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

Percentage Accurate: 70.3% → 77.4%

Time: 21.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

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));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

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));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
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]

Initial Program: 70.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

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));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

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));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
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]

Alternative 1: 77.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 0.3333333333333333 (* z t))) (t_2 (/ a (* 3.0 b))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 2.0)
     (- (* (* 2.0 (sqrt x)) (fma (cos t_1) (cos y) (* (sin y) (sin t_1)))) t_2)
     (- (* 2.0 (* (sqrt x) (- (cos y)))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 0.3333333333333333 * (z * t);
	double t_2 = a / (3.0 * b);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 2.0) {
		tmp = ((2.0 * sqrt(x)) * fma(cos(t_1), cos(y), (sin(y) * sin(t_1)))) - t_2;
	} else {
		tmp = (2.0 * (sqrt(x) * -cos(y))) - t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(0.3333333333333333 * Float64(z * t))
	t_2 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 2.0)
		tmp = Float64(Float64(Float64(2.0 * sqrt(x)) * fma(cos(t_1), cos(y), Float64(sin(y) * sin(t_1)))) - t_2);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(-cos(y)))) - t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(0.3333333333333333 * N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * (-N[Cos[y], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

      1. cos-diff 82.4%

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

      2. associate-/l* 81.9%

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

      3. associate-/l* 81.8%

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

   3. Applied egg-rr 81.8%

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

      1. cos-neg 81.8%

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

      2. *-commutative 81.8%

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

      3. cancel-sign-sub 81.8%

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

      4. sin-neg 81.8%

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

      5. *-commutative 81.8%

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

      6. *-commutative 81.8%

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

      7. fma-neg 81.8%

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

   5. Simplified 82.3%

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

   if 2 < (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. Step-by-step derivation

      1. add-cube-cbrt 0.0%

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

      2. pow3 0.0%

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

      3. associate-*l* 0.0%

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

      4. associate-/l* 0.0%

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

   3. Applied egg-rr 0.0%

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

   4. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{x}\right)\right)} - \frac{a}{b \cdot 3} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\left({1}^{0.16666666666666666} \cdot \sqrt{x}\right) \cdot \left(\cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} - \frac{a}{b \cdot 3} \]

      2. pow-base-1 0.0%

         \[\leadsto 2 \cdot \left(\left(\color{blue}{1} \cdot \sqrt{x}\right) \cdot \left(\cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) - \frac{a}{b \cdot 3} \]

      3. *-lft-identity 0.0%

         \[\leadsto 2 \cdot \left(\color{blue}{\sqrt{x}} \cdot \left(\cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) - \frac{a}{b \cdot 3} \]

      4. *-commutative 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)}\right) - \frac{a}{b \cdot 3} \]

      5. unpow2 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3} \]

      6. rem-square-sqrt 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(\color{blue}{-1} \cdot \cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3} \]

      7. sub-neg 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \color{blue}{\left(y + \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)}\right)\right) - \frac{a}{b \cdot 3} \]

      8. +-commutative 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right) + y\right)}\right)\right) - \frac{a}{b \cdot 3} \]

      9. distribute-lft-neg-in 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(\color{blue}{\left(-0.3333333333333333\right) \cdot \left(t \cdot z\right)} + y\right)\right)\right) - \frac{a}{b \cdot 3} \]

      10. metadata-eval 0.0%

          \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(\color{blue}{-0.3333333333333333} \cdot \left(t \cdot z\right) + y\right)\right)\right) - \frac{a}{b \cdot 3} \]

      11. *-commutative 0.0%

          \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(\color{blue}{\left(t \cdot z\right) \cdot -0.3333333333333333} + y\right)\right)\right) - \frac{a}{b \cdot 3} \]

      12. associate-*r* 0.0%

          \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(\color{blue}{t \cdot \left(z \cdot -0.3333333333333333\right)} + y\right)\right)\right) - \frac{a}{b \cdot 3} \]

      13. *-commutative 0.0%

          \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(t \cdot \color{blue}{\left(-0.3333333333333333 \cdot z\right)} + y\right)\right)\right) - \frac{a}{b \cdot 3} \]

      14. fma-udef 0.0%

          \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(t, -0.3333333333333333 \cdot z, y\right)\right)}\right)\right) - \frac{a}{b \cdot 3} \]

   6. Simplified 0.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(\mathsf{fma}\left(t, -0.3333333333333333 \cdot z, y\right)\right)\right)\right)} - \frac{a}{b \cdot 3} \]

   7. Taylor expanded in t around 0 56.1%

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

      1. mul-1-neg 56.1%

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

      2. distribute-rgt-neg-in 56.1%

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

   9. Simplified 56.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 2:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos \left(0.3333333333333333 \cdot \left(z \cdot t\right)\right), \cos y, \sin y \cdot \sin \left(0.3333333333333333 \cdot \left(z \cdot t\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(-\cos y\right)\right) - \frac{a}{3 \cdot b}\\ \end{array} \]

Alternative 2: 77.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ z (/ 3.0 t))) (t_2 (/ a (* 3.0 b))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 2.0)
     (-
      (* (* 2.0 (sqrt x)) (+ (* (cos y) (cos t_1)) (* (sin y) (sin t_1))))
      t_2)
     (- (* 2.0 (* (sqrt x) (- (cos y)))) t_2))))

C

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 / (3.0 * b);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 2.0) {
		tmp = ((2.0 * sqrt(x)) * ((cos(y) * cos(t_1)) + (sin(y) * sin(t_1)))) - t_2;
	} else {
		tmp = (2.0 * (sqrt(x) * -cos(y))) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = z / (3.0d0 / t)
    t_2 = a / (3.0d0 * b)
    if (cos((y - ((z * t) / 3.0d0))) <= 2.0d0) then
        tmp = ((2.0d0 * sqrt(x)) * ((cos(y) * cos(t_1)) + (sin(y) * sin(t_1)))) - t_2
    else
        tmp = (2.0d0 * (sqrt(x) * -cos(y))) - t_2
    end if
    code = tmp
end function

Java

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 / (3.0 * b);
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 2.0) {
		tmp = ((2.0 * Math.sqrt(x)) * ((Math.cos(y) * Math.cos(t_1)) + (Math.sin(y) * Math.sin(t_1)))) - t_2;
	} else {
		tmp = (2.0 * (Math.sqrt(x) * -Math.cos(y))) - t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = z / (3.0 / t)
	t_2 = a / (3.0 * b)
	tmp = 0
	if math.cos((y - ((z * t) / 3.0))) <= 2.0:
		tmp = ((2.0 * math.sqrt(x)) * ((math.cos(y) * math.cos(t_1)) + (math.sin(y) * math.sin(t_1)))) - t_2
	else:
		tmp = (2.0 * (math.sqrt(x) * -math.cos(y))) - t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z / Float64(3.0 / t))
	t_2 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 2.0)
		tmp = Float64(Float64(Float64(2.0 * sqrt(x)) * Float64(Float64(cos(y) * cos(t_1)) + Float64(sin(y) * sin(t_1)))) - t_2);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(-cos(y)))) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = z / (3.0 / t);
	t_2 = a / (3.0 * b);
	tmp = 0.0;
	if (cos((y - ((z * t) / 3.0))) <= 2.0)
		tmp = ((2.0 * sqrt(x)) * ((cos(y) * cos(t_1)) + (sin(y) * sin(t_1)))) - t_2;
	else
		tmp = (2.0 * (sqrt(x) * -cos(y))) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

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[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * (-N[Cos[y], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

      1. cos-diff 82.4%

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

      2. associate-/l* 81.9%

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

      3. associate-/l* 81.8%

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

   3. Applied egg-rr 81.8%

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

   if 2 < (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. Step-by-step derivation

      1. add-cube-cbrt 0.0%

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

      2. pow3 0.0%

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

      3. associate-*l* 0.0%

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

      4. associate-/l* 0.0%

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

   3. Applied egg-rr 0.0%

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

   4. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{x}\right)\right)} - \frac{a}{b \cdot 3} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\left({1}^{0.16666666666666666} \cdot \sqrt{x}\right) \cdot \left(\cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} - \frac{a}{b \cdot 3} \]

      2. pow-base-1 0.0%

         \[\leadsto 2 \cdot \left(\left(\color{blue}{1} \cdot \sqrt{x}\right) \cdot \left(\cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) - \frac{a}{b \cdot 3} \]

      3. *-lft-identity 0.0%

         \[\leadsto 2 \cdot \left(\color{blue}{\sqrt{x}} \cdot \left(\cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) - \frac{a}{b \cdot 3} \]

      4. *-commutative 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)}\right) - \frac{a}{b \cdot 3} \]

      5. unpow2 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3} \]

      6. rem-square-sqrt 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(\color{blue}{-1} \cdot \cos \left(y - 0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3} \]

      7. sub-neg 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \color{blue}{\left(y + \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)}\right)\right) - \frac{a}{b \cdot 3} \]

      8. +-commutative 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right) + y\right)}\right)\right) - \frac{a}{b \cdot 3} \]

      9. distribute-lft-neg-in 0.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(\color{blue}{\left(-0.3333333333333333\right) \cdot \left(t \cdot z\right)} + y\right)\right)\right) - \frac{a}{b \cdot 3} \]

      10. metadata-eval 0.0%

          \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(\color{blue}{-0.3333333333333333} \cdot \left(t \cdot z\right) + y\right)\right)\right) - \frac{a}{b \cdot 3} \]

      11. *-commutative 0.0%

          \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(\color{blue}{\left(t \cdot z\right) \cdot -0.3333333333333333} + y\right)\right)\right) - \frac{a}{b \cdot 3} \]

      12. associate-*r* 0.0%

          \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(\color{blue}{t \cdot \left(z \cdot -0.3333333333333333\right)} + y\right)\right)\right) - \frac{a}{b \cdot 3} \]

      13. *-commutative 0.0%

          \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(t \cdot \color{blue}{\left(-0.3333333333333333 \cdot z\right)} + y\right)\right)\right) - \frac{a}{b \cdot 3} \]

      14. fma-udef 0.0%

          \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(t, -0.3333333333333333 \cdot z, y\right)\right)}\right)\right) - \frac{a}{b \cdot 3} \]

   6. Simplified 0.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \left(-1 \cdot \cos \left(\mathsf{fma}\left(t, -0.3333333333333333 \cdot z, y\right)\right)\right)\right)} - \frac{a}{b \cdot 3} \]

   7. Taylor expanded in t around 0 56.1%

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

      1. mul-1-neg 56.1%

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

      2. distribute-rgt-neg-in 56.1%

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

   9. Simplified 56.1%

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

4. Final simplification 78.8%

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

Alternative 3: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+22} \lor \neg \left(t_1 \leq 5 \cdot 10^{-201}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (or (<= t_1 -4e+22) (not (<= t_1 5e-201)))
     (- (* 2.0 (sqrt x)) t_1)
     (* 2.0 (* (sqrt x) (cos y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if ((t_1 <= -4e+22) || !(t_1 <= 5e-201)) {
		tmp = (2.0 * sqrt(x)) - t_1;
	} else {
		tmp = 2.0 * (sqrt(x) * cos(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (3.0d0 * b)
    if ((t_1 <= (-4d+22)) .or. (.not. (t_1 <= 5d-201))) then
        tmp = (2.0d0 * sqrt(x)) - t_1
    else
        tmp = 2.0d0 * (sqrt(x) * cos(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if ((t_1 <= -4e+22) || !(t_1 <= 5e-201)) {
		tmp = (2.0 * Math.sqrt(x)) - t_1;
	} else {
		tmp = 2.0 * (Math.sqrt(x) * Math.cos(y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if (t_1 <= -4e+22) or not (t_1 <= 5e-201):
		tmp = (2.0 * math.sqrt(x)) - t_1
	else:
		tmp = 2.0 * (math.sqrt(x) * math.cos(y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if ((t_1 <= -4e+22) || !(t_1 <= 5e-201))
		tmp = Float64(Float64(2.0 * sqrt(x)) - t_1);
	else
		tmp = Float64(2.0 * Float64(sqrt(x) * cos(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	tmp = 0.0;
	if ((t_1 <= -4e+22) || ~((t_1 <= 5e-201)))
		tmp = (2.0 * sqrt(x)) - t_1;
	else
		tmp = 2.0 * (sqrt(x) * cos(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+22], N[Not[LessEqual[t$95$1, 5e-201]], $MachinePrecision]], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b 3)) < -4e22 or 4.9999999999999999e-201 < (/.f64 a (*.f64 b 3))

   1. Initial program 79.1%

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

   2. Taylor expanded in z around 0 87.6%

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

   3. Taylor expanded in y around 0 82.6%

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

   if -4e22 < (/.f64 a (*.f64 b 3)) < 4.9999999999999999e-201

   1. Initial program 59.7%

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

   2. Taylor expanded in z around 0 59.3%

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

   3. Taylor expanded in y around inf 59.3%

      \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right) - 0.3333333333333333 \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv 59.3%

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

      2. metadata-eval 59.3%

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

      3. *-commutative 59.3%

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

      4. associate-*r* 59.3%

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

      5. fma-def 59.3%

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

   5. Simplified 59.3%

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

   6. Taylor expanded in a around 0 57.4%

      \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 57.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \cos y\right)} \]

   8. Simplified 57.4%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -4 \cdot 10^{+22} \lor \neg \left(\frac{a}{3 \cdot b} \leq 5 \cdot 10^{-201}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right)\\ \end{array} \]

Alternative 4: 76.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos y)) (/ a (* 3.0 b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos(y)) - (a / (3.0 * b));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos(y)) - (a / (3.0d0 * b))
end function

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)) - (a / (3.0 * b));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos(y)) - (a / (3.0 * b))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(y)) - Float64(a / Float64(3.0 * b)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos(y)) - (a / (3.0 * b));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in z around 0 77.4%

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

3. Final simplification 77.4%

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

Alternative 5: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} - 0.3333333333333333 \cdot \frac{a}{b} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* 2.0 (sqrt x)) (* 0.3333333333333333 (/ a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (2.0 * sqrt(x)) - (0.3333333333333333 * (a / b));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (2.0d0 * sqrt(x)) - (0.3333333333333333d0 * (a / b))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (2.0 * Math.sqrt(x)) - (0.3333333333333333 * (a / b));
}

Python

def code(x, y, z, t, a, b):
	return (2.0 * math.sqrt(x)) - (0.3333333333333333 * (a / b))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(2.0 * sqrt(x)) - Float64(0.3333333333333333 * Float64(a / b)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (2.0 * sqrt(x)) - (0.3333333333333333 * (a / b));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in z around 0 77.4%

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

3. Taylor expanded in y around 0 64.8%

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

4. Final simplification 64.8%

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

Alternative 6: 65.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} - \frac{a}{3 \cdot b} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (- (* 2.0 (sqrt x)) (/ a (* 3.0 b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (2.0 * sqrt(x)) - (a / (3.0 * b));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (2.0d0 * sqrt(x)) - (a / (3.0d0 * b))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (2.0 * Math.sqrt(x)) - (a / (3.0 * b));
}

Python

def code(x, y, z, t, a, b):
	return (2.0 * math.sqrt(x)) - (a / (3.0 * b))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(2.0 * sqrt(x)) - Float64(a / Float64(3.0 * b)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (2.0 * sqrt(x)) - (a / (3.0 * b));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in z around 0 77.4%

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

3. Taylor expanded in y around 0 64.9%

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

4. Final simplification 64.9%

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

Alternative 7: 49.9% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \frac{a}{b} \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (/ a b) -0.3333333333333333))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (a / b) * -0.3333333333333333;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a / b) * (-0.3333333333333333d0)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (a / b) * -0.3333333333333333;
}

Python

def code(x, y, z, t, a, b):
	return (a / b) * -0.3333333333333333

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(a / b) * -0.3333333333333333)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (a / b) * -0.3333333333333333;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in z around 0 77.4%

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

3. Taylor expanded in x around 0 49.6%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]

4. Final simplification 49.6%

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

Alternative 8: 49.9% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \frac{a}{b \cdot -3} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ a (* b -3.0)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return a / (b * -3.0);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a / (b * (-3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return a / (b * -3.0);
}

Python

def code(x, y, z, t, a, b):
	return a / (b * -3.0)

Julia

function code(x, y, z, t, a, b)
	return Float64(a / Float64(b * -3.0))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = a / (b * -3.0);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in z around 0 77.4%

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

3. Taylor expanded in x around 0 49.6%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 26.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{a}{b}\right)\right)} \]

   2. expm1-udef 24.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{a}{b}\right)} - 1} \]

5. Applied egg-rr 24.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{a}{b}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 26.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{a}{b}\right)\right)} \]

   2. expm1-log1p 49.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]

   3. *-commutative 49.6%

      \[\leadsto \color{blue}{\frac{a}{b} \cdot -0.3333333333333333} \]

   4. associate-/r/ 49.7%

      \[\leadsto \color{blue}{\frac{a}{\frac{b}{-0.3333333333333333}}} \]

7. Simplified 49.7%

   \[\leadsto \color{blue}{\frac{a}{\frac{b}{-0.3333333333333333}}} \]
8. Step-by-step derivation

   1. div-inv 49.7%

      \[\leadsto \frac{a}{\color{blue}{b \cdot \frac{1}{-0.3333333333333333}}} \]

   2. metadata-eval 49.7%

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

9. Applied egg-rr 49.7%

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

10. Final simplification 49.7%

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

Developer target: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{0.3333333333333333}{z}}{t}\\ t_2 := \frac{\frac{a}{3}}{b}\\ t_3 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;t_3 \cdot \cos \left(\frac{1}{y} - t_1\right) - t_2\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - t_1\right) \cdot t_3 - \frac{\frac{a}{b}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ 0.3333333333333333 z) t))
        (t_2 (/ (/ a 3.0) b))
        (t_3 (* 2.0 (sqrt x))))
   (if (< z -1.3793337487235141e+129)
     (- (* t_3 (cos (- (/ 1.0 y) t_1))) t_2)
     (if (< z 3.516290613555987e+106)
       (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) t_2)
       (- (* (cos (- y t_1)) t_3) (/ (/ a b) 3.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (0.3333333333333333d0 / z) / t
    t_2 = (a / 3.0d0) / b
    t_3 = 2.0d0 * sqrt(x)
    if (z < (-1.3793337487235141d+129)) then
        tmp = (t_3 * cos(((1.0d0 / y) - t_1))) - t_2
    else if (z < 3.516290613555987d+106) then
        tmp = ((sqrt(x) * 2.0d0) * cos((y - ((t / 3.0d0) * z)))) - t_2
    else
        tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * Math.sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * Math.cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((Math.sqrt(x) * 2.0) * Math.cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (Math.cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (0.3333333333333333 / z) / t
	t_2 = (a / 3.0) / b
	t_3 = 2.0 * math.sqrt(x)
	tmp = 0
	if z < -1.3793337487235141e+129:
		tmp = (t_3 * math.cos(((1.0 / y) - t_1))) - t_2
	elif z < 3.516290613555987e+106:
		tmp = ((math.sqrt(x) * 2.0) * math.cos((y - ((t / 3.0) * z)))) - t_2
	else:
		tmp = (math.cos((y - t_1)) * t_3) - ((a / b) / 3.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(0.3333333333333333 / z) / t)
	t_2 = Float64(Float64(a / 3.0) / b)
	t_3 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (z < -1.3793337487235141e+129)
		tmp = Float64(Float64(t_3 * cos(Float64(Float64(1.0 / y) - t_1))) - t_2);
	elseif (z < 3.516290613555987e+106)
		tmp = Float64(Float64(Float64(sqrt(x) * 2.0) * cos(Float64(y - Float64(Float64(t / 3.0) * z)))) - t_2);
	else
		tmp = Float64(Float64(cos(Float64(y - t_1)) * t_3) - Float64(Float64(a / b) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (0.3333333333333333 / z) / t;
	t_2 = (a / 3.0) / b;
	t_3 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (z < -1.3793337487235141e+129)
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	elseif (z < 3.516290613555987e+106)
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	else
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.3793337487235141e+129], N[(N[(t$95$3 * N[Cos[N[(N[(1.0 / y), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[z, 3.516290613555987e+106], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(y - N[(N[(t / 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(N[Cos[N[(y - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2023228 
(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))))