Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.9s

Alternatives: 5

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

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{1 + {\tan x}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - (tan(x) * tan(x))) / (1.0d0 + (tan(x) ** 2.0d0))
end function

Java

public static double code(double x) {
	return (1.0 - (Math.tan(x) * Math.tan(x))) / (1.0 + Math.pow(Math.tan(x), 2.0));
}

Python

def code(x):
	return (1.0 - (math.tan(x) * math.tan(x))) / (1.0 + math.pow(math.tan(x), 2.0))

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / Float64(1.0 + (tan(x) ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. fma-undefine 99.5%

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

   2. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	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) ** 2.0d0
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$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. div-sub 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. add-sqr-sqrt 99.2%

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

   4. prod-diff 99.3%

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

4. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} + \mathsf{fma}\left(-\sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \sqrt{\frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}}, \frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\right)} \]
5. Step-by-step derivation

   1. fma-undefine 99.1%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}} \]
8. Add Preprocessing

Alternative 3: 55.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\tan x}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + (tan(x) ** 2.0d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 + Math.pow(Math.tan(x), 2.0));
}

Python

def code(x):
	return 1.0 / (1.0 + math.pow(math.tan(x), 2.0))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + (tan(x) ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 + (tan(x) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. fma-undefine 99.5%

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

   2. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 55.8%

   \[\leadsto \frac{\color{blue}{1}}{{\tan x}^{2} + 1} \]

8. Final simplification 55.8%

   \[\leadsto \frac{1}{1 + {\tan x}^{2}} \]
9. Add Preprocessing

Alternative 4: 59.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 - {\tan x}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 - (tan(x) ** 2.0d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 - Math.pow(Math.tan(x), 2.0));
}

Python

def code(x):
	return 1.0 / (1.0 - math.pow(math.tan(x), 2.0))

Julia

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

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 - (tan(x) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. fma-undefine 99.5%

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

   2. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 55.8%

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

   1. +-commutative 55.8%

      \[\leadsto \frac{1}{\color{blue}{1 + {\tan x}^{2}}} \]

   2. add-sqr-sqrt 55.8%

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

   3. add-cube-cbrt 55.8%

      \[\leadsto \frac{1}{1 + \sqrt{\color{blue}{\left(\sqrt[3]{{\tan x}^{2}} \cdot \sqrt[3]{{\tan x}^{2}}\right) \cdot \sqrt[3]{{\tan x}^{2}}}} \cdot \sqrt{{\tan x}^{2}}} \]

   4. unpow3 55.8%

      \[\leadsto \frac{1}{1 + \sqrt{\color{blue}{{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{3}}} \cdot \sqrt{{\tan x}^{2}}} \]

   5. sqrt-pow1 55.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)}} \cdot \sqrt{{\tan x}^{2}}} \]

   6. add-cube-cbrt 55.8%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot \sqrt{\color{blue}{\left(\sqrt[3]{{\tan x}^{2}} \cdot \sqrt[3]{{\tan x}^{2}}\right) \cdot \sqrt[3]{{\tan x}^{2}}}}} \]

   7. unpow3 55.8%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{3}}}} \]

   8. sqr-pow 55.8%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}}} \]

   9. sqr-neg 55.8%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot \sqrt{\color{blue}{\left(-{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left(-{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)}\right)}}} \]

   10. sqrt-pow1 55.8%

       \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot \sqrt{\left(-{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left(-\color{blue}{\sqrt{{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{3}}}\right)}} \]

   11. rem-cube-cbrt 55.8%

       \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot \sqrt{\left(-{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left(-\sqrt{\color{blue}{{\tan x}^{2}}}\right)}} \]

   12. sqrt-pow1 45.2%

       \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot \sqrt{\left(-{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left(-\color{blue}{{\tan x}^{\left(\frac{2}{2}\right)}}\right)}} \]

   13. metadata-eval 45.2%

       \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot \sqrt{\left(-{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left(-{\tan x}^{\color{blue}{1}}\right)}} \]

   14. pow1 45.2%

       \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot \sqrt{\left(-{\left(\sqrt[3]{{\tan x}^{2}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left(-\color{blue}{\tan x}\right)}} \]

9. Applied egg-rr 59.0%

   \[\leadsto \frac{1}{\color{blue}{1 - {\tan x}^{2}}} \]

10. Final simplification 59.0%

    \[\leadsto \frac{1}{1 - {\tan x}^{2}} \]
11. Add Preprocessing

Alternative 5: 55.4% accurate, 411.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. fma-undefine 99.5%

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

   2. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 55.4%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 55.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))