64(Float64(-3.0 / z) / t)) ^ 3.0)))) - Float64(sin(y) * sin(Float64(t * Float64(z / -3.0))))))) - t_1);
	else
		tmp = Float64(Float64(cos(y) * Float64(2.0 * sqrt(x))) + Float64(Float64(a / b) / -3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+197], N[(N[Sqrt[N[(N[Power[N[Cos[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(x * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+180], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[N[(1.0 / N[Power[N[Power[N[(N[(-3.0 / z), $MachinePrecision] / t), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * N[Sin[N[(t * N[(z / -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[Cos[y], $MachinePrecision] * N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	67.3%
Target	70.3%
Herbie	73.4%

\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.9999999999999999e197

   1. Initial program 21.3%

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

   2. Taylor expanded in z around 0 46.7%

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

   3. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\sqrt{{\cos y}^{2} \cdot \left(x \cdot 4\right)}} - \frac{a}{b \cdot 3} \]
      Proof

      [Start] 46.7	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3} \]
      add-sqr-sqrt [=>] 36.2	\[ \color{blue}{\sqrt{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \cdot \sqrt{\left(2 \cdot \sqrt{x}\right) \cdot \cos y}} - \frac{a}{b \cdot 3} \]
      sqrt-unprod [=>] 47.0	\[ \color{blue}{\sqrt{\left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right) \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right)}} - \frac{a}{b \cdot 3} \]
      *-commutative [=>] 47.0	\[ \sqrt{\color{blue}{\left(\cos y \cdot \left(2 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \cos y\right)} - \frac{a}{b \cdot 3} \]
      *-commutative [=>] 47.0	\[ \sqrt{\left(\cos y \cdot \left(2 \cdot \sqrt{x}\right)\right) \cdot \color{blue}{\left(\cos y \cdot \left(2 \cdot \sqrt{x}\right)\right)}} - \frac{a}{b \cdot 3} \]
      swap-sqr [=>] 47.0	\[ \sqrt{\color{blue}{\left(\cos y \cdot \cos y\right) \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(2 \cdot \sqrt{x}\right)\right)}} - \frac{a}{b \cdot 3} \]
      pow2 [=>] 47.0	\[ \sqrt{\color{blue}{{\cos y}^{2}} \cdot \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(2 \cdot \sqrt{x}\right)\right)} - \frac{a}{b \cdot 3} \]
      *-commutative [=>] 47.0	\[ \sqrt{{\cos y}^{2} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot 2\right)} \cdot \left(2 \cdot \sqrt{x}\right)\right)} - \frac{a}{b \cdot 3} \]
      *-commutative [=>] 47.0	\[ \sqrt{{\cos y}^{2} \cdot \left(\left(\sqrt{x} \cdot 2\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 2\right)}\right)} - \frac{a}{b \cdot 3} \]
      swap-sqr [=>] 47.0	\[ \sqrt{{\cos y}^{2} \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(2 \cdot 2\right)\right)}} - \frac{a}{b \cdot 3} \]
      add-sqr-sqrt [<=] 47.0	\[ \sqrt{{\cos y}^{2} \cdot \left(\color{blue}{x} \cdot \left(2 \cdot 2\right)\right)} - \frac{a}{b \cdot 3} \]
      metadata-eval [=>] 47.0	\[ \sqrt{{\cos y}^{2} \cdot \left(x \cdot \color{blue}{4}\right)} - \frac{a}{b \cdot 3} \]

   if -1.9999999999999999e197 < (*.f64 z t) < 1e180

   1. Initial program 81.5%

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

   2. Simplified 81.4%

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

      [Start] 81.5	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
      associate-*l* [=>] 81.5	\[ \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]
      associate-/l* [=>] 81.4	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \color{blue}{\frac{z}{\frac{3}{t}}}\right)\right) - \frac{a}{b \cdot 3} \]
      *-commutative [=>] 81.4	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{\frac{3}{t}}\right)\right) - \frac{a}{\color{blue}{3 \cdot b}} \]

   3. Applied egg-rr 82.1%

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

      [Start] 81.4	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{\frac{3}{t}}\right)\right) - \frac{a}{3 \cdot b} \]
      sub-neg [=>] 81.4	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \color{blue}{\left(y + \left(-\frac{z}{\frac{3}{t}}\right)\right)}\right) - \frac{a}{3 \cdot b} \]
      cos-sum [=>] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(\cos y \cdot \cos \left(-\frac{z}{\frac{3}{t}}\right) - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)}\right) - \frac{a}{3 \cdot b} \]
      associate-/r/ [=>] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(-\color{blue}{\frac{z}{3} \cdot t}\right) - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      distribute-rgt-neg-in [=>] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \color{blue}{\left(\frac{z}{3} \cdot \left(-t\right)\right)} - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      associate-/r/ [<=] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \color{blue}{\left(\frac{z}{\frac{3}{-t}}\right)} - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      metadata-eval [<=] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{\frac{\color{blue}{--3}}{-t}}\right) - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      metadata-eval [<=] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{\frac{-\color{blue}{\left(-3\right)}}{-t}}\right) - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      distribute-neg-frac [<=] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{\color{blue}{-\frac{-3}{-t}}}\right) - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      frac-2neg [<=] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{-\color{blue}{\frac{3}{t}}}\right) - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      distribute-neg-frac [=>] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{\color{blue}{\frac{-3}{t}}}\right) - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      associate-/r/ [=>] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \color{blue}{\left(\frac{z}{-3} \cdot t\right)} - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      metadata-eval [=>] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{\color{blue}{-3}} \cdot t\right) - \sin y \cdot \sin \left(-\frac{z}{\frac{3}{t}}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      associate-/r/ [=>] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{-3} \cdot t\right) - \sin y \cdot \sin \left(-\color{blue}{\frac{z}{3} \cdot t}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      distribute-rgt-neg-in [=>] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{-3} \cdot t\right) - \sin y \cdot \sin \color{blue}{\left(\frac{z}{3} \cdot \left(-t\right)\right)}\right)\right) - \frac{a}{3 \cdot b} \]
      associate-/r/ [<=] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{-3} \cdot t\right) - \sin y \cdot \sin \color{blue}{\left(\frac{z}{\frac{3}{-t}}\right)}\right)\right) - \frac{a}{3 \cdot b} \]

   4. Applied egg-rr 82.0%

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

      [Start] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{-3} \cdot t\right) - \sin y \cdot \sin \left(\frac{z}{-3} \cdot t\right)\right)\right) - \frac{a}{3 \cdot b} \]
      associate-*l/ [=>] 82.1	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \color{blue}{\left(\frac{z \cdot t}{-3}\right)} - \sin y \cdot \sin \left(\frac{z}{-3} \cdot t\right)\right)\right) - \frac{a}{3 \cdot b} \]
      clear-num [=>] 82.0	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \color{blue}{\left(\frac{1}{\frac{-3}{z \cdot t}}\right)} - \sin y \cdot \sin \left(\frac{z}{-3} \cdot t\right)\right)\right) - \frac{a}{3 \cdot b} \]

   5. Applied egg-rr 82.0%

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

      [Start] 82.0	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{\frac{-3}{z \cdot t}}\right) - \sin y \cdot \sin \left(\frac{z}{-3} \cdot t\right)\right)\right) - \frac{a}{3 \cdot b} \]
      add-cube-cbrt [=>] 82.0	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{\color{blue}{\left(\sqrt[3]{\frac{-3}{z \cdot t}} \cdot \sqrt[3]{\frac{-3}{z \cdot t}}\right) \cdot \sqrt[3]{\frac{-3}{z \cdot t}}}}\right) - \sin y \cdot \sin \left(\frac{z}{-3} \cdot t\right)\right)\right) - \frac{a}{3 \cdot b} \]
      pow3 [=>] 82.0	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{-3}{z \cdot t}}\right)}^{3}}}\right) - \sin y \cdot \sin \left(\frac{z}{-3} \cdot t\right)\right)\right) - \frac{a}{3 \cdot b} \]
      associate-/r* [=>] 82.0	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{{\left(\sqrt[3]{\color{blue}{\frac{\frac{-3}{z}}{t}}}\right)}^{3}}\right) - \sin y \cdot \sin \left(\frac{z}{-3} \cdot t\right)\right)\right) - \frac{a}{3 \cdot b} \]

   if 1e180 < (*.f64 z t)

   1. Initial program 24.1%

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

   2. Taylor expanded in z around 0 45.4%

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

   3. Applied egg-rr 45.5%

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

      [Start] 45.4	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3} \]
      frac-2neg [=>] 45.4	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{-a}{-b \cdot 3}} \]
      distribute-frac-neg [=>] 45.4	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\left(-\frac{a}{-b \cdot 3}\right)} \]
      distribute-rgt-neg-in [=>] 45.4	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \left(-\frac{a}{\color{blue}{b \cdot \left(-3\right)}}\right) \]
      associate-/r* [=>] 45.5	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \left(-\color{blue}{\frac{\frac{a}{b}}{-3}}\right) \]
      metadata-eval [=>] 45.5	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \left(-\frac{\frac{a}{b}}{\color{blue}{-3}}\right) \]

   4. Simplified 45.5%

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

      [Start] 45.5	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \left(-\frac{\frac{a}{b}}{-3}\right) \]
      distribute-neg-frac [=>] 45.5	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{-\frac{a}{b}}{-3}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{{\cos y}^{2} \cdot \left(x \cdot 4\right)} - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \leq 10^{+180}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{{\left(\sqrt[3]{\frac{\frac{-3}{z}}{t}}\right)}^{3}}\right) - \sin y \cdot \sin \left(t \cdot \frac{z}{-3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot \left(2 \cdot \sqrt{x}\right) + \frac{\frac{a}{b}}{-3}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.4%
Cost	34376

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+19}:\\ \;\;\;\;2 \cdot \sqrt{{\cos y}^{2} \cdot x} - t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{\frac{-3}{z} \cdot \frac{1}{t}}\right) - \sin y \cdot \sin \left(t \cdot \frac{z}{-3}\right)\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot \left(2 \cdot \sqrt{x}\right) + a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \]

Alternative 2
Accuracy	73.4%
Cost	34248

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+19}:\\ \;\;\;\;2 \cdot \sqrt{{\cos y}^{2} \cdot x} - t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{1}{\frac{-3}{z \cdot t}}\right) - \sin y \cdot \sin \left(t \cdot \frac{z}{-3}\right)\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot \left(2 \cdot \sqrt{x}\right) + a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \]

Alternative 3
Accuracy	73.5%
Cost	34120

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+19}:\\ \;\;\;\;2 \cdot \sqrt{{\cos y}^{2} \cdot x} - t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+32}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \left(z \cdot -0.3333333333333333\right)\right) - \sin y \cdot \sin \left(t \cdot \frac{z}{-3}\right)\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot \left(2 \cdot \sqrt{x}\right) + \frac{\frac{a}{b}}{-3}\\ \end{array} \]

Alternative 4
Accuracy	73.4%
Cost	34120

\[\begin{array}{l} t_1 := t \cdot \frac{z}{-3}\\ t_2 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+19}:\\ \;\;\;\;2 \cdot \sqrt{{\cos y}^{2} \cdot x} - t_2\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos t_1 - \sin y \cdot \sin t_1\right)\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot \left(2 \cdot \sqrt{x}\right) + a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \]

Alternative 5
Accuracy	73.4%
Cost	34120

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{{\cos y}^{2} \cdot \left(x \cdot 4\right)} - t_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+180}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{-3}\right) - \sin y \cdot \sin \left(\frac{z}{\frac{-3}{t}}\right)\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot \left(2 \cdot \sqrt{x}\right) + \frac{\frac{a}{b}}{-3}\\ \end{array} \]

Alternative 6
Accuracy	73.4%
Cost	34120

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{b \cdot 3}\\ t_3 := z \cdot \left(t \cdot 0.3333333333333333\right)\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{{\cos y}^{2} \cdot \left(x \cdot 4\right)} - t_2\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+150}:\\ \;\;\;\;t_1 \cdot \left(\sin y \cdot \sin t_3 + \cos y \cdot \cos t_3\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot t_1 + a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \]

Alternative 7
Accuracy	67.6%
Cost	13897

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-71} \lor \neg \left(t_1 \leq 5 \cdot 10^{-166}\right):\\ \;\;\;\;2 \cdot \sqrt{x} + \frac{\frac{a}{b}}{-3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos y \cdot \sqrt{x}\right)\\ \end{array} \]

Alternative 8
Accuracy	72.3%
Cost	13504

\[\frac{-0.3333333333333333}{\frac{b}{a}} + 2 \cdot \left(\cos y \cdot \sqrt{x}\right) \]

Alternative 9
Accuracy	72.3%
Cost	13504

\[\cos y \cdot \left(2 \cdot \sqrt{x}\right) + \frac{a}{b} \cdot -0.3333333333333333 \]

Alternative 10
Accuracy	72.4%
Cost	13504

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

Alternative 11
Accuracy	72.3%
Cost	13504

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

Alternative 12
Accuracy	60.1%
Cost	6976

\[2 \cdot \sqrt{x} + \frac{a}{b} \cdot -0.3333333333333333 \]

Alternative 13
Accuracy	60.2%
Cost	6976

\[2 \cdot \sqrt{x} - \frac{a}{b \cdot 3} \]

Alternative 14
Accuracy	60.2%
Cost	6976

\[2 \cdot \sqrt{x} + \frac{\frac{a}{b}}{-3} \]

Alternative 15
Accuracy	43.1%
Cost	320

\[\frac{a}{b} \cdot -0.3333333333333333 \]

Alternative 16
Accuracy	43.1%
Cost	320

\[\frac{a}{\frac{b}{-0.3333333333333333}} \]

Alternative 17
Accuracy	43.1%
Cost	320

\[\frac{a \cdot -0.3333333333333333}{b} \]

Alternative 18
Accuracy	43.2%
Cost	320

\[\frac{\frac{a}{-3}}{b} \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.3793337487235141e+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))))