Trigonometry B

Percentage Accurate: 99.5% → 99.4%

Time: 11.0s

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (1.0 - (tan(x) / (1.0 / 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) / (1.0d0 / tan(x)))) / (1.0d0 + (tan(x) ** 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[x], $MachinePrecision]), $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. tan-quot 99.4%

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

   2. associate-*r/ 99.3%

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

3. Applied egg-rr 99.3%

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

   1. associate-/l* 99.4%

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

5. Simplified 99.4%

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

   1. expm1-log1p-u 61.6%

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

   2. expm1-udef 61.3%

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

   3. clear-num 61.4%

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

   4. tan-quot 61.4%

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

7. Applied egg-rr 61.4%

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

   1. expm1-def 61.6%

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

   2. expm1-log1p 99.5%

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

9. Simplified 99.5%

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

    1. add-log-exp 98.7%

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

    2. *-un-lft-identity 98.7%

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

    3. log-prod 98.7%

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

    4. metadata-eval 98.7%

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

    5. add-log-exp 99.5%

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

    6. pow2 99.5%

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

11. Applied egg-rr 99.5%

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

    1. +-lft-identity 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

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. Step-by-step derivation

   1. add-log-exp 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. log-prod 98.7%

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

   4. metadata-eval 98.7%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

3. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

5. Simplified 99.5%

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

   1. expm1-log1p-u 99.4%

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

   2. expm1-udef 99.4%

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

   3. pow2 99.4%

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

7. Applied egg-rr 99.4%

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

   1. expm1-def 99.4%

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

   2. expm1-log1p 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 56.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \frac{\tan x}{x \cdot -0.3333333333333333 + \frac{1}{x}}}{1 + \tan x \cdot \tan x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (/ (tan x) (+ (* x -0.3333333333333333) (/ 1.0 x))))
  (+ 1.0 (* (tan x) (tan x)))))

C

double code(double x) {
	return (1.0 - (tan(x) / ((x * -0.3333333333333333) + (1.0 / x)))) / (1.0 + (tan(x) * tan(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - (tan(x) / ((x * (-0.3333333333333333d0)) + (1.0d0 / x)))) / (1.0d0 + (tan(x) * tan(x)))
end function

Java

public static double code(double x) {
	return (1.0 - (Math.tan(x) / ((x * -0.3333333333333333) + (1.0 / x)))) / (1.0 + (Math.tan(x) * Math.tan(x)));
}

Python

def code(x):
	return (1.0 - (math.tan(x) / ((x * -0.3333333333333333) + (1.0 / x)))) / (1.0 + (math.tan(x) * math.tan(x)))

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) / Float64(Float64(x * -0.3333333333333333) + Float64(1.0 / x)))) / Float64(1.0 + Float64(tan(x) * tan(x))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - (tan(x) / ((x * -0.3333333333333333) + (1.0 / x)))) / (1.0 + (tan(x) * tan(x)));
end

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(N[(x * -0.3333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $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. tan-quot 99.4%

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

   2. associate-*r/ 99.3%

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

3. Applied egg-rr 99.3%

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

   1. associate-/l* 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0 56.8%

   \[\leadsto \frac{1 - \frac{\tan x}{\color{blue}{-0.3333333333333333 \cdot x + \frac{1}{x}}}}{1 + \tan x \cdot \tan x} \]

7. Final simplification 56.8%

   \[\leadsto \frac{1 - \frac{\tan x}{x \cdot -0.3333333333333333 + \frac{1}{x}}}{1 + \tan x \cdot \tan x} \]

Alternative 4: 55.6% 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. add-log-exp 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. log-prod 98.7%

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

   4. metadata-eval 98.7%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

3. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

5. Simplified 99.5%

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

   1. expm1-log1p-u 99.4%

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

   2. expm1-udef 99.4%

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

   3. pow2 99.4%

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

7. Applied egg-rr 99.4%

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

   1. expm1-def 99.4%

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

   2. expm1-log1p 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in x around 0 56.8%

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

11. Final simplification 56.8%

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

Alternative 5: 55.3% 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. add-log-exp 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. log-prod 98.7%

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

   4. metadata-eval 98.7%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

3. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

5. Simplified 99.5%

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

   1. expm1-log1p-u 99.4%

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

   2. expm1-udef 99.4%

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

   3. pow2 99.4%

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

7. Applied egg-rr 99.4%

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

   1. expm1-def 99.4%

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

   2. expm1-log1p 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in x around 0 56.4%

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

11. Final simplification 56.4%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023314 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))