Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.6s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * (-N[Tan[x], $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. 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. sqr-neg 99.3%

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

   3. tan-neg 99.3%

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

   4. tan-neg 99.3%

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

   5. div-sub 99.5%

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

   6. sqr-neg 99.5%

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

   7. tan-neg 99.5%

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

   8. tan-neg 99.5%

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

   9. div-sub 99.3%

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

   10. sub-neg 99.3%

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

   11. +-commutative 99.3%

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

3. Simplified 99.6%

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

4. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \color{blue}{\tan x \cdot \tan x}}{1 + \tan x \cdot \tan x} \]
5. Step-by-step derivation

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

   1. pow2 99.5%

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

   2. +-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 99.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \color{blue}{\tan x \cdot \tan x}}{1 + \tan x \cdot \tan x} \]
5. Step-by-step derivation

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

   1. pow2 99.5%

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

   2. +-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

   1. expm1-log1p-u 79.2%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - {\tan x}^{2}\right)\right)}}{{\tan x}^{2} + 1} \]

   2. expm1-undefine 79.2%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(1 - {\tan x}^{2}\right)} - 1}}{{\tan x}^{2} + 1} \]

10. Applied egg-rr 79.2%

    \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(1 - {\tan x}^{2}\right)} - 1}}{{\tan x}^{2} + 1} \]
11. Step-by-step derivation

    1. sub-neg 79.2%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(1 - {\tan x}^{2}\right)} + \left(-1\right)}}{{\tan x}^{2} + 1} \]

    2. metadata-eval 79.2%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(1 - {\tan x}^{2}\right)} + \color{blue}{-1}}{{\tan x}^{2} + 1} \]

    3. +-commutative 79.2%

       \[\leadsto \frac{\color{blue}{-1 + e^{\mathsf{log1p}\left(1 - {\tan x}^{2}\right)}}}{{\tan x}^{2} + 1} \]

    4. log1p-undefine 79.2%

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

    5. rem-exp-log 99.4%

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

    6. associate-+r- 99.4%

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

    7. metadata-eval 99.4%

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

12. Simplified 99.4%

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

    1. associate-+r- 99.5%

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

    2. metadata-eval 99.5%

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

    3. div-sub 99.3%

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

    4. add-exp-log 99.3%

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

    5. +-commutative 99.3%

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

    6. log1p-define 99.3%

       \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{log1p}\left({\tan x}^{2}\right)}}} - \frac{{\tan x}^{2}}{{\tan x}^{2} + 1} \]

    7. rec-exp 99.3%

       \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left({\tan x}^{2}\right)}} - \frac{{\tan x}^{2}}{{\tan x}^{2} + 1} \]

    8. +-commutative 99.3%

       \[\leadsto e^{-\mathsf{log1p}\left({\tan x}^{2}\right)} - \frac{{\tan x}^{2}}{\color{blue}{1 + {\tan x}^{2}}} \]

14. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left({\tan x}^{2}\right)} - \frac{{\tan x}^{2}}{1 + {\tan x}^{2}}} \]
15. Step-by-step derivation

    1. exp-neg 99.3%

       \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left({\tan x}^{2}\right)}}} - \frac{{\tan x}^{2}}{1 + {\tan x}^{2}} \]

    2. log1p-undefine 99.3%

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

    3. rem-exp-log 99.3%

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

    4. *-rgt-identity 99.3%

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

    5. *-rgt-identity 99.3%

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

    6. div-sub 99.5%

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

    7. *-rgt-identity 99.5%

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

    8. metadata-eval 99.5%

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

    9. associate--r+ 99.4%

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

    10. div-sub 99.3%

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

    11. sub-neg 99.3%

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

    12. *-inverses 99.3%

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

    13. metadata-eval 99.3%

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

16. Simplified 99.3%

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

17. Final simplification 99.3%

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

Alternative 4: 55.4% 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. Add Preprocessing
3. Step-by-step derivation

4. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \color{blue}{\tan x \cdot \tan x}}{1 + \tan x \cdot \tan x} \]
5. Step-by-step derivation

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

   1. pow2 99.5%

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

   2. +-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

9. Taylor expanded in x around 0 57.4%

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

10. Final simplification 57.4%

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

Alternative 5: 55.0% 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. Add Preprocessing
3. Step-by-step derivation

4. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \color{blue}{\tan x \cdot \tan x}}{1 + \tan x \cdot \tan x} \]
5. Step-by-step derivation

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

   1. pow2 99.5%

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

   2. +-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

9. Taylor expanded in x around 0 57.1%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))