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

Average Error: 21.0 → 16.7

Time: 29.0s

Precision: binary64

Cost: 59588

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{z}{\frac{-3}{t}}\\ t_2 := \frac{a}{b \cdot 3}\\ t_3 := {\left(4 \cdot \left(x \cdot {\cos y}^{2}\right)\right)}^{0.16666666666666666}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+189}:\\ \;\;\;\;\left(t_3 \cdot t_3\right) \cdot \sqrt[3]{\sqrt{x} \cdot \left(\cos y \cdot 2\right)} - t_2\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos t_1 - \sin y \cdot {\left(\sqrt[3]{\sin t_1}\right)}^{3}\right)\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - t_2\\ \end{array} \]

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 (/ z (/ -3.0 t)))
        (t_2 (/ a (* b 3.0)))
        (t_3 (pow (* 4.0 (* x (pow (cos y) 2.0))) 0.16666666666666666)))
   (if (<= (* z t) -5e+189)
     (- (* (* t_3 t_3) (cbrt (* (sqrt x) (* (cos y) 2.0)))) t_2)
     (if (<= (* z t) 5e+94)
       (-
        (*
         2.0
         (*
          (sqrt x)
          (- (* (cos y) (cos t_1)) (* (sin y) (pow (cbrt (sin t_1)) 3.0)))))
        t_2)
       (- (* 2.0 (sqrt x)) t_2)))))

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 = z / (-3.0 / t);
	double t_2 = a / (b * 3.0);
	double t_3 = pow((4.0 * (x * pow(cos(y), 2.0))), 0.16666666666666666);
	double tmp;
	if ((z * t) <= -5e+189) {
		tmp = ((t_3 * t_3) * cbrt((sqrt(x) * (cos(y) * 2.0)))) - t_2;
	} else if ((z * t) <= 5e+94) {
		tmp = (2.0 * (sqrt(x) * ((cos(y) * cos(t_1)) - (sin(y) * pow(cbrt(sin(t_1)), 3.0))))) - t_2;
	} else {
		tmp = (2.0 * sqrt(x)) - t_2;
	}
	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 = z / (-3.0 / t);
	double t_2 = a / (b * 3.0);
	double t_3 = Math.pow((4.0 * (x * Math.pow(Math.cos(y), 2.0))), 0.16666666666666666);
	double tmp;
	if ((z * t) <= -5e+189) {
		tmp = ((t_3 * t_3) * Math.cbrt((Math.sqrt(x) * (Math.cos(y) * 2.0)))) - t_2;
	} else if ((z * t) <= 5e+94) {
		tmp = (2.0 * (Math.sqrt(x) * ((Math.cos(y) * Math.cos(t_1)) - (Math.sin(y) * Math.pow(Math.cbrt(Math.sin(t_1)), 3.0))))) - t_2;
	} else {
		tmp = (2.0 * Math.sqrt(x)) - t_2;
	}
	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(z / Float64(-3.0 / t))
	t_2 = Float64(a / Float64(b * 3.0))
	t_3 = Float64(4.0 * Float64(x * (cos(y) ^ 2.0))) ^ 0.16666666666666666
	tmp = 0.0
	if (Float64(z * t) <= -5e+189)
		tmp = Float64(Float64(Float64(t_3 * t_3) * cbrt(Float64(sqrt(x) * Float64(cos(y) * 2.0)))) - t_2);
	elseif (Float64(z * t) <= 5e+94)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(cos(y) * cos(t_1)) - Float64(sin(y) * (cbrt(sin(t_1)) ^ 3.0))))) - t_2);
	else
		tmp = Float64(Float64(2.0 * sqrt(x)) - t_2);
	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[(z / N[(-3.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(4.0 * N[(x * N[Power[N[Cos[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+189], N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] * N[Power[N[(N[Sqrt[x], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+94], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * N[Power[N[Power[N[Sin[t$95$1], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Target

Original	21.0
Target	19.2
Herbie	16.7

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000004e189

   1. Initial program 50.6

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

   2. Taylor expanded in z around 0 34.5

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

   3. Applied egg-rr 43.0

      \[\leadsto \color{blue}{{\left(8 \cdot {\left(\sqrt{x} \cdot \cos y\right)}^{3}\right)}^{0.3333333333333333}} - \frac{a}{b \cdot 3} \]

   4. Simplified 37.6

      \[\leadsto \color{blue}{\sqrt[3]{8 \cdot {\left(\sqrt{x} \cdot \cos y\right)}^{3}}} - \frac{a}{b \cdot 3} \]
      Proof

      [Start] 43.0	\[ {\left(8 \cdot {\left(\sqrt{x} \cdot \cos y\right)}^{3}\right)}^{0.3333333333333333} - \frac{a}{b \cdot 3} \]
      unpow1/3 [=>] 37.6	\[ \color{blue}{\sqrt[3]{8 \cdot {\left(\sqrt{x} \cdot \cos y\right)}^{3}}} - \frac{a}{b \cdot 3} \]

   5. Applied egg-rr 34.5

      \[\leadsto \color{blue}{\left({\left(4 \cdot \left(x \cdot {\cos y}^{2}\right)\right)}^{0.16666666666666666} \cdot {\left(4 \cdot \left(x \cdot {\cos y}^{2}\right)\right)}^{0.16666666666666666}\right) \cdot \sqrt[3]{\sqrt{x} \cdot \left(\cos y \cdot 2\right)}} - \frac{a}{b \cdot 3} \]

   if -5.0000000000000004e189 < (*.f64 z t) < 5.0000000000000001e94

   1. Initial program 9.9

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

   2. Simplified 9.9

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

      [Start] 9.9	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
      associate-*l* [=>] 9.9	\[ \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]
      associate-/l* [=>] 9.9	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \color{blue}{\frac{z}{\frac{3}{t}}}\right)\right) - \frac{a}{b \cdot 3} \]
      *-commutative [=>] 9.9	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{\frac{3}{t}}\right)\right) - \frac{a}{\color{blue}{3 \cdot b}} \]

   3. Applied egg-rr 9.2

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

   4. Applied egg-rr 9.3

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

   5. Applied egg-rr 9.3

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

   if 5.0000000000000001e94 < (*.f64 z t)

   1. Initial program 43.8

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

   2. Taylor expanded in z around 0 33.4

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

   3. Taylor expanded in y around 0 33.4

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

4. Final simplification 16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+189}:\\ \;\;\;\;\left({\left(4 \cdot \left(x \cdot {\cos y}^{2}\right)\right)}^{0.16666666666666666} \cdot {\left(4 \cdot \left(x \cdot {\cos y}^{2}\right)\right)}^{0.16666666666666666}\right) \cdot \sqrt[3]{\sqrt{x} \cdot \left(\cos y \cdot 2\right)} - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{\frac{-3}{t}}\right) - \sin y \cdot {\left(\sqrt[3]{\sin \left(\frac{z}{\frac{-3}{t}}\right)}\right)}^{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{b \cdot 3}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.7
Cost	46984

\[\begin{array}{l} t_1 := \frac{z}{\frac{-3}{t}}\\ t_2 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+189}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\cos y \cdot \sqrt{x}\right)}\right)}^{3} - t_2\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos t_1 - \sin y \cdot {\left(\sqrt[3]{\sin t_1}\right)}^{3}\right)\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - t_2\\ \end{array} \]

Alternative 2
Error	16.7
Cost	34120

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+189}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\cos y \cdot \sqrt{x}\right)}\right)}^{3} - t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+179}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{z}{\frac{-3}{t}}\right) - \sin y \cdot \sin \left(t \cdot \frac{z}{-3}\right)\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} + \frac{a}{3} \cdot \frac{-1}{b}\\ \end{array} \]

Alternative 3
Error	17.4
Cost	19776

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

Alternative 4
Error	20.5
Cost	14025

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-130} \lor \neg \left(t_1 \leq 2 \cdot 10^{-95}\right):\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, \frac{\frac{a}{-3}}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\cos y \cdot 2\right)\\ \end{array} \]

Alternative 5
Error	20.5
Cost	13896

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-130}:\\ \;\;\;\;t_2 - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{x} \cdot \left(\cos y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 - t_1\\ \end{array} \]

Alternative 6
Error	17.5
Cost	13632

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

Alternative 7
Error	17.5
Cost	13504

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

Alternative 8
Error	25.7
Cost	7104

\[2 \cdot \sqrt{x} + \frac{a}{3} \cdot \frac{-1}{b} \]

Alternative 9
Error	25.7
Cost	6976

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

Alternative 10
Error	25.7
Cost	6976

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

Alternative 11
Error	36.8
Cost	320

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

Alternative 12
Error	36.8
Cost	320

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

Alternative 13
Error	36.8
Cost	320

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

Reproduce

herbie shell --seed 2023018 
(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))))