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

Average Error: 21.0 → 16.9

Time: 50.9s

Precision: binary64

Cost: 72968

Math

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

↓

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ t_3 := \sin y \cdot \sin \left(z \cdot \left(0.3333333333333333 \cdot t\right)\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+68}:\\ \;\;\;\;t_2 - t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_2 \cdot \mathsf{fma}\left(\sqrt[3]{t_3}, \sqrt[3]{{t_3}^{2}}, \cos y \cdot \cos \left(\left(t \cdot z\right) \cdot -0.3333333333333333\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{4 \cdot {\left(\sqrt{x}\right)}^{2}} \cdot \sqrt[3]{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} - t_1\\ \end{array} \]

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 (/ a (* b 3.0)))
        (t_2 (* 2.0 (sqrt x)))
        (t_3 (* (sin y) (sin (* z (* 0.3333333333333333 t))))))
   (if (<= (* z t) -1e+68)
     (- t_2 t_1)
     (if (<= (* z t) 5e+300)
       (-
        (*
         t_2
         (fma
          (cbrt t_3)
          (cbrt (pow t_3 2.0))
          (* (cos y) (cos (* (* t z) -0.3333333333333333)))))
        t_1)
       (-
        (*
         (cbrt (* 4.0 (pow (sqrt x) 2.0)))
         (cbrt (* 2.0 (* (sqrt x) (cos y)))))
        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 = a / (b * 3.0);
	double t_2 = 2.0 * sqrt(x);
	double t_3 = sin(y) * sin((z * (0.3333333333333333 * t)));
	double tmp;
	if ((z * t) <= -1e+68) {
		tmp = t_2 - t_1;
	} else if ((z * t) <= 5e+300) {
		tmp = (t_2 * fma(cbrt(t_3), cbrt(pow(t_3, 2.0)), (cos(y) * cos(((t * z) * -0.3333333333333333))))) - t_1;
	} else {
		tmp = (cbrt((4.0 * pow(sqrt(x), 2.0))) * cbrt((2.0 * (sqrt(x) * cos(y))))) - 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(a / Float64(b * 3.0))
	t_2 = Float64(2.0 * sqrt(x))
	t_3 = Float64(sin(y) * sin(Float64(z * Float64(0.3333333333333333 * t))))
	tmp = 0.0
	if (Float64(z * t) <= -1e+68)
		tmp = Float64(t_2 - t_1);
	elseif (Float64(z * t) <= 5e+300)
		tmp = Float64(Float64(t_2 * fma(cbrt(t_3), cbrt((t_3 ^ 2.0)), Float64(cos(y) * cos(Float64(Float64(t * z) * -0.3333333333333333))))) - t_1);
	else
		tmp = Float64(Float64(cbrt(Float64(4.0 * (sqrt(x) ^ 2.0))) * cbrt(Float64(2.0 * Float64(sqrt(x) * cos(y))))) - 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[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] * N[Sin[N[(z * N[(0.3333333333333333 * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+68], N[(t$95$2 - t$95$1), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+300], N[(N[(t$95$2 * N[(N[Power[t$95$3, 1/3], $MachinePrecision] * N[Power[N[Power[t$95$3, 2.0], $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[Cos[N[(N[(t * z), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[Power[N[(4.0 * N[Power[N[Sqrt[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Target

Original	21.0
Target	19.1
Herbie	16.9

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.99999999999999953e67

   1. Initial program 43.3

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

   2. Taylor expanded in z around 0 33.8

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

   3. Taylor expanded in y around 0 33.7

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

   if -9.99999999999999953e67 < (*.f64 z t) < 5.00000000000000026e300

   1. Initial program 11.2

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

   2. Applied egg-rr 10.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin y \cdot \sin \left(\left(z \cdot 0.3333333333333333\right) \cdot t\right)}, \sqrt[3]{{\left(\sin y \cdot \sin \left(\left(z \cdot 0.3333333333333333\right) \cdot t\right)\right)}^{2}}, \cos y \cdot \cos \left(\left(z \cdot t\right) \cdot -0.3333333333333333\right)\right)} - \frac{a}{b \cdot 3} \]

   3. Simplified 10.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin y \cdot \sin \left(z \cdot \left(0.3333333333333333 \cdot t\right)\right)}, \sqrt[3]{{\left(\sin y \cdot \sin \left(z \cdot \left(0.3333333333333333 \cdot t\right)\right)\right)}^{2}}, \cos y \cdot \cos \left(\left(t \cdot z\right) \cdot -0.3333333333333333\right)\right)} - \frac{a}{b \cdot 3} \]
      Proof

   if 5.00000000000000026e300 < (*.f64 z t)

   1. Initial program 62.6

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

   2. Taylor expanded in z around 0 35.2

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

   3. Applied egg-rr 35.2

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

   4. Taylor expanded in y around 0 35.1

      \[\leadsto \sqrt[3]{\color{blue}{4 \cdot {\left(\sqrt{x}\right)}^{2}}} \cdot \sqrt[3]{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} - \frac{a}{b \cdot 3} \]
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	16.9
Cost	39944

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+68}:\\ \;\;\;\;t_2 - t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_2 \cdot \left(\sin y \cdot \sin \left(\left(z \cdot 0.3333333333333333\right) \cdot t\right) + \cos y \cdot \cos \left(\left(z \cdot t\right) \cdot -0.3333333333333333\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{4 \cdot {\left(\sqrt{x}\right)}^{2}} \cdot \sqrt[3]{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} - t_1\\ \end{array} \]

Alternative 2
Error	16.9
Cost	34120

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+68}:\\ \;\;\;\;t_2 - t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_2 \cdot \left(\sin y \cdot \sin \left(\left(z \cdot 0.3333333333333333\right) \cdot t\right) + \cos y \cdot \cos \left(\left(z \cdot t\right) \cdot -0.3333333333333333\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot {\left(\sqrt[3]{\cos y}\right)}^{3} - t_1\\ \end{array} \]

Alternative 3
Error	20.8
Cost	14280

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-119}:\\ \;\;\;\;t_2 - t_1\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-63}:\\ \;\;\;\;2 \cdot \left(\cos \left(-0.3333333333333333 \cdot \left(t \cdot z\right) + y\right) \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 - \frac{\frac{-a}{3}}{-b}\\ \end{array} \]

Alternative 4
Error	20.6
Cost	13896

\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-119}:\\ \;\;\;\;t_2 - t_1\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-63}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 - \frac{\frac{-a}{3}}{-b}\\ \end{array} \]

Alternative 5
Error	17.7
Cost	13504

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

Alternative 6
Error	17.7
Cost	13504

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

Alternative 7
Error	25.6
Cost	6976

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

Alternative 8
Error	36.3
Cost	320

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

Alternative 9
Error	36.3
Cost	320

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

Reproduce

herbie shell --seed 2023010 
(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))))