arccos

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))

C

double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 * Math.atan(Math.sqrt(((1.0 - x) / (1.0 + x))));
}

Python

def code(x):
	return 2.0 * math.atan(math.sqrt(((1.0 - x) / (1.0 + x))))

Julia

function code(x)
	return Float64(2.0 * atan(sqrt(Float64(Float64(1.0 - x) / Float64(1.0 + x)))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))

C

double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 * Math.atan(Math.sqrt(((1.0 - x) / (1.0 + x))));
}

Python

def code(x):
	return 2.0 * math.atan(math.sqrt(((1.0 - x) / (1.0 + x))))

Julia

function code(x)
	return Float64(2.0 * atan(sqrt(Float64(Float64(1.0 - x) / Float64(1.0 + x)))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
end

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left(e^{\left(\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)\right) \cdot 0.5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 2.0 (atan (exp (* (- (log1p (- x)) (log1p x)) 0.5)))))

C

double code(double x) {
	return 2.0 * atan(exp(((log1p(-x) - log1p(x)) * 0.5)));
}

Java

public static double code(double x) {
	return 2.0 * Math.atan(Math.exp(((Math.log1p(-x) - Math.log1p(x)) * 0.5)));
}

Python

def code(x):
	return 2.0 * math.atan(math.exp(((math.log1p(-x) - math.log1p(x)) * 0.5)))

Julia

function code(x)
	return Float64(2.0 * atan(exp(Float64(Float64(log1p(Float64(-x)) - log1p(x)) * 0.5))))
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[Exp[N[(N[(N[Log[1 + (-x)], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
2. Step-by-step derivation

   1. pow1/2 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left(\frac{1 - x}{1 + x}\right)}^{0.5}\right)} \]

   2. pow-to-exp 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(e^{\log \left(\frac{1 - x}{1 + x}\right) \cdot 0.5}\right)} \]

   3. log-div 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\color{blue}{\left(\log \left(1 - x\right) - \log \left(1 + x\right)\right)} \cdot 0.5}\right) \]

   4. sub-neg 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\left(\log \color{blue}{\left(1 + \left(-x\right)\right)} - \log \left(1 + x\right)\right) \cdot 0.5}\right) \]

   5. log1p-def 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\left(\color{blue}{\mathsf{log1p}\left(-x\right)} - \log \left(1 + x\right)\right) \cdot 0.5}\right) \]

   6. log1p-udef 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\left(\mathsf{log1p}\left(-x\right) - \color{blue}{\mathsf{log1p}\left(x\right)}\right) \cdot 0.5}\right) \]

3. Applied egg-rr 100.0%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(e^{\left(\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)\right) \cdot 0.5}\right)} \]

4. Final simplification 100.0%

   \[\leadsto 2 \cdot \tan^{-1} \left(e^{\left(\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)\right) \cdot 0.5}\right) \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* 2.0 (atan (sqrt (* (- 1.0 x) (/ 1.0 (+ x 1.0)))))))

C

double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) * (1.0 / (x + 1.0)))));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 * Math.atan(Math.sqrt(((1.0 - x) * (1.0 / (x + 1.0)))));
}

Python

def code(x):
	return 2.0 * math.atan(math.sqrt(((1.0 - x) * (1.0 / (x + 1.0)))))

Julia

function code(x)
	return Float64(2.0 * atan(sqrt(Float64(Float64(1.0 - x) * Float64(1.0 / Float64(x + 1.0))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) * (1.0 / (x + 1.0)))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
2. Step-by-step derivation

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* 2.0 (atan (/ 1.0 (sqrt (/ (+ x 1.0) (- 1.0 x)))))))

C

double code(double x) {
	return 2.0 * atan((1.0 / sqrt(((x + 1.0) / (1.0 - x)))));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 * Math.atan((1.0 / Math.sqrt(((x + 1.0) / (1.0 - x)))));
}

Python

def code(x):
	return 2.0 * math.atan((1.0 / math.sqrt(((x + 1.0) / (1.0 - x)))))

Julia

function code(x)
	return Float64(2.0 * atan(Float64(1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(1.0 - x))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan((1.0 / sqrt(((x + 1.0) / (1.0 - x)))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
2. Step-by-step derivation

   1. clear-num 100.0%

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

   2. sqrt-div 100.0%

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

   3. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ x 1.0))))))

C

double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) / (x + 1.0))));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 * Math.atan(Math.sqrt(((1.0 - x) / (x + 1.0))));
}

Python

def code(x):
	return 2.0 * math.atan(math.sqrt(((1.0 - x) / (x + 1.0))))

Julia

function code(x)
	return Float64(2.0 * atan(sqrt(Float64(Float64(1.0 - x) / Float64(x + 1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) / (x + 1.0))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (exp (- x)))))

C

double code(double x) {
	return 2.0 * atan(exp(-x));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 * Math.atan(Math.exp(-x));
}

Python

def code(x):
	return 2.0 * math.atan(math.exp(-x))

Julia

function code(x)
	return Float64(2.0 * atan(exp(Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(exp(-x));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[Exp[(-x)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
2. Step-by-step derivation

   1. pow1/2 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left(\frac{1 - x}{1 + x}\right)}^{0.5}\right)} \]

   2. pow-to-exp 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(e^{\log \left(\frac{1 - x}{1 + x}\right) \cdot 0.5}\right)} \]

   3. log-div 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\color{blue}{\left(\log \left(1 - x\right) - \log \left(1 + x\right)\right)} \cdot 0.5}\right) \]

   4. sub-neg 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\left(\log \color{blue}{\left(1 + \left(-x\right)\right)} - \log \left(1 + x\right)\right) \cdot 0.5}\right) \]

   5. log1p-def 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\left(\color{blue}{\mathsf{log1p}\left(-x\right)} - \log \left(1 + x\right)\right) \cdot 0.5}\right) \]

   6. log1p-udef 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\left(\mathsf{log1p}\left(-x\right) - \color{blue}{\mathsf{log1p}\left(x\right)}\right) \cdot 0.5}\right) \]

3. Applied egg-rr 100.0%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(e^{\left(\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)\right) \cdot 0.5}\right)} \]

4. Taylor expanded in x around 0 99.3%

   \[\leadsto 2 \cdot \tan^{-1} \left(e^{\color{blue}{\left(-2 \cdot x\right)} \cdot 0.5}\right) \]
5. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\color{blue}{\left(x \cdot -2\right)} \cdot 0.5}\right) \]

6. Simplified 99.3%

   \[\leadsto 2 \cdot \tan^{-1} \left(e^{\color{blue}{\left(x \cdot -2\right)} \cdot 0.5}\right) \]

7. Taylor expanded in x around 0 99.3%

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

   1. mul-1-neg 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 6: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left(1 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (+ 1.0 (- (* x (* x 0.5)) x)))))

C

double code(double x) {
	return 2.0 * atan((1.0 + ((x * (x * 0.5)) - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan((1.0d0 + ((x * (x * 0.5d0)) - x)))
end function

Java

public static double code(double x) {
	return 2.0 * Math.atan((1.0 + ((x * (x * 0.5)) - x)));
}

Python

def code(x):
	return 2.0 * math.atan((1.0 + ((x * (x * 0.5)) - x)))

Julia

function code(x)
	return Float64(2.0 * atan(Float64(1.0 + Float64(Float64(x * Float64(x * 0.5)) - x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan((1.0 + ((x * (x * 0.5)) - x)));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(1.0 + N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 99.3%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(1 + \left(0.5 \cdot {x}^{2} + -1 \cdot x\right)\right)} \]
3. Step-by-step derivation

   1. neg-mul-1 99.3%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(0.5 \cdot {x}^{2} + \color{blue}{\left(-x\right)}\right)\right) \]

   2. unsub-neg 99.3%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \color{blue}{\left(0.5 \cdot {x}^{2} - x\right)}\right) \]

   3. *-commutative 99.3%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(\color{blue}{{x}^{2} \cdot 0.5} - x\right)\right) \]

   4. unpow2 99.3%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.5 - x\right)\right) \]

   5. associate-*l* 99.3%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(\color{blue}{x \cdot \left(x \cdot 0.5\right)} - x\right)\right) \]

4. Simplified 99.3%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(1 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)\right)} \]

5. Final simplification 99.3%

   \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)\right) \]

Alternative 7: 98.9% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (+ (- 1.0 x) (* x x)))))

C

double code(double x) {
	return 2.0 * atan(((1.0 - x) + (x * x)));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 * Math.atan(((1.0 - x) + (x * x)));
}

Python

def code(x):
	return 2.0 * math.atan(((1.0 - x) + (x * x)))

Julia

function code(x)
	return Float64(2.0 * atan(Float64(Float64(1.0 - x) + Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(((1.0 - x) + (x * x)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
2. Step-by-step derivation

   1. clear-num 100.0%

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

   2. sqrt-div 100.0%

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

   3. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in x around 0 99.1%

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

   1. +-commutative 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in x around 0 99.1%

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

   1. associate-+r+ 99.1%

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

   2. mul-1-neg 99.1%

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

   3. sub-neg 99.1%

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

   4. unpow2 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 8: 98.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (/ 1.0 (+ x 1.0)))))

C

double code(double x) {
	return 2.0 * atan((1.0 / (x + 1.0)));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 * Math.atan((1.0 / (x + 1.0)));
}

Python

def code(x):
	return 2.0 * math.atan((1.0 / (x + 1.0)))

Julia

function code(x)
	return Float64(2.0 * atan(Float64(1.0 / Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan((1.0 / (x + 1.0)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
2. Step-by-step derivation

   1. clear-num 100.0%

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

   2. sqrt-div 100.0%

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

   3. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in x around 0 99.1%

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

   1. +-commutative 99.1%

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

6. Simplified 99.1%

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

7. Final simplification 99.1%

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

Alternative 9: 98.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (- 1.0 x))))

C

double code(double x) {
	return 2.0 * atan((1.0 - x));
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 * Math.atan((1.0 - x));
}

Python

def code(x):
	return 2.0 * math.atan((1.0 - x))

Julia

function code(x)
	return Float64(2.0 * atan(Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan((1.0 - x));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 99.1%

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

   1. neg-mul-1 99.1%

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

   2. sub-neg 99.1%

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

4. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 10: 97.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan 1.0)))

C

double code(double x) {
	return 2.0 * atan(1.0);
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 * Math.atan(1.0);
}

Python

def code(x):
	return 2.0 * math.atan(1.0)

Julia

function code(x)
	return Float64(2.0 * atan(1.0))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(1.0);
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
2. Step-by-step derivation

   1. pow1/2 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left(\frac{1 - x}{1 + x}\right)}^{0.5}\right)} \]

   2. pow-to-exp 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(e^{\log \left(\frac{1 - x}{1 + x}\right) \cdot 0.5}\right)} \]

   3. log-div 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\color{blue}{\left(\log \left(1 - x\right) - \log \left(1 + x\right)\right)} \cdot 0.5}\right) \]

   4. sub-neg 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\left(\log \color{blue}{\left(1 + \left(-x\right)\right)} - \log \left(1 + x\right)\right) \cdot 0.5}\right) \]

   5. log1p-def 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\left(\color{blue}{\mathsf{log1p}\left(-x\right)} - \log \left(1 + x\right)\right) \cdot 0.5}\right) \]

   6. log1p-udef 100.0%

      \[\leadsto 2 \cdot \tan^{-1} \left(e^{\left(\mathsf{log1p}\left(-x\right) - \color{blue}{\mathsf{log1p}\left(x\right)}\right) \cdot 0.5}\right) \]

3. Applied egg-rr 100.0%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(e^{\left(\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)\right) \cdot 0.5}\right)} \]

5. Applied egg-rr 97.9%

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

   1. expm1-def 96.4%

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

   2. expm1-log1p 97.9%

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

   3. *-inverses 97.9%

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

   4. metadata-eval 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

   \[\leadsto 2 \cdot \tan^{-1} 1 \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))