Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 12.2s

Alternatives: 4

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, 0.8× 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 99.6%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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. expm1-log1p-u 99.3%

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

   2. log1p-def 99.1%

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

   3. expm1-udef 99.1%

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

   4. add-exp-log 99.4%

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

   5. +-commutative 99.4%

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

   6. associate--l+ 99.6%

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

   7. pow2 99.6%

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

   8. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

   1. +-rgt-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. Final simplification 99.6%

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

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 99.6%

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

   1. frac-2neg 99.6%

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

   2. +-commutative 99.6%

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

   3. fma-udef 99.6%

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

   4. fma-udef 99.6%

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

   5. distribute-neg-in 99.6%

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

   6. metadata-eval 99.6%

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

   7. +-commutative 99.6%

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

   8. sub-neg 99.6%

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

   9. distribute-frac-neg 99.6%

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

   10. flip-- 99.4%

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

   11. associate-/l/ 99.4%

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

3. Applied egg-rr 99.6%

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

   1. distribute-neg-frac 99.6%

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

   2. distribute-neg-in 99.6%

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

   3. metadata-eval 99.6%

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

   4. +-commutative 99.6%

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

   5. sub-neg 99.6%

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

   6. +-commutative 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 59.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. expm1-log1p-u 99.3%

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

   2. log1p-def 99.1%

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

   3. expm1-udef 99.1%

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

   4. add-exp-log 99.4%

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

   5. +-commutative 99.4%

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

   6. associate--l+ 99.6%

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

   7. pow2 99.6%

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

   8. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

   1. +-rgt-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. frac-2neg 99.6%

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

   2. div-inv 99.5%

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

   3. neg-sub0 99.5%

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

   4. associate--r- 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. fma-udef 99.5%

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

   8. unpow2 99.4%

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

   9. distribute-neg-in 99.4%

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

   10. metadata-eval 99.4%

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

   11. +-commutative 99.4%

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

   12. sub-neg 99.4%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around 0 58.2%

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

11. Final simplification 58.2%

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

Alternative 4: 55.5% 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 99.6%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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. expm1-log1p-u 99.3%

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

   2. log1p-def 99.1%

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

   3. expm1-udef 99.1%

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

   4. add-exp-log 99.4%

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

   5. +-commutative 99.4%

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

   6. associate--l+ 99.6%

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

   7. pow2 99.6%

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

   8. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

   1. +-rgt-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 53.7%

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

11. Final simplification 53.7%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023305 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))