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

Percentage Accurate: 70.5% → 77.5%

Time: 19.8s

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: 70.5% 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: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := t \cdot \left(z \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.916:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos t_2 + \left(e^{\mathsf{log1p}\left(\sin t_2 \cdot \sin y\right)} + -1\right)\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right) - t_1\\ \end{array} \end{array} \]

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 (/ a (* 3.0 b))) (t_2 (* t (* z 0.3333333333333333))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 0.916)
     (-
      (*
       2.0
       (*
        (sqrt x)
        (+
         (* (cos y) (cos t_2))
         (+ (exp (log1p (* (sin t_2) (sin y)))) -1.0))))
      t_1)
     (- (* 2.0 (* (sqrt x) (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 = a / (3.0 * b);
	double t_2 = t * (z * 0.3333333333333333);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 0.916) {
		tmp = (2.0 * (sqrt(x) * ((cos(y) * cos(t_2)) + (exp(log1p((sin(t_2) * sin(y)))) + -1.0)))) - t_1;
	} else {
		tmp = (2.0 * (sqrt(x) * cos(y))) - t_1;
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = t * (z * 0.3333333333333333);
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 0.916) {
		tmp = (2.0 * (Math.sqrt(x) * ((Math.cos(y) * Math.cos(t_2)) + (Math.exp(Math.log1p((Math.sin(t_2) * Math.sin(y)))) + -1.0)))) - t_1;
	} else {
		tmp = (2.0 * (Math.sqrt(x) * Math.cos(y))) - t_1;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = t * (z * 0.3333333333333333)
	tmp = 0
	if math.cos((y - ((z * t) / 3.0))) <= 0.916:
		tmp = (2.0 * (math.sqrt(x) * ((math.cos(y) * math.cos(t_2)) + (math.exp(math.log1p((math.sin(t_2) * math.sin(y)))) + -1.0)))) - t_1
	else:
		tmp = (2.0 * (math.sqrt(x) * math.cos(y))) - t_1
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(t * Float64(z * 0.3333333333333333))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 0.916)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(cos(y) * cos(t_2)) + Float64(exp(log1p(Float64(sin(t_2) * sin(y)))) + -1.0)))) - t_1);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * cos(y))) - t_1);
	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[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.916], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[Log[1 + N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))) < 0.916000000000000036

   1. Initial program 71.1%

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

      1. associate-*l* 71.1%

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

      2. fma-neg 71.1%

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

      3. remove-double-neg 71.1%

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

      4. fma-neg 71.1%

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

      5. remove-double-neg 71.1%

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

      6. associate-/l* 71.5%

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

      7. *-commutative 71.5%

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

   3. Simplified 71.5%

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

      1. cos-diff 73.1%

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

      2. associate-/r/ 72.8%

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

      3. *-commutative 72.8%

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

      4. div-inv 72.8%

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

      5. metadata-eval 72.8%

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

      6. associate-/r/ 72.9%

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

      7. *-commutative 72.9%

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

      8. div-inv 72.9%

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

      9. metadata-eval 72.9%

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

   5. Applied egg-rr 72.9%

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

      1. expm1-log1p-u 72.9%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y \cdot \sin \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right)\right)\right)}\right)\right) - \frac{a}{3 \cdot b} \]

      2. expm1-udef 72.9%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\sin y \cdot \sin \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right)\right)} - 1\right)}\right)\right) - \frac{a}{3 \cdot b} \]

      3. *-commutative 72.9%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\sin \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) \cdot \sin y}\right)} - 1\right)\right)\right) - \frac{a}{3 \cdot b} \]

   7. Applied egg-rr 72.9%

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) \cdot \sin y\right)} - 1\right)}\right)\right) - \frac{a}{3 \cdot b} \]

   if 0.916000000000000036 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))

   1. Initial program 64.5%

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

      1. associate-*l* 64.5%

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

      2. fma-neg 64.5%

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

      3. remove-double-neg 64.5%

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

      4. fma-neg 64.5%

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

      5. remove-double-neg 64.5%

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

      6. associate-/l* 64.5%

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

      7. *-commutative 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in z around 0 81.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.916:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) + \left(e^{\mathsf{log1p}\left(\sin \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) \cdot \sin y\right)} + -1\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{a}{3 \cdot b}\\ \end{array} \]

Alternative 2: 77.7% accurate, 0.3× speedup

Math

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

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 (/ a (* 3.0 b))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 1.0)
     (-
      (*
       2.0
       (*
        (sqrt x)
        (+
         (* (sin (* t (* z 0.3333333333333333))) (sin y))
         (* (cos y) (cos (* 0.3333333333333333 (* z t)))))))
      t_1)
     (- (* 2.0 (sqrt x)) t_1))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 1.0) {
		tmp = (2.0 * (sqrt(x) * ((sin((t * (z * 0.3333333333333333))) * sin(y)) + (cos(y) * cos((0.3333333333333333 * (z * t))))))) - t_1;
	} else {
		tmp = (2.0 * sqrt(x)) - 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) :: tmp
    t_1 = a / (3.0d0 * b)
    if (cos((y - ((z * t) / 3.0d0))) <= 1.0d0) then
        tmp = (2.0d0 * (sqrt(x) * ((sin((t * (z * 0.3333333333333333d0))) * sin(y)) + (cos(y) * cos((0.3333333333333333d0 * (z * t))))))) - t_1
    else
        tmp = (2.0d0 * sqrt(x)) - 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 = a / (3.0 * b);
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 1.0) {
		tmp = (2.0 * (Math.sqrt(x) * ((Math.sin((t * (z * 0.3333333333333333))) * Math.sin(y)) + (Math.cos(y) * Math.cos((0.3333333333333333 * (z * t))))))) - t_1;
	} else {
		tmp = (2.0 * Math.sqrt(x)) - t_1;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if math.cos((y - ((z * t) / 3.0))) <= 1.0:
		tmp = (2.0 * (math.sqrt(x) * ((math.sin((t * (z * 0.3333333333333333))) * math.sin(y)) + (math.cos(y) * math.cos((0.3333333333333333 * (z * t))))))) - t_1
	else:
		tmp = (2.0 * math.sqrt(x)) - t_1
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 1.0)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(sin(Float64(t * Float64(z * 0.3333333333333333))) * sin(y)) + Float64(cos(y) * cos(Float64(0.3333333333333333 * Float64(z * t))))))) - t_1);
	else
		tmp = Float64(Float64(2.0 * sqrt(x)) - 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 = a / (3.0 * b);
	tmp = 0.0;
	if (cos((y - ((z * t) / 3.0))) <= 1.0)
		tmp = (2.0 * (sqrt(x) * ((sin((t * (z * 0.3333333333333333))) * sin(y)) + (cos(y) * cos((0.3333333333333333 * (z * t))))))) - t_1;
	else
		tmp = (2.0 * sqrt(x)) - 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[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Sin[N[(t * N[(z * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[Cos[N[(0.3333333333333333 * N[(z * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.6%

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

      1. associate-*l* 78.6%

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

      2. fma-neg 78.6%

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

      3. remove-double-neg 78.6%

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

      4. fma-neg 78.6%

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

      5. remove-double-neg 78.6%

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

      6. associate-/l* 78.8%

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

      7. *-commutative 78.8%

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

   3. Simplified 78.8%

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

      1. cos-diff 80.1%

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

      2. associate-/r/ 79.8%

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

      3. *-commutative 79.8%

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

      4. div-inv 79.7%

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

      5. metadata-eval 79.7%

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

      6. associate-/r/ 79.8%

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

      7. *-commutative 79.8%

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

      8. div-inv 79.8%

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

      9. metadata-eval 79.8%

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

   5. Applied egg-rr 79.8%

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

   6. Taylor expanded in t around inf 80.0%

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

   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 y around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l* 0.0%

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

      3. distribute-lft-neg-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. *-commutative 0.0%

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

      6. associate-*r* 0.0%

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

      7. *-commutative 0.0%

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

   4. Simplified 0.0%

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

   5. Taylor expanded in t around 0 53.0%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
   6. Step-by-step derivation

      1. *-commutative 53.0%

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

   7. Simplified 53.0%

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

4. Final simplification 76.5%

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

Alternative 3: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := t \cdot \left(z \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.916:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos t_2 + \sin t_2 \cdot \sin y\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right) - t_1\\ \end{array} \end{array} \]

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 (/ a (* 3.0 b))) (t_2 (* t (* z 0.3333333333333333))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 0.916)
     (-
      (* 2.0 (* (sqrt x) (+ (* (cos y) (cos t_2)) (* (sin t_2) (sin y)))))
      t_1)
     (- (* 2.0 (* (sqrt x) (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 = a / (3.0 * b);
	double t_2 = t * (z * 0.3333333333333333);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 0.916) {
		tmp = (2.0 * (sqrt(x) * ((cos(y) * cos(t_2)) + (sin(t_2) * sin(y))))) - t_1;
	} else {
		tmp = (2.0 * (sqrt(x) * 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 = a / (3.0d0 * b)
    t_2 = t * (z * 0.3333333333333333d0)
    if (cos((y - ((z * t) / 3.0d0))) <= 0.916d0) then
        tmp = (2.0d0 * (sqrt(x) * ((cos(y) * cos(t_2)) + (sin(t_2) * sin(y))))) - t_1
    else
        tmp = (2.0d0 * (sqrt(x) * 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 = a / (3.0 * b);
	double t_2 = t * (z * 0.3333333333333333);
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 0.916) {
		tmp = (2.0 * (Math.sqrt(x) * ((Math.cos(y) * Math.cos(t_2)) + (Math.sin(t_2) * Math.sin(y))))) - t_1;
	} else {
		tmp = (2.0 * (Math.sqrt(x) * Math.cos(y))) - t_1;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = t * (z * 0.3333333333333333)
	tmp = 0
	if math.cos((y - ((z * t) / 3.0))) <= 0.916:
		tmp = (2.0 * (math.sqrt(x) * ((math.cos(y) * math.cos(t_2)) + (math.sin(t_2) * math.sin(y))))) - t_1
	else:
		tmp = (2.0 * (math.sqrt(x) * math.cos(y))) - t_1
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(t * Float64(z * 0.3333333333333333))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 0.916)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(cos(y) * cos(t_2)) + Float64(sin(t_2) * sin(y))))) - t_1);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * 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 = a / (3.0 * b);
	t_2 = t * (z * 0.3333333333333333);
	tmp = 0.0;
	if (cos((y - ((z * t) / 3.0))) <= 0.916)
		tmp = (2.0 * (sqrt(x) * ((cos(y) * cos(t_2)) + (sin(t_2) * sin(y))))) - t_1;
	else
		tmp = (2.0 * (sqrt(x) * 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[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(z * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.916], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))) < 0.916000000000000036

   1. Initial program 71.1%

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

      1. associate-*l* 71.1%

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

      2. fma-neg 71.1%

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

      3. remove-double-neg 71.1%

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

      4. fma-neg 71.1%

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

      5. remove-double-neg 71.1%

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

      6. associate-/l* 71.5%

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

      7. *-commutative 71.5%

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

   3. Simplified 71.5%

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

      1. cos-diff 73.1%

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

      2. associate-/r/ 72.8%

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

      3. *-commutative 72.8%

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

      4. div-inv 72.8%

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

      5. metadata-eval 72.8%

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

      6. associate-/r/ 72.9%

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

      7. *-commutative 72.9%

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

      8. div-inv 72.9%

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

      9. metadata-eval 72.9%

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

   5. Applied egg-rr 72.9%

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

   if 0.916000000000000036 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))

   1. Initial program 64.5%

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

      1. associate-*l* 64.5%

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

      2. fma-neg 64.5%

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

      3. remove-double-neg 64.5%

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

      4. fma-neg 64.5%

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

      5. remove-double-neg 64.5%

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

      6. associate-/l* 64.5%

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

      7. *-commutative 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in z around 0 81.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.916:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) + \sin \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) \cdot \sin y\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{a}{3 \cdot b}\\ \end{array} \]

Alternative 4: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-69} \lor \neg \left(t_1 \leq 5 \cdot 10^{-58}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \end{array} \end{array} \]

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 (/ a (* 3.0 b))))
   (if (or (<= t_1 -1e-69) (not (<= t_1 5e-58)))
     (- (* 2.0 (sqrt x)) t_1)
     (* (sqrt x) (* 2.0 (cos y))))))

C

assert(z < t);
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 <= -1e-69) || !(t_1 <= 5e-58)) {
		tmp = (2.0 * sqrt(x)) - t_1;
	} else {
		tmp = sqrt(x) * (2.0 * cos(y));
	}
	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) :: tmp
    t_1 = a / (3.0d0 * b)
    if ((t_1 <= (-1d-69)) .or. (.not. (t_1 <= 5d-58))) then
        tmp = (2.0d0 * sqrt(x)) - t_1
    else
        tmp = sqrt(x) * (2.0d0 * cos(y))
    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 = a / (3.0 * b);
	double tmp;
	if ((t_1 <= -1e-69) || !(t_1 <= 5e-58)) {
		tmp = (2.0 * Math.sqrt(x)) - t_1;
	} else {
		tmp = Math.sqrt(x) * (2.0 * Math.cos(y));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if (t_1 <= -1e-69) or not (t_1 <= 5e-58):
		tmp = (2.0 * math.sqrt(x)) - t_1
	else:
		tmp = math.sqrt(x) * (2.0 * math.cos(y))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if ((t_1 <= -1e-69) || !(t_1 <= 5e-58))
		tmp = Float64(Float64(2.0 * sqrt(x)) - t_1);
	else
		tmp = Float64(sqrt(x) * Float64(2.0 * cos(y)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	tmp = 0.0;
	if ((t_1 <= -1e-69) || ~((t_1 <= 5e-58)))
		tmp = (2.0 * sqrt(x)) - t_1;
	else
		tmp = sqrt(x) * (2.0 * cos(y));
	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[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-69], N[Not[LessEqual[t$95$1, 5e-58]], $MachinePrecision]], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(2.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b 3)) < -9.9999999999999996e-70 or 4.99999999999999977e-58 < (/.f64 a (*.f64 b 3))

   1. Initial program 76.7%

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

   2. Taylor expanded in y around 0 74.7%

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

      1. *-commutative 74.7%

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

      2. associate-*l* 74.7%

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

      3. distribute-lft-neg-in 74.7%

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

      4. metadata-eval 74.7%

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

      5. *-commutative 74.7%

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

      6. associate-*r* 74.4%

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

      7. *-commutative 74.4%

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

   4. Simplified 74.4%

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

   5. Taylor expanded in t around 0 84.9%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
   6. Step-by-step derivation

      1. *-commutative 84.9%

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

   7. Simplified 84.9%

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

   if -9.9999999999999996e-70 < (/.f64 a (*.f64 b 3)) < 4.99999999999999977e-58

   1. Initial program 54.6%

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

      1. associate-*l* 54.6%

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

      2. fma-neg 54.6%

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

      3. remove-double-neg 54.6%

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

      4. fma-neg 54.6%

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

      5. remove-double-neg 54.6%

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

      6. associate-/l* 55.2%

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

      7. *-commutative 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in z around 0 52.9%

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

   5. Taylor expanded in a around 0 51.4%

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

      1. *-commutative 51.4%

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

      2. *-commutative 51.4%

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

      3. associate-*l* 51.4%

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

   7. Simplified 51.4%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -1 \cdot 10^{-69} \lor \neg \left(\frac{a}{3 \cdot b} \leq 5 \cdot 10^{-58}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \end{array} \]

Alternative 5: 76.8% 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 68.4%

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

   1. associate-*l* 68.4%

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

   2. fma-neg 68.4%

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

   3. remove-double-neg 68.4%

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

   4. fma-neg 68.4%

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

   5. remove-double-neg 68.4%

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

   6. associate-/l* 68.7%

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

   7. *-commutative 68.7%

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

3. Simplified 68.7%

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

4. Taylor expanded in z around 0 74.5%

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

5. Final simplification 74.5%

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

Alternative 6: 65.6% 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 68.4%

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

2. Taylor expanded in y around 0 58.7%

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

   1. *-commutative 58.7%

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

   2. associate-*l* 58.7%

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

   3. distribute-lft-neg-in 58.7%

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

   4. metadata-eval 58.7%

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

   5. *-commutative 58.7%

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

   6. associate-*r* 58.3%

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

   7. *-commutative 58.3%

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

4. Simplified 58.3%

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

5. Taylor expanded in t around 0 64.8%

   \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
6. Step-by-step derivation

   1. *-commutative 64.8%

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

7. Simplified 64.8%

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

8. Final simplification 64.8%

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

Alternative 7: 50.9% 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 68.4%

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

   1. associate-*l* 68.4%

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

   2. fma-neg 68.4%

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

   3. remove-double-neg 68.4%

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

   4. fma-neg 68.4%

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

   5. remove-double-neg 68.4%

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

   6. associate-/l* 68.7%

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

   7. *-commutative 68.7%

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

3. Simplified 68.7%

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

4. Taylor expanded in z around 0 74.5%

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

5. Taylor expanded in x around 0 51.4%

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

6. Final simplification 51.4%

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

Alternative 8: 50.9% accurate, 43.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a, b)
	tmp = 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[(a / N[(b / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.4%

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

   1. associate-*l* 68.4%

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

   2. fma-neg 68.4%

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

   3. remove-double-neg 68.4%

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

   4. fma-neg 68.4%

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

   5. remove-double-neg 68.4%

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

   6. associate-/l* 68.7%

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

   7. *-commutative 68.7%

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

3. Simplified 68.7%

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

4. Taylor expanded in z around 0 74.5%

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

5. Taylor expanded in x around 0 51.4%

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

   1. associate-*r/ 51.3%

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

   2. *-commutative 51.3%

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

7. Applied egg-rr 51.3%

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

   1. *-un-lft-identity 51.3%

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

   2. associate-/l* 51.4%

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

9. Applied egg-rr 51.4%

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

10. Final simplification 51.4%

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

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 2023252 
(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))))