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

Percentage Accurate: 70.5% → 77.7%

Time: 19.6s

Precision: binary64

Cost: 53828

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

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{\frac{3}{z}}\\ t_2 := \frac{a}{3 \cdot b}\\ t_3 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t_3 \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - t_2 \leq 5 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \mathsf{fma}\left(\cos y, \cos t_1, \sin y \cdot \sin t_1\right)\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_3 + \frac{a}{3} \cdot \frac{-1}{b}\\ \end{array} \]

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 (/ t (/ 3.0 z))) (t_2 (/ a (* 3.0 b))) (t_3 (* 2.0 (sqrt x))))
   (if (<= (- (* t_3 (cos (- y (/ (* z t) 3.0)))) t_2) 5e+150)
     (- (* 2.0 (* (sqrt x) (fma (cos y) (cos t_1) (* (sin y) (sin t_1))))) t_2)
     (+ t_3 (* (/ 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 = t / (3.0 / z);
	double t_2 = a / (3.0 * b);
	double t_3 = 2.0 * sqrt(x);
	double tmp;
	if (((t_3 * cos((y - ((z * t) / 3.0)))) - t_2) <= 5e+150) {
		tmp = (2.0 * (sqrt(x) * fma(cos(y), cos(t_1), (sin(y) * sin(t_1))))) - t_2;
	} else {
		tmp = t_3 + ((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(t / Float64(3.0 / z))
	t_2 = Float64(a / Float64(3.0 * b))
	t_3 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (Float64(Float64(t_3 * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - t_2) <= 5e+150)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * fma(cos(y), cos(t_1), Float64(sin(y) * sin(t_1))))) - t_2);
	else
		tmp = Float64(t_3 + 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[(t / N[(3.0 / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$3 * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], 5e+150], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(t$95$3 + N[(N[(a / 3.0), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	70.5%
Target	74.4%
Herbie	77.7%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 a (*.f64 b 3))) < 5.00000000000000009e150

   1. Initial program 78.3%

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

   2. Simplified 78.5%

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

      [Start] 78.3%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
      associate-*l* [=>] 78.3%	\[ \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]
      fma-neg [=>] 78.3%	\[ \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right), -\frac{a}{b \cdot 3}\right)} \]
      remove-double-neg [<=] 78.3%	\[ \mathsf{fma}\left(2, \color{blue}{-\left(-\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)}, -\frac{a}{b \cdot 3}\right) \]
      fma-neg [<=] 78.3%	\[ \color{blue}{2 \cdot \left(-\left(-\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}} \]
      remove-double-neg [=>] 78.3%	\[ 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]
      associate-/l* [=>] 78.5%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \color{blue}{\frac{z}{\frac{3}{t}}}\right)\right) - \frac{a}{b \cdot 3} \]
      *-commutative [=>] 78.5%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{\frac{3}{t}}\right)\right) - \frac{a}{\color{blue}{3 \cdot b}} \]

   3. Applied egg-rr 78.2%

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

      [Start] 78.5%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{\frac{3}{t}}\right)\right) - \frac{a}{3 \cdot b} \]
      associate-/r/ [=>] 78.1%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \color{blue}{\frac{z}{3} \cdot t}\right)\right) - \frac{a}{3 \cdot b} \]
      add-cube-cbrt [=>] 78.1%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \color{blue}{\left(\sqrt[3]{\frac{z}{3} \cdot t} \cdot \sqrt[3]{\frac{z}{3} \cdot t}\right) \cdot \sqrt[3]{\frac{z}{3} \cdot t}}\right)\right) - \frac{a}{3 \cdot b} \]
      pow3 [=>] 78.2%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \color{blue}{{\left(\sqrt[3]{\frac{z}{3} \cdot t}\right)}^{3}}\right)\right) - \frac{a}{3 \cdot b} \]
      *-commutative [=>] 78.2%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - {\left(\sqrt[3]{\color{blue}{t \cdot \frac{z}{3}}}\right)}^{3}\right)\right) - \frac{a}{3 \cdot b} \]

   4. Applied egg-rr 79.5%

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

      [Start] 78.2%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - {\left(\sqrt[3]{t \cdot \frac{z}{3}}\right)}^{3}\right)\right) - \frac{a}{3 \cdot b} \]
      cos-diff [=>] 78.9%	\[ 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(\cos y \cdot \cos \left({\left(\sqrt[3]{t \cdot \frac{z}{3}}\right)}^{3}\right) + \sin y \cdot \sin \left({\left(\sqrt[3]{t \cdot \frac{z}{3}}\right)}^{3}\right)\right)}\right) - \frac{a}{3 \cdot b} \]
      unpow3 [=>] 78.8%	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \color{blue}{\left(\left(\sqrt[3]{t \cdot \frac{z}{3}} \cdot \sqrt[3]{t \cdot \frac{z}{3}}\right) \cdot \sqrt[3]{t \cdot \frac{z}{3}}\right)} + \sin y \cdot \sin \left({\left(\sqrt[3]{t \cdot \frac{z}{3}}\right)}^{3}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      add-cube-cbrt [<=] 79.1%	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \color{blue}{\left(t \cdot \frac{z}{3}\right)} + \sin y \cdot \sin \left({\left(\sqrt[3]{t \cdot \frac{z}{3}}\right)}^{3}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      associate-*r/ [=>] 79.4%	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \color{blue}{\left(\frac{t \cdot z}{3}\right)} + \sin y \cdot \sin \left({\left(\sqrt[3]{t \cdot \frac{z}{3}}\right)}^{3}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      unpow3 [=>] 79.4%	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right) + \sin y \cdot \sin \color{blue}{\left(\left(\sqrt[3]{t \cdot \frac{z}{3}} \cdot \sqrt[3]{t \cdot \frac{z}{3}}\right) \cdot \sqrt[3]{t \cdot \frac{z}{3}}\right)}\right)\right) - \frac{a}{3 \cdot b} \]
      add-cube-cbrt [<=] 79.5%	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right) + \sin y \cdot \sin \color{blue}{\left(t \cdot \frac{z}{3}\right)}\right)\right) - \frac{a}{3 \cdot b} \]
      associate-*r/ [=>] 79.5%	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right) + \sin y \cdot \sin \color{blue}{\left(\frac{t \cdot z}{3}\right)}\right)\right) - \frac{a}{3 \cdot b} \]

   5. Simplified 79.8%

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

      [Start] 79.5%	\[ 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\frac{t \cdot z}{3}\right) + \sin y \cdot \sin \left(\frac{t \cdot z}{3}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      fma-def [=>] 79.5%	\[ 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{fma}\left(\cos y, \cos \left(\frac{t \cdot z}{3}\right), \sin y \cdot \sin \left(\frac{t \cdot z}{3}\right)\right)}\right) - \frac{a}{3 \cdot b} \]
      associate-/l* [=>] 79.4%	\[ 2 \cdot \left(\sqrt{x} \cdot \mathsf{fma}\left(\cos y, \cos \color{blue}{\left(\frac{t}{\frac{3}{z}}\right)}, \sin y \cdot \sin \left(\frac{t \cdot z}{3}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      associate-/l* [=>] 79.8%	\[ 2 \cdot \left(\sqrt{x} \cdot \mathsf{fma}\left(\cos y, \cos \left(\frac{t}{\frac{3}{z}}\right), \sin y \cdot \sin \color{blue}{\left(\frac{t}{\frac{3}{z}}\right)}\right)\right) - \frac{a}{3 \cdot b} \]

   if 5.00000000000000009e150 < (-.f64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 a (*.f64 b 3)))

   1. Initial program 54.8%

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

   2. Simplified 54.8%

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

      [Start] 54.8%	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
      associate-*l* [=>] 54.8%	\[ \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]
      fma-neg [=>] 54.8%	\[ \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right), -\frac{a}{b \cdot 3}\right)} \]
      remove-double-neg [<=] 54.8%	\[ \mathsf{fma}\left(2, \color{blue}{-\left(-\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)}, -\frac{a}{b \cdot 3}\right) \]
      fma-neg [<=] 54.8%	\[ \color{blue}{2 \cdot \left(-\left(-\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}} \]
      remove-double-neg [=>] 54.8%	\[ 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]
      associate-/l* [=>] 54.8%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \color{blue}{\frac{z}{\frac{3}{t}}}\right)\right) - \frac{a}{b \cdot 3} \]
      *-commutative [=>] 54.8%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{\frac{3}{t}}\right)\right) - \frac{a}{\color{blue}{3 \cdot b}} \]

   3. Taylor expanded in z around 0 74.6%

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

   4. Applied egg-rr 74.8%

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

      [Start] 74.6%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{a}{3 \cdot b} \]
      *-commutative [=>] 74.6%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{a}{\color{blue}{b \cdot 3}} \]
      *-un-lft-identity [=>] 74.6%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{\color{blue}{1 \cdot a}}{b \cdot 3} \]
      times-frac [=>] 74.8%	\[ 2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \color{blue}{\frac{1}{b} \cdot \frac{a}{3}} \]

   5. Taylor expanded in y around 0 75.7%

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

4. Final simplification 78.7%

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

Alternatives

Alternative 1
Accuracy	77.7%
Cost	53828

\[\begin{array}{l} t_1 := \frac{t}{\frac{3}{z}}\\ t_2 := \frac{a}{3 \cdot b}\\ t_3 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t_3 \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - t_2 \leq 5 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \mathsf{fma}\left(\cos y, \cos t_1, \sin y \cdot \sin t_1\right)\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_3 + \frac{a}{3} \cdot \frac{-1}{b}\\ \end{array} \]

Alternative 2
Accuracy	76.7%
Cost	13504

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

Alternative 3
Accuracy	66.5%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+160}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \end{array} \]

Alternative 4
Accuracy	65.3%
Cost	6976

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

Alternative 5
Accuracy	65.4%
Cost	6976

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

Alternative 6
Accuracy	50.9%
Cost	320

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

Reproduce

herbie shell --seed 2023171 
(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))))