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

Percentage Accurate: 70.1% → 77.4%

Time: 23.6s

Alternatives: 10

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: 70.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := t \cdot \left(z \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.957:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos t_2 + \sin y \cdot \sin t_2\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left|\cos y\right|\right) - t_1\\ \end{array} \end{array} \]

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))) 0.957)
     (-
      (* 2.0 (* (sqrt x) (+ (* (cos y) (cos t_2)) (* (sin y) (sin t_2)))))
      t_1)
     (- (* 2.0 (* (sqrt x) (fabs (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))) <= 0.957) {
		tmp = (2.0 * (sqrt(x) * ((cos(y) * cos(t_2)) + (sin(y) * sin(t_2))))) - t_1;
	} else {
		tmp = (2.0 * (sqrt(x) * fabs(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))) <= 0.957d0) then
        tmp = (2.0d0 * (sqrt(x) * ((cos(y) * cos(t_2)) + (sin(y) * sin(t_2))))) - t_1
    else
        tmp = (2.0d0 * (sqrt(x) * abs(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))) <= 0.957) {
		tmp = (2.0 * (Math.sqrt(x) * ((Math.cos(y) * Math.cos(t_2)) + (Math.sin(y) * Math.sin(t_2))))) - t_1;
	} else {
		tmp = (2.0 * (Math.sqrt(x) * Math.abs(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))) <= 0.957:
		tmp = (2.0 * (math.sqrt(x) * ((math.cos(y) * math.cos(t_2)) + (math.sin(y) * math.sin(t_2))))) - t_1
	else:
		tmp = (2.0 * (math.sqrt(x) * math.fabs(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))) <= 0.957)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(cos(y) * cos(t_2)) + Float64(sin(y) * sin(t_2))))) - t_1);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * abs(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))) <= 0.957)
		tmp = (2.0 * (sqrt(x) * ((cos(y) * cos(t_2)) + (sin(y) * sin(t_2))))) - t_1;
	else
		tmp = (2.0 * (sqrt(x) * abs(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], 0.957], 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[y], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Abs[N[Cos[y], $MachinePrecision]], $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))) < 0.956999999999999962

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. fma-neg 73.7%

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

      3. remove-double-neg 73.7%

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

      4. fma-neg 73.7%

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

      5. remove-double-neg 73.7%

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

      6. associate-*l/ 73.6%

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

      7. *-commutative 73.6%

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

   3. Simplified 73.6%

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

      1. cos-diff 74.5%

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

      2. div-inv 74.6%

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

      3. metadata-eval 74.6%

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

      4. div-inv 74.4%

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

      5. metadata-eval 74.4%

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

   5. Applied egg-rr 74.4%

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

   if 0.956999999999999962 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))

   1. Initial program 67.1%

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

      1. associate-*l* 67.1%

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

      2. fma-neg 67.1%

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

      3. remove-double-neg 67.1%

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

      4. fma-neg 67.1%

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

      5. remove-double-neg 67.1%

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

      6. associate-*l/ 67.3%

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

      7. *-commutative 67.3%

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

   3. Simplified 67.3%

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

      1. add-sqr-sqrt 67.1%

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

      2. sqrt-unprod 67.1%

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

      3. pow2 67.1%

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

      4. div-inv 67.1%

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

      5. metadata-eval 67.1%

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

   5. Applied egg-rr 67.1%

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

      1. unpow2 67.1%

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

      2. rem-sqrt-square 67.1%

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

      3. sub-neg 67.1%

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

      4. +-commutative 67.1%

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

      5. distribute-lft-neg-in 67.1%

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

      6. distribute-rgt-neg-in 67.1%

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

      7. metadata-eval 67.1%

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

      8. *-commutative 67.1%

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

      9. associate-*r* 67.1%

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

      10. fma-def 67.1%

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

      11. *-commutative 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in t around 0 85.1%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.957:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right) + \sin y \cdot \sin \left(t \cdot \left(z \cdot 0.3333333333333333\right)\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left|\cos y\right|\right) - \frac{a}{3 \cdot b}\\ \end{array} \]

Alternative 2: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot t_1 - t_2 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y - {\left(\sqrt[3]{z} \cdot \sqrt[3]{t \cdot 0.3333333333333333}\right)}^{3}\right)\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 - t_2\\ \end{array} \end{array} \]

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 (<= (- (* (cos (- y (/ (* z t) 3.0))) t_1) t_2) 2e+139)
     (-
      (*
       2.0
       (*
        (sqrt x)
        (cos (- y (pow (* (cbrt z) (cbrt (* t 0.3333333333333333))) 3.0)))))
      t_2)
     (- t_1 t_2))))

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 (((cos((y - ((z * t) / 3.0))) * t_1) - t_2) <= 2e+139) {
		tmp = (2.0 * (sqrt(x) * cos((y - pow((cbrt(z) * cbrt((t * 0.3333333333333333))), 3.0))))) - t_2;
	} else {
		tmp = t_1 - t_2;
	}
	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 = 2.0 * Math.sqrt(x);
	double t_2 = a / (3.0 * b);
	double tmp;
	if (((Math.cos((y - ((z * t) / 3.0))) * t_1) - t_2) <= 2e+139) {
		tmp = (2.0 * (Math.sqrt(x) * Math.cos((y - Math.pow((Math.cbrt(z) * Math.cbrt((t * 0.3333333333333333))), 3.0))))) - t_2;
	} else {
		tmp = t_1 - t_2;
	}
	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 (Float64(Float64(cos(Float64(y - Float64(Float64(z * t) / 3.0))) * t_1) - t_2) <= 2e+139)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * cos(Float64(y - (Float64(cbrt(z) * cbrt(Float64(t * 0.3333333333333333))) ^ 3.0))))) - t_2);
	else
		tmp = Float64(t_1 - t_2);
	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[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], 2e+139], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[N[(y - N[Power[N[(N[Power[z, 1/3], $MachinePrecision] * N[Power[N[(t * 0.3333333333333333), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(t$95$1 - t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 a (*.f64 b 3))) < 2.00000000000000007e139

   1. Initial program 77.8%

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

      1. associate-*l* 77.8%

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

      2. fma-neg 77.8%

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

      3. remove-double-neg 77.8%

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

      4. fma-neg 77.8%

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

      5. remove-double-neg 77.8%

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

      6. associate-*l/ 77.8%

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

      7. *-commutative 77.8%

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

   3. Simplified 77.8%

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

      1. add-cube-cbrt 77.9%

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

      2. pow3 77.7%

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

      3. div-inv 77.7%

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

      4. metadata-eval 77.7%

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

   5. Applied egg-rr 77.7%

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

      1. pow1/3 41.1%

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

      2. associate-*l* 41.1%

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

      3. unpow-prod-down 17.7%

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

   7. Applied egg-rr 17.7%

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

      1. unpow1/3 39.9%

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

      2. unpow1/3 78.0%

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

   9. Simplified 78.0%

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

   if 2.00000000000000007e139 < (-.f64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 a (*.f64 b 3)))

   1. Initial program 46.7%

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

   2. Taylor expanded in z around 0 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. *-commutative 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in y around 0 80.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{3 \cdot b} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y - {\left(\sqrt[3]{z} \cdot \sqrt[3]{t \cdot 0.3333333333333333}\right)}^{3}\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array} \]

Alternative 3: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot t_1 - t_2 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y - {\left(\sqrt[3]{\left(z \cdot t\right) \cdot 0.3333333333333333}\right)}^{3}\right)\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 - t_2\\ \end{array} \end{array} \]

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 (<= (- (* (cos (- y (/ (* z t) 3.0))) t_1) t_2) 2e+139)
     (-
      (*
       2.0
       (*
        (sqrt x)
        (cos (- y (pow (cbrt (* (* z t) 0.3333333333333333)) 3.0)))))
      t_2)
     (- t_1 t_2))))

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 (((cos((y - ((z * t) / 3.0))) * t_1) - t_2) <= 2e+139) {
		tmp = (2.0 * (sqrt(x) * cos((y - pow(cbrt(((z * t) * 0.3333333333333333)), 3.0))))) - t_2;
	} else {
		tmp = t_1 - t_2;
	}
	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 = 2.0 * Math.sqrt(x);
	double t_2 = a / (3.0 * b);
	double tmp;
	if (((Math.cos((y - ((z * t) / 3.0))) * t_1) - t_2) <= 2e+139) {
		tmp = (2.0 * (Math.sqrt(x) * Math.cos((y - Math.pow(Math.cbrt(((z * t) * 0.3333333333333333)), 3.0))))) - t_2;
	} else {
		tmp = t_1 - t_2;
	}
	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 (Float64(Float64(cos(Float64(y - Float64(Float64(z * t) / 3.0))) * t_1) - t_2) <= 2e+139)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * cos(Float64(y - (cbrt(Float64(Float64(z * t) * 0.3333333333333333)) ^ 3.0))))) - t_2);
	else
		tmp = Float64(t_1 - t_2);
	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[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], 2e+139], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[N[(y - N[Power[N[Power[N[(N[(z * t), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(t$95$1 - t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 a (*.f64 b 3))) < 2.00000000000000007e139

   1. Initial program 77.8%

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

      1. associate-*l* 77.8%

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

      2. fma-neg 77.8%

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

      3. remove-double-neg 77.8%

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

      4. fma-neg 77.8%

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

      5. remove-double-neg 77.8%

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

      6. associate-*l/ 77.8%

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

      7. *-commutative 77.8%

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

   3. Simplified 77.8%

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

      1. add-cube-cbrt 77.9%

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

      2. pow3 77.7%

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

      3. div-inv 77.7%

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

      4. metadata-eval 77.7%

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

   5. Applied egg-rr 77.7%

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

      1. expm1-log1p-u 62.9%

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

      2. expm1-udef 63.1%

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

      3. *-commutative 63.1%

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

      4. *-commutative 63.1%

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

   7. Applied egg-rr 63.1%

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - {\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{t \cdot \left(0.3333333333333333 \cdot z\right)}\right)} - 1\right)}}^{3}\right)\right) - \frac{a}{3 \cdot b} \]
   8. Step-by-step derivation

      1. expm1-def 62.9%

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

      2. expm1-log1p 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*l* 78.0%

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

      5. *-commutative 78.0%

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

   9. Simplified 78.0%

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

   if 2.00000000000000007e139 < (-.f64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 a (*.f64 b 3)))

   1. Initial program 46.7%

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

   2. Taylor expanded in z around 0 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. *-commutative 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in y around 0 80.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{3 \cdot b} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y - {\left(\sqrt[3]{\left(z \cdot t\right) \cdot 0.3333333333333333}\right)}^{3}\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array} \]

Alternative 4: 77.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := \left(z \cdot t\right) \cdot 0.3333333333333333\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.957:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos t_2 + \sin y \cdot \sin t_2\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left|\cos y\right|\right) - t_1\\ \end{array} \end{array} \]

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 (* (* z t) 0.3333333333333333)))
   (if (<= (cos (- y (/ (* z t) 3.0))) 0.957)
     (-
      (* 2.0 (* (sqrt x) (+ (* (cos y) (cos t_2)) (* (sin y) (sin t_2)))))
      t_1)
     (- (* 2.0 (* (sqrt x) (fabs (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 = (z * t) * 0.3333333333333333;
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 0.957) {
		tmp = (2.0 * (sqrt(x) * ((cos(y) * cos(t_2)) + (sin(y) * sin(t_2))))) - t_1;
	} else {
		tmp = (2.0 * (sqrt(x) * fabs(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 = (z * t) * 0.3333333333333333d0
    if (cos((y - ((z * t) / 3.0d0))) <= 0.957d0) then
        tmp = (2.0d0 * (sqrt(x) * ((cos(y) * cos(t_2)) + (sin(y) * sin(t_2))))) - t_1
    else
        tmp = (2.0d0 * (sqrt(x) * abs(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 = (z * t) * 0.3333333333333333;
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 0.957) {
		tmp = (2.0 * (Math.sqrt(x) * ((Math.cos(y) * Math.cos(t_2)) + (Math.sin(y) * Math.sin(t_2))))) - t_1;
	} else {
		tmp = (2.0 * (Math.sqrt(x) * Math.abs(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 = (z * t) * 0.3333333333333333
	tmp = 0
	if math.cos((y - ((z * t) / 3.0))) <= 0.957:
		tmp = (2.0 * (math.sqrt(x) * ((math.cos(y) * math.cos(t_2)) + (math.sin(y) * math.sin(t_2))))) - t_1
	else:
		tmp = (2.0 * (math.sqrt(x) * math.fabs(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(Float64(z * t) * 0.3333333333333333)
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 0.957)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(cos(y) * cos(t_2)) + Float64(sin(y) * sin(t_2))))) - t_1);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * abs(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 = (z * t) * 0.3333333333333333;
	tmp = 0.0;
	if (cos((y - ((z * t) / 3.0))) <= 0.957)
		tmp = (2.0 * (sqrt(x) * ((cos(y) * cos(t_2)) + (sin(y) * sin(t_2))))) - t_1;
	else
		tmp = (2.0 * (sqrt(x) * abs(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[(N[(z * t), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.957], 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[y], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Abs[N[Cos[y], $MachinePrecision]], $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))) < 0.956999999999999962

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. fma-neg 73.7%

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

      3. remove-double-neg 73.7%

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

      4. fma-neg 73.7%

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

      5. remove-double-neg 73.7%

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

      6. associate-*l/ 73.6%

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

      7. *-commutative 73.6%

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

   3. Simplified 73.6%

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

      1. associate-*l/ 73.7%

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

      2. clear-num 73.6%

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

   5. Applied egg-rr 73.6%

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

      1. cos-diff 74.9%

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

      2. associate-/r/ 74.7%

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

      3. metadata-eval 74.7%

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

      4. associate-/r/ 74.4%

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

      5. metadata-eval 74.4%

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

   7. Applied egg-rr 74.4%

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

   if 0.956999999999999962 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))

   1. Initial program 67.1%

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

      1. associate-*l* 67.1%

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

      2. fma-neg 67.1%

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

      3. remove-double-neg 67.1%

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

      4. fma-neg 67.1%

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

      5. remove-double-neg 67.1%

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

      6. associate-*l/ 67.3%

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

      7. *-commutative 67.3%

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

   3. Simplified 67.3%

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

      1. add-sqr-sqrt 67.1%

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

      2. sqrt-unprod 67.1%

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

      3. pow2 67.1%

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

      4. div-inv 67.1%

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

      5. metadata-eval 67.1%

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

   5. Applied egg-rr 67.1%

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

      1. unpow2 67.1%

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

      2. rem-sqrt-square 67.1%

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

      3. sub-neg 67.1%

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

      4. +-commutative 67.1%

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

      5. distribute-lft-neg-in 67.1%

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

      6. distribute-rgt-neg-in 67.1%

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

      7. metadata-eval 67.1%

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

      8. *-commutative 67.1%

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

      9. associate-*r* 67.1%

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

      10. fma-def 67.1%

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

      11. *-commutative 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in t around 0 85.1%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.957:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333\right) + \sin y \cdot \sin \left(\left(z \cdot t\right) \cdot 0.3333333333333333\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left|\cos y\right|\right) - \frac{a}{3 \cdot b}\\ \end{array} \]

Alternative 5: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot t_1 - t_2 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;t_1 \cdot \cos \left(y + t \cdot \left(z \cdot -0.3333333333333333\right)\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 - t_2\\ \end{array} \end{array} \]

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 (<= (- (* (cos (- y (/ (* z t) 3.0))) t_1) t_2) 2e+139)
     (- (* t_1 (cos (+ y (* t (* z -0.3333333333333333))))) t_2)
     (- t_1 t_2))))

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 (((cos((y - ((z * t) / 3.0))) * t_1) - t_2) <= 2e+139) {
		tmp = (t_1 * cos((y + (t * (z * -0.3333333333333333))))) - t_2;
	} else {
		tmp = t_1 - t_2;
	}
	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 (((cos((y - ((z * t) / 3.0d0))) * t_1) - t_2) <= 2d+139) then
        tmp = (t_1 * cos((y + (t * (z * (-0.3333333333333333d0)))))) - t_2
    else
        tmp = t_1 - t_2
    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 (((Math.cos((y - ((z * t) / 3.0))) * t_1) - t_2) <= 2e+139) {
		tmp = (t_1 * Math.cos((y + (t * (z * -0.3333333333333333))))) - t_2;
	} else {
		tmp = t_1 - t_2;
	}
	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 ((math.cos((y - ((z * t) / 3.0))) * t_1) - t_2) <= 2e+139:
		tmp = (t_1 * math.cos((y + (t * (z * -0.3333333333333333))))) - t_2
	else:
		tmp = t_1 - t_2
	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 (Float64(Float64(cos(Float64(y - Float64(Float64(z * t) / 3.0))) * t_1) - t_2) <= 2e+139)
		tmp = Float64(Float64(t_1 * cos(Float64(y + Float64(t * Float64(z * -0.3333333333333333))))) - t_2);
	else
		tmp = Float64(t_1 - t_2);
	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 (((cos((y - ((z * t) / 3.0))) * t_1) - t_2) <= 2e+139)
		tmp = (t_1 * cos((y + (t * (z * -0.3333333333333333))))) - t_2;
	else
		tmp = t_1 - t_2;
	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[LessEqual[N[(N[(N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], 2e+139], N[(N[(t$95$1 * N[Cos[N[(y + N[(t * N[(z * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(t$95$1 - t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 a (*.f64 b 3))) < 2.00000000000000007e139

   1. Initial program 77.8%

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

   2. Taylor expanded in y around inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. *-commutative 77.7%

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

      3. associate-*r* 77.9%

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

      4. *-commutative 77.9%

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

      5. *-commutative 77.9%

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

      6. associate-*r* 77.9%

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

      7. cancel-sign-sub-inv 77.9%

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

      8. distribute-rgt-neg-in 77.9%

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

      9. metadata-eval 77.9%

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

      10. *-commutative 77.9%

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

   4. Simplified 77.9%

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

   if 2.00000000000000007e139 < (-.f64 (*.f64 (*.f64 2 (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))) (/.f64 a (*.f64 b 3)))

   1. Initial program 46.7%

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

   2. Taylor expanded in z around 0 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. *-commutative 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in y around 0 80.3%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{3 \cdot b} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + t \cdot \left(z \cdot -0.3333333333333333\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array} \]

Alternative 6: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-119} \lor \neg \left(t_1 \leq 2 \cdot 10^{-103}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right)\\ \end{array} \end{array} \]

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 -2e-119) (not (<= t_1 2e-103)))
     (- (* 2.0 (sqrt x)) t_1)
     (* 2.0 (* (sqrt x) (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 <= -2e-119) || !(t_1 <= 2e-103)) {
		tmp = (2.0 * sqrt(x)) - t_1;
	} else {
		tmp = 2.0 * (sqrt(x) * 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 <= (-2d-119)) .or. (.not. (t_1 <= 2d-103))) then
        tmp = (2.0d0 * sqrt(x)) - t_1
    else
        tmp = 2.0d0 * (sqrt(x) * 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 <= -2e-119) || !(t_1 <= 2e-103)) {
		tmp = (2.0 * Math.sqrt(x)) - t_1;
	} else {
		tmp = 2.0 * (Math.sqrt(x) * 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 <= -2e-119) or not (t_1 <= 2e-103):
		tmp = (2.0 * math.sqrt(x)) - t_1
	else:
		tmp = 2.0 * (math.sqrt(x) * 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 <= -2e-119) || !(t_1 <= 2e-103))
		tmp = Float64(Float64(2.0 * sqrt(x)) - t_1);
	else
		tmp = Float64(2.0 * Float64(sqrt(x) * 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 <= -2e-119) || ~((t_1 <= 2e-103)))
		tmp = (2.0 * sqrt(x)) - t_1;
	else
		tmp = 2.0 * (sqrt(x) * 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, -2e-119], N[Not[LessEqual[t$95$1, 2e-103]], $MachinePrecision]], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b 3)) < -2.00000000000000003e-119 or 1.99999999999999992e-103 < (/.f64 a (*.f64 b 3))

   1. Initial program 77.1%

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

   2. Taylor expanded in z around 0 86.7%

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

      1. *-commutative 86.7%

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

      2. associate-*l* 86.7%

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

      3. *-commutative 86.7%

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

   4. Simplified 86.7%

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

   5. Taylor expanded in y around 0 80.6%

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

   if -2.00000000000000003e-119 < (/.f64 a (*.f64 b 3)) < 1.99999999999999992e-103

   1. Initial program 59.9%

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

   2. Taylor expanded in z around 0 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*l* 58.4%

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

      3. *-commutative 58.4%

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

   4. Simplified 58.4%

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

      1. add-cube-cbrt 57.3%

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in a around 0 56.7%

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

      1. *-commutative 56.7%

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

   9. Simplified 56.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-119} \lor \neg \left(\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-103}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right)\\ \end{array} \]

Alternative 7: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

Derivation

1. Initial program 71.0%

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

2. Taylor expanded in z around 0 76.7%

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

   1. *-commutative 76.7%

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

   2. associate-*l* 76.7%

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

   3. *-commutative 76.7%

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

4. Simplified 76.7%

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

5. Final simplification 76.7%

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

Alternative 8: 65.8% 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 71.0%

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

2. Taylor expanded in z around 0 76.7%

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

   1. *-commutative 76.7%

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

   2. associate-*l* 76.7%

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

   3. *-commutative 76.7%

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

4. Simplified 76.7%

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

5. Taylor expanded in y around 0 64.9%

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

6. Final simplification 64.9%

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

Alternative 9: 51.0% accurate, 43.4× speedup

Math

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

FPCore

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

C

assert(z < t);
double code(double x, double y, double z, double t, double a, double b) {
	return a * (-0.3333333333333333 / 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 = a * ((-0.3333333333333333d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 71.0%

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

2. Taylor expanded in z around 0 76.7%

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

   1. *-commutative 76.7%

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

   2. associate-*l* 76.7%

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

   3. *-commutative 76.7%

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

4. Simplified 76.7%

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

5. Taylor expanded in x around 0 49.3%

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

   1. metadata-eval 49.3%

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

   2. distribute-lft-neg-in 49.3%

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

   3. *-commutative 49.3%

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

   4. metadata-eval 49.3%

      \[\leadsto -\frac{a}{b} \cdot \color{blue}{\frac{1}{3}} \]

   5. times-frac 49.4%

      \[\leadsto -\color{blue}{\frac{a \cdot 1}{b \cdot 3}} \]

   6. associate-*r/ 49.3%

      \[\leadsto -\color{blue}{a \cdot \frac{1}{b \cdot 3}} \]

   7. distribute-rgt-neg-in 49.3%

      \[\leadsto \color{blue}{a \cdot \left(-\frac{1}{b \cdot 3}\right)} \]

   8. distribute-neg-frac 49.3%

      \[\leadsto a \cdot \color{blue}{\frac{-1}{b \cdot 3}} \]

   9. metadata-eval 49.3%

      \[\leadsto a \cdot \frac{\color{blue}{-1}}{b \cdot 3} \]

   10. *-commutative 49.3%

       \[\leadsto a \cdot \frac{-1}{\color{blue}{3 \cdot b}} \]

   11. associate-/r* 49.3%

       \[\leadsto a \cdot \color{blue}{\frac{\frac{-1}{3}}{b}} \]

   12. metadata-eval 49.3%

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

7. Simplified 49.3%

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

8. Final simplification 49.3%

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

Alternative 10: 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 71.0%

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

2. Taylor expanded in z around 0 76.7%

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

   1. *-commutative 76.7%

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

   2. associate-*l* 76.7%

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

   3. *-commutative 76.7%

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

4. Simplified 76.7%

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

5. Taylor expanded in x around 0 49.3%

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

   1. *-commutative 49.3%

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

7. Simplified 49.3%

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

8. Final simplification 49.3%

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

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 2023199 
(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))))