Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 15.7s

Alternatives: 6

Speedup: 0.8×

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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 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. Step-by-step derivation

   1. add-log-exp 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. log-prod 98.3%

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

   4. metadata-eval 98.3%

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

   5. add-log-exp 99.4%

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

   6. pow2 99.4%

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

5. Applied egg-rr 99.4%

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

   1. +-lft-identity 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 2: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1:\\ \;\;\;\;\frac{1}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\log 2 - x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.0)
   (/ 1.0 (pow (hypot 1.0 (tan x)) 2.0))
   (+ -1.0 (exp (- (log 2.0) (* x x))))))

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.0) {
		tmp = 1.0 / pow(hypot(1.0, tan(x)), 2.0);
	} else {
		tmp = -1.0 + exp((log(2.0) - (x * x)));
	}
	return tmp;
}

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 1.0)
		tmp = Float64(1.0 / (hypot(1.0, tan(x)) ^ 2.0));
	else
		tmp = Float64(-1.0 + exp(Float64(log(2.0) - Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((tan(x) * tan(x)) <= 1.0)
		tmp = 1.0 / (hypot(1.0, tan(x)) ^ 2.0);
	else
		tmp = -1.0 + exp((log(2.0) - (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 1.0], N[(1.0 / N[Power[N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Exp[N[(N[Log[2.0], $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1

   1. Initial program 99.6%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-udef 99.4%

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

   3. Applied egg-rr 99.4%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\tan x, \tan x, 1\right)\right)} - 1}} \]
   4. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\tan x, \tan x, 1\right)\right)\right)}} \]

      2. expm1-log1p 99.6%

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

      3. rem-square-sqrt 99.4%

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

      4. fma-udef 99.4%

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

      5. unpow2 99.4%

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

      6. +-commutative 99.4%

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

      7. unpow2 99.4%

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

      8. hypot-1-def 99.5%

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

      9. fma-udef 99.5%

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

      10. unpow2 99.5%

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

      11. +-commutative 99.5%

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

      12. unpow2 99.5%

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

      13. hypot-1-def 99.5%

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

      14. unpow2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 72.6%

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

   if 1 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-def 99.0%

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

   3. Simplified 99.0%

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

      1. add-log-exp 94.7%

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

      2. *-un-lft-identity 94.7%

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

      3. log-prod 94.7%

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

      4. metadata-eval 94.7%

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

      5. add-log-exp 99.0%

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

      6. pow2 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. +-lft-identity 99.0%

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

   7. Simplified 99.0%

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

      1. fma-udef 99.0%

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

      2. pow2 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. expm1-log1p-u 98.9%

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

       2. expm1-udef 99.0%

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

       3. unpow2 99.0%

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

       4. fma-udef 99.0%

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

   11. Applied egg-rr 99.0%

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

   12. Taylor expanded in x around 0 20.9%

       \[\leadsto e^{\color{blue}{\log 2 + -1 \cdot {x}^{2}}} - 1 \]
   13. Step-by-step derivation

       1. mul-1-neg 20.9%

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

       2. unsub-neg 20.9%

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

       3. unpow2 20.9%

          \[\leadsto e^{\log 2 - \color{blue}{x \cdot x}} - 1 \]

   14. Simplified 20.9%

       \[\leadsto e^{\color{blue}{\log 2 - x \cdot x}} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1:\\ \;\;\;\;\frac{1}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\log 2 - x \cdot x}\\ \end{array} \]

Alternative 3: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\log 2 - x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.0)
   (/ 1.0 (+ 1.0 (pow (tan x) 2.0)))
   (+ -1.0 (exp (- (log 2.0) (* x x))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. add-log-exp 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. log-prod 99.5%

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

      4. metadata-eval 99.5%

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

      5. add-log-exp 99.6%

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

      6. pow2 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. +-lft-identity 99.6%

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

   7. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. pow2 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in x around 0 72.6%

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

   if 1 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-def 99.0%

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

   3. Simplified 99.0%

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

      1. add-log-exp 94.7%

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

      2. *-un-lft-identity 94.7%

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

      3. log-prod 94.7%

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

      4. metadata-eval 94.7%

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

      5. add-log-exp 99.0%

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

      6. pow2 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. +-lft-identity 99.0%

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

   7. Simplified 99.0%

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

      1. fma-udef 99.0%

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

      2. pow2 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. expm1-log1p-u 98.9%

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

       2. expm1-udef 99.0%

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

       3. unpow2 99.0%

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

       4. fma-udef 99.0%

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

   11. Applied egg-rr 99.0%

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

   12. Taylor expanded in x around 0 20.9%

       \[\leadsto e^{\color{blue}{\log 2 + -1 \cdot {x}^{2}}} - 1 \]
   13. Step-by-step derivation

       1. mul-1-neg 20.9%

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

       2. unsub-neg 20.9%

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

       3. unpow2 20.9%

          \[\leadsto e^{\log 2 - \color{blue}{x \cdot x}} - 1 \]

   14. Simplified 20.9%

       \[\leadsto e^{\color{blue}{\log 2 - x \cdot x}} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\log 2 - x \cdot x}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. frac-2neg 99.4%

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

   2. div-inv 99.4%

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

   3. pow2 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. neg-mul-1 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-def 99.4%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. neg-sub0 99.4%

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

   4. associate--r- 99.4%

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

   5. metadata-eval 99.4%

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

   6. fma-udef 99.4%

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

   7. neg-mul-1 99.4%

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

   8. +-commutative 99.4%

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

   9. unsub-neg 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 55.2% 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 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. Step-by-step derivation

   1. add-log-exp 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. log-prod 98.3%

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

   4. metadata-eval 98.3%

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

   5. add-log-exp 99.4%

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

   6. pow2 99.4%

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

5. Applied egg-rr 99.4%

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

   1. +-lft-identity 99.4%

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

7. Simplified 99.4%

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

   1. fma-udef 99.4%

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

   2. pow2 99.4%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around 0 54.0%

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

11. Final simplification 54.0%

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

Alternative 6: 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 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. Step-by-step derivation

   1. add-log-exp 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. log-prod 98.3%

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

   4. metadata-eval 98.3%

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

   5. add-log-exp 99.4%

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

   6. pow2 99.4%

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

5. Applied egg-rr 99.4%

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

   1. +-lft-identity 99.4%

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

7. Simplified 99.4%

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

   1. fma-udef 99.4%

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

   2. pow2 99.4%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around 0 53.7%

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

11. Final simplification 53.7%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023283 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))