arccos

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 7

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, 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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.6

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

   \[\leadsto 2 \cdot \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(x, -0.5, 0.5\right), \color{blue}{x \cdot x}, 1 - x\right)\right) \]
8. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.5, 0.5\right), -1\right), 1\right)\right)} \]
6. Add Preprocessing

Alternative 4: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

   \[\leadsto 2 \cdot \tan^{-1} \left(\mathsf{fma}\left(x \cdot x, \color{blue}{0.5}, 1 - x\right)\right) \]
8. Add Preprocessing

Alternative 5: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\right)} \]
6. Add Preprocessing

Alternative 6: 98.9% accurate, 1.2× 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. sub-neg N/A

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

   3. lower--.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 7: 97.9% accurate, 1.3× 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. Add Preprocessing

4. Applied rewrites 97.9%

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

5. Final simplification 97.9%

   \[\leadsto 2 \cdot \tan^{-1} 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))