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

Percentage Accurate: 70.2% → 77.1%

Time: 28.9s

Alternatives: 7

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: 70.2% 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: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \sqrt[3]{t \cdot -0.3333333333333333}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(y + z \cdot \left(t_1 \cdot \left(t_1 \cdot t_1\right)\right)\right), -0.3333333333333333 \cdot \frac{a}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{a}{3 \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cbrt (* t -0.3333333333333333))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 1.0)
     (fma
      2.0
      (* (sqrt x) (cos (+ y (* z (* t_1 (* t_1 t_1))))))
      (* -0.3333333333333333 (/ a b)))
     (- (* 2.0 (* (sqrt x) (cos y))) (/ a (* 3.0 b))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cbrt((t * -0.3333333333333333));
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 1.0) {
		tmp = fma(2.0, (sqrt(x) * cos((y + (z * (t_1 * (t_1 * t_1)))))), (-0.3333333333333333 * (a / b)));
	} else {
		tmp = (2.0 * (sqrt(x) * cos(y))) - (a / (3.0 * b));
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t, a, b)
	t_1 = cbrt(Float64(t * -0.3333333333333333))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 1.0)
		tmp = fma(2.0, Float64(sqrt(x) * cos(Float64(y + Float64(z * Float64(t_1 * Float64(t_1 * t_1)))))), Float64(-0.3333333333333333 * Float64(a / b)));
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * cos(y))) - Float64(a / Float64(3.0 * b)));
	end
	return tmp
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[N[(t * -0.3333333333333333), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[N[(y + N[(z * N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))) < 1

   1. Initial program 79.3%

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

      1. associate-*l* 79.3%

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

      2. fma-neg 79.3%

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

      3. *-commutative 79.3%

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

      4. associate-*r/ 79.4%

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

      5. cancel-sign-sub-inv 79.4%

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

      6. *-commutative 79.4%

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

      7. associate-/r/ 79.4%

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

      8. neg-mul-1 79.4%

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

      9. associate-/r* 79.4%

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

      10. metadata-eval 79.4%

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

      11. distribute-frac-neg 79.4%

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

      12. *-commutative 79.4%

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

      13. neg-mul-1 79.4%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(y + \frac{z}{\frac{-3}{t}}\right), -0.3333333333333333 \cdot \frac{a}{b}\right)} \]
   4. Step-by-step derivation

      1. div-inv 79.2%

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

   5. Applied egg-rr 79.2%

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

      1. add-cube-cbrt 79.7%

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

      2. associate-/r/ 79.8%

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

      3. metadata-eval 79.8%

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

      4. associate-/r/ 79.7%

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

      5. metadata-eval 79.7%

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

      6. associate-/r/ 79.9%

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

      7. metadata-eval 79.9%

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

   7. Applied egg-rr 79.9%

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

   if 1 < (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. Taylor expanded in z around 0 62.1%

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

4. Final simplification 77.9%

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

Alternative 2: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-144} \lor \neg \left(t_2 \leq 2 \cdot 10^{-114}\right):\\ \;\;\;\;t_1 - t_2\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot t_1\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 2.0 (sqrt x))) (t_2 (/ a (* 3.0 b))))
   (if (or (<= t_2 -5e-144) (not (<= t_2 2e-114)))
     (- t_1 t_2)
     (* (cos y) t_1))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * sqrt(x);
	double t_2 = a / (3.0 * b);
	double tmp;
	if ((t_2 <= -5e-144) || !(t_2 <= 2e-114)) {
		tmp = t_1 - t_2;
	} else {
		tmp = cos(y) * t_1;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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 = 2.0d0 * sqrt(x)
    t_2 = a / (3.0d0 * b)
    if ((t_2 <= (-5d-144)) .or. (.not. (t_2 <= 2d-114))) then
        tmp = t_1 - t_2
    else
        tmp = cos(y) * t_1
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * Math.sqrt(x);
	double t_2 = a / (3.0 * b);
	double tmp;
	if ((t_2 <= -5e-144) || !(t_2 <= 2e-114)) {
		tmp = t_1 - t_2;
	} else {
		tmp = Math.cos(y) * t_1;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a, b):
	t_1 = 2.0 * math.sqrt(x)
	t_2 = a / (3.0 * b)
	tmp = 0
	if (t_2 <= -5e-144) or not (t_2 <= 2e-114):
		tmp = t_1 - t_2
	else:
		tmp = math.cos(y) * t_1
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a, b)
	t_1 = Float64(2.0 * sqrt(x))
	t_2 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if ((t_2 <= -5e-144) || !(t_2 <= 2e-114))
		tmp = Float64(t_1 - t_2);
	else
		tmp = Float64(cos(y) * t_1);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 2.0 * sqrt(x);
	t_2 = a / (3.0 * b);
	tmp = 0.0;
	if ((t_2 <= -5e-144) || ~((t_2 <= 2e-114)))
		tmp = t_1 - t_2;
	else
		tmp = cos(y) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5e-144], N[Not[LessEqual[t$95$2, 2e-114]], $MachinePrecision]], N[(t$95$1 - t$95$2), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b 3)) < -4.9999999999999998e-144 or 2.0000000000000001e-114 < (/.f64 a (*.f64 b 3))

   1. Initial program 79.8%

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

   2. Taylor expanded in z around 0 89.7%

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

   3. Taylor expanded in y around 0 84.6%

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

   if -4.9999999999999998e-144 < (/.f64 a (*.f64 b 3)) < 2.0000000000000001e-114

   1. Initial program 50.7%

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

   2. Taylor expanded in z around 0 48.4%

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

   3. Taylor expanded in a around 0 48.4%

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

      1. associate-*r* 48.4%

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

      2. *-commutative 48.4%

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

      3. associate-*r* 48.4%

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

   5. Simplified 48.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -5 \cdot 10^{-144} \lor \neg \left(\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-114}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot \left(2 \cdot \sqrt{x}\right)\\ \end{array} \]

Alternative 3: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ -0.3333333333333333 \cdot \frac{a}{b} + 2 \cdot \left(\sqrt{x} \cdot \cos y\right) \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (+ (* -0.3333333333333333 (/ a b)) (* 2.0 (* (sqrt x) (cos y)))))

C

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

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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 = ((-0.3333333333333333d0) * (a / b)) + (2.0d0 * (sqrt(x) * cos(y)))
end function

Java

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

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a, b):
	return (-0.3333333333333333 * (a / b)) + (2.0 * (math.sqrt(x) * math.cos(y)))

Julia

z, t = sort([z, t])
function code(x, y, z, t, a, b)
	return Float64(Float64(-0.3333333333333333 * Float64(a / b)) + Float64(2.0 * Float64(sqrt(x) * cos(y))))
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = (-0.3333333333333333 * (a / b)) + (2.0 * (sqrt(x) * cos(y)));
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

   1. fma-neg 70.4%

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

   2. distribute-frac-neg 70.4%

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

   3. *-commutative 70.4%

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

3. Simplified 70.4%

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

4. Taylor expanded in z around 0 76.2%

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

5. Final simplification 76.2%

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

Alternative 4: 76.9% accurate, 1.0× speedup

Math

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

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (- (* 2.0 (* (sqrt x) (cos y))) (/ a (* 3.0 b))))

C

assert(z < t);
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

NOTE: z and t should be sorted in increasing order before calling this function.
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

assert z < t;
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

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

Julia

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

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = (2.0 * (sqrt(x) * cos(y))) - (a / (3.0 * b));
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

2. Taylor expanded in z around 0 76.3%

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

3. Final simplification 76.3%

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

Alternative 5: 65.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ 2 \cdot \sqrt{x} - \frac{a}{b} \cdot 0.3333333333333333 \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a b)
 :precision binary64
 (- (* 2.0 (sqrt x)) (* (/ a b) 0.3333333333333333)))

C

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

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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 / b) * 0.3333333333333333d0)
end function

Java

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

Python

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

Julia

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

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = (2.0 * sqrt(x)) - ((a / b) * 0.3333333333333333);
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

2. Taylor expanded in z around 0 76.3%

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

3. Taylor expanded in y around 0 66.7%

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

4. Final simplification 66.7%

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

Alternative 6: 65.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ 2 \cdot \sqrt{x} - \frac{a}{3 \cdot b} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (- (* 2.0 (sqrt x)) (/ a (* 3.0 b))))

C

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

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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

assert z < t;
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

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

Julia

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

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = (2.0 * sqrt(x)) - (a / (3.0 * b));
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
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 70.4%

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

2. Taylor expanded in z around 0 76.3%

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

3. Taylor expanded in y around 0 66.8%

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

4. Final simplification 66.8%

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

Alternative 7: 50.5% accurate, 43.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ -0.3333333333333333 \cdot \frac{a}{b} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a b) :precision binary64 (* -0.3333333333333333 (/ a b)))

C

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

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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 = (-0.3333333333333333d0) * (a / b)
end function

Java

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

Python

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

Julia

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

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = -0.3333333333333333 * (a / b);
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_] := N[(-0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

2. Taylor expanded in z around 0 76.3%

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

3. Taylor expanded in x around 0 52.4%

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

   1. *-commutative 52.4%

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

5. Simplified 52.4%

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

6. Final simplification 52.4%

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

Developer target: 74.4% 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 2023279 
(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))))