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

Percentage Accurate: 70.4% → 77.8%

Time: 42.7s

Precision: binary64

Cost: 79428

• Unsound rule application detected in e-graph. Results may not be sound.

\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

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

↓

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

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 (* 2.0 (sqrt x)))
        (t_2 (* (pow (cbrt z) 2.0) (* t (/ (- (cbrt z)) 3.0)))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 1.0)
     (-
      (* t_1 (- (* (cos y) (cos t_2)) (* (sin y) (sin t_2))))
      (/ a (* 3.0 b)))
     (+ t_1 (* (/ a 3.0) (/ -1.0 b))))))

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 = 2.0 * sqrt(x);
	double t_2 = pow(cbrt(z), 2.0) * (t * (-cbrt(z) / 3.0));
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 1.0) {
		tmp = (t_1 * ((cos(y) * cos(t_2)) - (sin(y) * sin(t_2)))) - (a / (3.0 * b));
	} else {
		tmp = t_1 + ((a / 3.0) * (-1.0 / b));
	}
	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 = 2.0 * Math.sqrt(x);
	double t_2 = Math.pow(Math.cbrt(z), 2.0) * (t * (-Math.cbrt(z) / 3.0));
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 1.0) {
		tmp = (t_1 * ((Math.cos(y) * Math.cos(t_2)) - (Math.sin(y) * Math.sin(t_2)))) - (a / (3.0 * b));
	} else {
		tmp = t_1 + ((a / 3.0) * (-1.0 / b));
	}
	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(2.0 * sqrt(x))
	t_2 = Float64((cbrt(z) ^ 2.0) * Float64(t * Float64(Float64(-cbrt(z)) / 3.0)))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 1.0)
		tmp = Float64(Float64(t_1 * Float64(Float64(cos(y) * cos(t_2)) - Float64(sin(y) * sin(t_2)))) - Float64(a / Float64(3.0 * b)));
	else
		tmp = Float64(t_1 + Float64(Float64(a / 3.0) * Float64(-1.0 / b)));
	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[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Power[z, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[((-N[Power[z, 1/3], $MachinePrecision]) / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0], N[(N[(t$95$1 * N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(a / 3.0), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	70.4%
Target	74.6%
Herbie	77.8%

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))) < 1

   1. Initial program 82.3%

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

   2. Applied egg-rr 82.0%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\log \left(e^{\cos \left(y - \frac{z}{\frac{3}{t}}\right)}\right)} - \frac{a}{b \cdot 3} \]
      Step-by-step derivation

      [Start] 82.3%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
      associate-/l* [=>] 82.1%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\frac{z}{\frac{3}{t}}}\right) - \frac{a}{b \cdot 3} \]
      add-log-exp [=>] 82.0%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\log \left(e^{\cos \left(y - \frac{z}{\frac{3}{t}}\right)}\right)} - \frac{a}{b \cdot 3} \]

   3. Applied egg-rr 82.9%

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

      [Start] 82.0%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \log \left(e^{\cos \left(y - \frac{z}{\frac{3}{t}}\right)}\right) - \frac{a}{b \cdot 3} \]
      add-log-exp [<=] 82.1%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos \left(y - \frac{z}{\frac{3}{t}}\right)} - \frac{a}{b \cdot 3} \]
      add-cube-cbrt [=>] 81.9%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\frac{3}{t}}\right) - \frac{a}{b \cdot 3} \]
      unpow2 [<=] 81.9%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{\color{blue}{{\left(\sqrt[3]{z}\right)}^{2}} \cdot \sqrt[3]{z}}{\frac{3}{t}}\right) - \frac{a}{b \cdot 3} \]
      associate-*r/ [<=] 82.0%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{{\left(\sqrt[3]{z}\right)}^{2} \cdot \frac{\sqrt[3]{z}}{\frac{3}{t}}}\right) - \frac{a}{b \cdot 3} \]
      cancel-sign-sub-inv [=>] 82.0%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(y + \left(-{\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \frac{\sqrt[3]{z}}{\frac{3}{t}}\right)} - \frac{a}{b \cdot 3} \]
      cos-sum [=>] 82.6%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\left(-{\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \frac{\sqrt[3]{z}}{\frac{3}{t}}\right) - \sin y \cdot \sin \left(\left(-{\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \frac{\sqrt[3]{z}}{\frac{3}{t}}\right)\right)} - \frac{a}{b \cdot 3} \]
      associate-/r/ [=>] 82.7%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(-{\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{z}}{3} \cdot t\right)}\right) - \sin y \cdot \sin \left(\left(-{\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \frac{\sqrt[3]{z}}{\frac{3}{t}}\right)\right) - \frac{a}{b \cdot 3} \]
      associate-/r/ [=>] 82.9%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(-{\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \left(\frac{\sqrt[3]{z}}{3} \cdot t\right)\right) - \sin y \cdot \sin \left(\left(-{\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{z}}{3} \cdot t\right)}\right)\right) - \frac{a}{b \cdot 3} \]

   if 1 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))

   1. Initial program 0.0%

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

   2. Taylor expanded in z around 0 59.8%

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

   3. Applied egg-rr 60.0%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{1}{b} \cdot \frac{a}{3}} \]
      Step-by-step derivation

      [Start] 59.8%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3} \]
      *-un-lft-identity [=>] 59.8%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\color{blue}{1 \cdot a}}{b \cdot 3} \]
      times-frac [=>] 60.0%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{1}{b} \cdot \frac{a}{3}} \]

   4. Taylor expanded in y around 0 63.1%

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

4. Final simplification 80.5%

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

Alternatives

Alternative 1
Accuracy	77.8%
Cost	79428

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

Alternative 2
Accuracy	77.7%
Cost	47172

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{z}{\frac{3}{t}}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot t_1 \leq 5 \cdot 10^{+148}:\\ \;\;\;\;t_1 \cdot \left(\cos y \cdot \cos t_2 + \sin y \cdot \sin t_2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{a}{3} \cdot \frac{-1}{b}\\ \end{array} \]

Alternative 3
Accuracy	71.8%
Cost	13896

\[\begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-134}:\\ \;\;\;\;t_2 + \frac{a}{3} \cdot \frac{-1}{b}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-192}:\\ \;\;\;\;t_2 \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t_2 - t_1\\ \end{array} \]

Alternative 4
Accuracy	76.9%
Cost	13504

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

Alternative 5
Accuracy	66.0%
Cost	6976

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

Alternative 6
Accuracy	66.1%
Cost	6976

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

Alternative 7
Accuracy	50.5%
Cost	320

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

Alternative 8
Accuracy	50.6%
Cost	320

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

Alternative 9
Accuracy	50.6%
Cost	320

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

Reproduce

herbie shell --seed 2023263 
(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))))