Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 9.5s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{{\tan x}^{2} - -1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (tan x) (- (tan x)) 1.0) (- (pow (tan x) 2.0) -1.0)))

C

double code(double x) {
	return fma(tan(x), -tan(x), 1.0) / (pow(tan(x), 2.0) - -1.0);
}

Julia

function code(x)
	return Float64(fma(tan(x), Float64(-tan(x)), 1.0) / Float64((tan(x) ^ 2.0) - -1.0))
end

Wolfram

code[x_] := N[(N[(N[Tan[x], $MachinePrecision] * (-N[Tan[x], $MachinePrecision]) + 1.0), $MachinePrecision] / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \tan x \cdot \tan x}}{1 + \tan x \cdot \tan x} \]

   2. sub-neg N/A

      \[\leadsto \frac{\color{blue}{1 + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}}{1 + \tan x \cdot \tan x} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right) + 1}}{1 + \tan x \cdot \tan x} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\tan x \cdot \tan x}\right)\right) + 1}{1 + \tan x \cdot \tan x} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\color{blue}{\tan x \cdot \left(\mathsf{neg}\left(\tan x\right)\right)} + 1}{1 + \tan x \cdot \tan x} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, \mathsf{neg}\left(\tan x\right), 1\right)}}{1 + \tan x \cdot \tan x} \]

   7. lower-neg.f64 99.5

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \color{blue}{-\tan x}, 1\right)}{1 + \tan x \cdot \tan x} \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \mathsf{neg}\left(\tan x\right), 1\right)}{\color{blue}{1 + \tan x \cdot \tan x}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \mathsf{neg}\left(\tan x\right), 1\right)}{\color{blue}{\tan x \cdot \tan x + 1}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \mathsf{neg}\left(\tan x\right), 1\right)}{\tan x \cdot \tan x + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}} \]

   4. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \mathsf{neg}\left(\tan x\right), 1\right)}{\color{blue}{\tan x \cdot \tan x - -1}} \]

   5. lower--.f64 99.5

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{\color{blue}{\tan x \cdot \tan x - -1}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \mathsf{neg}\left(\tan x\right), 1\right)}{\color{blue}{\tan x \cdot \tan x} - -1} \]

   7. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \mathsf{neg}\left(\tan x\right), 1\right)}{\color{blue}{{\tan x}^{2}} - -1} \]

   8. lift-pow.f64 99.5

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{\color{blue}{{\tan x}^{2}} - -1} \]

6. Applied rewrites 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{\color{blue}{{\tan x}^{2} - -1}} \]
7. Add Preprocessing

Alternative 2: 61.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ t_1 := {\tan x}^{2}\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq 0.2:\\ \;\;\;\;\left(t\_1 + -1\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(1 + t\_1\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))) (t_1 (pow (tan x) 2.0)))
   (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) 0.2)
     (* (+ t_1 -1.0) -1.0)
     (* 1.0 (pow (+ 1.0 t_1) -2.0)))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double t_1 = pow(tan(x), 2.0);
	double tmp;
	if (((1.0 - t_0) / (1.0 + t_0)) <= 0.2) {
		tmp = (t_1 + -1.0) * -1.0;
	} else {
		tmp = 1.0 * pow((1.0 + t_1), -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(x) * tan(x)
    t_1 = tan(x) ** 2.0d0
    if (((1.0d0 - t_0) / (1.0d0 + t_0)) <= 0.2d0) then
        tmp = (t_1 + (-1.0d0)) * (-1.0d0)
    else
        tmp = 1.0d0 * ((1.0d0 + t_1) ** (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	double t_1 = Math.pow(Math.tan(x), 2.0);
	double tmp;
	if (((1.0 - t_0) / (1.0 + t_0)) <= 0.2) {
		tmp = (t_1 + -1.0) * -1.0;
	} else {
		tmp = 1.0 * Math.pow((1.0 + t_1), -2.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	t_1 = math.pow(math.tan(x), 2.0)
	tmp = 0
	if ((1.0 - t_0) / (1.0 + t_0)) <= 0.2:
		tmp = (t_1 + -1.0) * -1.0
	else:
		tmp = 1.0 * math.pow((1.0 + t_1), -2.0)
	return tmp

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	t_1 = tan(x) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0)) <= 0.2)
		tmp = Float64(Float64(t_1 + -1.0) * -1.0);
	else
		tmp = Float64(1.0 * (Float64(1.0 + t_1) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = tan(x) * tan(x);
	t_1 = tan(x) ^ 2.0;
	tmp = 0.0;
	if (((1.0 - t_0) / (1.0 + t_0)) <= 0.2)
		tmp = (t_1 + -1.0) * -1.0;
	else
		tmp = 1.0 * ((1.0 + t_1) ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 0.2], N[(N[(t$95$1 + -1.0), $MachinePrecision] * -1.0), $MachinePrecision], N[(1.0 * N[Power[N[(1.0 + t$95$1), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x)))) < 0.20000000000000001

   1. Initial program 98.9%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{\color{blue}{{\sin x}^{2}}}{{\cos x}^{2}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{{\color{blue}{\sin x}}^{2}}{{\cos x}^{2}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{{\sin x}^{2}}{\color{blue}{{\cos x}^{2}}}} \]

      5. lower-cos.f64 98.2

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{{\sin x}^{2}}{{\color{blue}{\cos x}}^{2}}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{1 - \tan x \cdot \tan x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{1 + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

      5. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(-1 + \tan x \cdot \tan x\right)\right)}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\tan x \cdot \tan x + -1\right)}\right)}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{\tan x \cdot \tan x} + -1\right)\right)}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{{\tan x}^{2}} + -1\right)\right)}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{{\tan x}^{2}} + -1\right)\right)}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left({\tan x}^{2} + -1\right)}\right)}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

      11. distribute-frac-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{\tan x}^{2} + -1}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)} \]

      12. distribute-frac-neg2 N/A

          \[\leadsto \color{blue}{\frac{{\tan x}^{2} + -1}{\mathsf{neg}\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}} \]

      13. div-inv N/A

          \[\leadsto \color{blue}{\left({\tan x}^{2} + -1\right) \cdot \frac{1}{\mathsf{neg}\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}} \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left({\tan x}^{2} + -1\right) \cdot \frac{1}{-1 - {\tan x}^{2}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left({\tan x}^{2} + -1\right) \cdot \color{blue}{-1} \]
   9. Step-by-step derivation

   10. Applied rewrites 16.6%

       \[\leadsto \left({\tan x}^{2} + -1\right) \cdot \color{blue}{-1} \]

   if 0.20000000000000001 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))))

   1. Initial program 99.7%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{1 - \tan x \cdot \tan x}}{1 + \tan x \cdot \tan x} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{1 + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}}{1 + \tan x \cdot \tan x} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right) + 1}}{1 + \tan x \cdot \tan x} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\tan x \cdot \tan x}\right)\right) + 1}{1 + \tan x \cdot \tan x} \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\color{blue}{\tan x \cdot \left(\mathsf{neg}\left(\tan x\right)\right)} + 1}{1 + \tan x \cdot \tan x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, \mathsf{neg}\left(\tan x\right), 1\right)}}{1 + \tan x \cdot \tan x} \]

      7. lower-neg.f64 99.8

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \color{blue}{-\tan x}, 1\right)}{1 + \tan x \cdot \tan x} \]

   4. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\tan x, \mathsf{neg}\left(\tan x\right), 1\right)}{1 + \tan x \cdot \tan x}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x, \mathsf{neg}\left(\tan x\right), 1\right) \cdot \frac{1}{1 + \tan x \cdot \tan x}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\tan x \cdot \left(\mathsf{neg}\left(\tan x\right)\right) + 1\right)} \cdot \frac{1}{1 + \tan x \cdot \tan x} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \tan x \cdot \left(\mathsf{neg}\left(\tan x\right)\right)\right)} \cdot \frac{1}{1 + \tan x \cdot \tan x} \]

      5. *-commutative N/A

         \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(\tan x\right)\right) \cdot \tan x}\right) \cdot \frac{1}{1 + \tan x \cdot \tan x} \]

      6. lift-neg.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(\tan x\right)\right)} \cdot \tan x\right) \cdot \frac{1}{1 + \tan x \cdot \tan x} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(1 - \tan x \cdot \tan x\right)} \cdot \frac{1}{1 + \tan x \cdot \tan x} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(1 - \color{blue}{\tan x \cdot \tan x}\right) \cdot \frac{1}{1 + \tan x \cdot \tan x} \]

      9. flip-- N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}{1 + \tan x \cdot \tan x}} \cdot \frac{1}{1 + \tan x \cdot \tan x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}{\color{blue}{1 + \tan x \cdot \tan x}} \cdot \frac{1}{1 + \tan x \cdot \tan x} \]

      11. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right) \cdot \frac{1}{1 + \tan x \cdot \tan x}\right)} \cdot \frac{1}{1 + \tan x \cdot \tan x} \]

      12. associate-*l* N/A

          \[\leadsto \color{blue}{\left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right) \cdot \left(\frac{1}{1 + \tan x \cdot \tan x} \cdot \frac{1}{1 + \tan x \cdot \tan x}\right)} \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(1 - {\tan x}^{4}\right) \cdot {\left(1 + {\tan x}^{2}\right)}^{-2}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot {\left(1 + {\tan x}^{2}\right)}^{-2} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.7%

      \[\leadsto \color{blue}{1} \cdot {\left(1 + {\tan x}^{2}\right)}^{-2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))