Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 12.1s

Alternatives: 3

Speedup: N/A×

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

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

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. add-log-exp 99.2%

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

   2. *-un-lft-identity 99.2%

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

   3. log-prod 99.2%

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

   4. metadata-eval 99.2%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}}\right) \]

   5. add-log-exp 99.5%

      \[\leadsto 0 + \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}} \]

   6. pow2 99.5%

      \[\leadsto 0 + \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \tan x \cdot \tan x} \]

   7. add-sqr-sqrt 99.2%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{\color{blue}{\sqrt{1 + \tan x \cdot \tan x} \cdot \sqrt{1 + \tan x \cdot \tan x}}} \]

   8. pow2 99.2%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{\color{blue}{{\left(\sqrt{1 + \tan x \cdot \tan x}\right)}^{2}}} \]

   9. hypot-1-def 99.2%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{{\color{blue}{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}}^{2}} \]

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{0 + \frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}} \]
5. Step-by-step derivation

   1. +-lft-identity 99.2%

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

   2. unpow2 99.2%

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

   3. hypot-undefine 99.3%

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

   4. metadata-eval 99.3%

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

   5. unpow2 99.3%

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

   6. rem-exp-log 99.3%

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

   7. log1p-undefine 99.3%

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

   8. hypot-undefine 99.2%

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

   9. metadata-eval 99.2%

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

   10. unpow2 99.2%

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

   11. rem-exp-log 99.2%

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

   12. log1p-undefine 99.2%

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

   13. rem-square-sqrt 99.3%

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

   14. log1p-undefine 99.3%

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

   15. rem-exp-log 99.5%

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

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 2: 59.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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. add-log-exp 99.2%

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

   2. *-un-lft-identity 99.2%

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

   3. log-prod 99.2%

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

   4. metadata-eval 99.2%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}}\right) \]

   5. add-log-exp 99.5%

      \[\leadsto 0 + \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}} \]

   6. pow2 99.5%

      \[\leadsto 0 + \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \tan x \cdot \tan x} \]

   7. add-sqr-sqrt 99.2%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{\color{blue}{\sqrt{1 + \tan x \cdot \tan x} \cdot \sqrt{1 + \tan x \cdot \tan x}}} \]

   8. pow2 99.2%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{\color{blue}{{\left(\sqrt{1 + \tan x \cdot \tan x}\right)}^{2}}} \]

   9. hypot-1-def 99.2%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{{\color{blue}{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}}^{2}} \]

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{0 + \frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}} \]
5. Step-by-step derivation

   1. +-lft-identity 99.2%

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

   2. unpow2 99.2%

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

   3. hypot-undefine 99.3%

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

   4. metadata-eval 99.3%

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

   5. unpow2 99.3%

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

   6. rem-exp-log 99.3%

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

   7. log1p-undefine 99.3%

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

   8. hypot-undefine 99.2%

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

   9. metadata-eval 99.2%

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

   10. unpow2 99.2%

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

   11. rem-exp-log 99.2%

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

   12. log1p-undefine 99.2%

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

   13. rem-square-sqrt 99.3%

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

   14. log1p-undefine 99.3%

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

   15. rem-exp-log 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 53.8%

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

   1. add-exp-log 53.8%

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

   2. log-rec 53.8%

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

   3. log1p-define 53.8%

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

9. Applied egg-rr 53.8%

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

    1. add-sqr-sqrt 20.7%

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

    2. sqrt-unprod 53.4%

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

    3. sqr-neg 53.4%

       \[\leadsto e^{\sqrt{\color{blue}{\mathsf{log1p}\left({\tan x}^{2}\right) \cdot \mathsf{log1p}\left({\tan x}^{2}\right)}}} \]

    4. sqrt-unprod 53.4%

       \[\leadsto e^{\color{blue}{\sqrt{\mathsf{log1p}\left({\tan x}^{2}\right)} \cdot \sqrt{\mathsf{log1p}\left({\tan x}^{2}\right)}}} \]

    5. add-sqr-sqrt 53.4%

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

    6. log1p-undefine 53.4%

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

    7. add-exp-log 53.4%

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

    8. +-commutative 53.4%

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

    9. unpow2 53.4%

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

    10. fma-define 53.4%

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

    11. add-sqr-sqrt 27.5%

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

    12. sqrt-prod 55.6%

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

    13. sqr-neg 55.6%

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

    14. sqrt-unprod 28.1%

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

    15. add-sqr-sqrt 58.4%

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

    16. fma-undefine 58.4%

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

    17. +-commutative 58.4%

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

    18. *-commutative 58.4%

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

    19. cancel-sign-sub-inv 58.4%

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

    20. unpow2 58.4%

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

11. Applied egg-rr 58.4%

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

12. Final simplification 58.4%

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

Alternative 3: 54.8% 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

   1. add-log-exp 99.2%

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

   2. *-un-lft-identity 99.2%

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

   3. log-prod 99.2%

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

   4. metadata-eval 99.2%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}}\right) \]

   5. add-log-exp 99.5%

      \[\leadsto 0 + \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}} \]

   6. pow2 99.5%

      \[\leadsto 0 + \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \tan x \cdot \tan x} \]

   7. add-sqr-sqrt 99.2%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{\color{blue}{\sqrt{1 + \tan x \cdot \tan x} \cdot \sqrt{1 + \tan x \cdot \tan x}}} \]

   8. pow2 99.2%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{\color{blue}{{\left(\sqrt{1 + \tan x \cdot \tan x}\right)}^{2}}} \]

   9. hypot-1-def 99.2%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{{\color{blue}{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}}^{2}} \]

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{0 + \frac{1 - {\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}} \]
5. Step-by-step derivation

   1. +-lft-identity 99.2%

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

   2. unpow2 99.2%

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

   3. hypot-undefine 99.3%

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

   4. metadata-eval 99.3%

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

   5. unpow2 99.3%

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

   6. rem-exp-log 99.3%

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

   7. log1p-undefine 99.3%

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

   8. hypot-undefine 99.2%

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

   9. metadata-eval 99.2%

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

   10. unpow2 99.2%

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

   11. rem-exp-log 99.2%

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

   12. log1p-undefine 99.2%

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

   13. rem-square-sqrt 99.3%

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

   14. log1p-undefine 99.3%

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

   15. rem-exp-log 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 53.8%

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

8. Taylor expanded in x around 0 53.5%

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

9. Final simplification 53.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))