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

Average Error: 20.7 → 17.3

Time: 11.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x \cdot {\cos y}^{2}} - \frac{1}{b} \cdot \frac{a}{3}\\ t_2 := y - \frac{z \cdot t}{3}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+297}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - {\left(\sqrt[3]{z \cdot \left(t \cdot 0.3333333333333333\right)}\right)}^{3}\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (pow (cos y) 2.0)))) (* (/ 1.0 b) (/ a 3.0))))
        (t_2 (- y (/ (* z t) 3.0))))
   (if (<= t_2 -1e+293)
     t_1
     (if (<= t_2 1e+297)
       (-
        (*
         (* 2.0 (sqrt x))
         (cos (- y (pow (cbrt (* z (* t 0.3333333333333333))) 3.0))))
        (/ a (* 3.0 b)))
       t_1))))

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 * pow(cos(y), 2.0)))) - ((1.0 / b) * (a / 3.0));
	double t_2 = y - ((z * t) / 3.0);
	double tmp;
	if (t_2 <= -1e+293) {
		tmp = t_1;
	} else if (t_2 <= 1e+297) {
		tmp = ((2.0 * sqrt(x)) * cos((y - pow(cbrt((z * (t * 0.3333333333333333))), 3.0)))) - (a / (3.0 * b));
	} else {
		tmp = t_1;
	}
	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 * Math.pow(Math.cos(y), 2.0)))) - ((1.0 / b) * (a / 3.0));
	double t_2 = y - ((z * t) / 3.0);
	double tmp;
	if (t_2 <= -1e+293) {
		tmp = t_1;
	} else if (t_2 <= 1e+297) {
		tmp = ((2.0 * Math.sqrt(x)) * Math.cos((y - Math.pow(Math.cbrt((z * (t * 0.3333333333333333))), 3.0)))) - (a / (3.0 * b));
	} else {
		tmp = t_1;
	}
	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(Float64(2.0 * sqrt(Float64(x * (cos(y) ^ 2.0)))) - Float64(Float64(1.0 / b) * Float64(a / 3.0)))
	t_2 = Float64(y - Float64(Float64(z * t) / 3.0))
	tmp = 0.0
	if (t_2 <= -1e+293)
		tmp = t_1;
	elseif (t_2 <= 1e+297)
		tmp = Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - (cbrt(Float64(z * Float64(t * 0.3333333333333333))) ^ 3.0)))) - Float64(a / Float64(3.0 * b)));
	else
		tmp = t_1;
	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[(N[(2.0 * N[Sqrt[N[(x * N[Power[N[Cos[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / b), $MachinePrecision] * N[(a / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+293], t$95$1, If[LessEqual[t$95$2, 1e+297], N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[Power[N[Power[N[(z * N[(t * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	20.7
Target	19.1
Herbie	17.3

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 y (/.f64 (*.f64 z t) 3)) < -9.9999999999999992e292 or 1e297 < (-.f64 y (/.f64 (*.f64 z t) 3))

   1. Initial program 56.2

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

   2. Taylor expanded in z around 0 31.3

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

   3. Applied egg-rr 31.4

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

   4. Applied egg-rr 32.7

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

   if -9.9999999999999992e292 < (-.f64 y (/.f64 (*.f64 z t) 3)) < 1e297

   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. Applied egg-rr 14.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{{\left(\sqrt[3]{z \cdot \left(t \cdot 0.3333333333333333\right)}\right)}^{3}}\right) - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -1 \cdot 10^{+293}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot {\cos y}^{2}} - \frac{1}{b} \cdot \frac{a}{3}\\ \mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 10^{+297}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - {\left(\sqrt[3]{z \cdot \left(t \cdot 0.3333333333333333\right)}\right)}^{3}\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot {\cos y}^{2}} - \frac{1}{b} \cdot \frac{a}{3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(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))))