arccos

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 5

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[ArcTan[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] * 2.0), $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 99.6

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

5. Applied rewrites 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 97.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[ArcTan[1.0], $MachinePrecision] * 2.0), $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 98.0%

   \[\leadsto \color{blue}{\tan^{-1} 1 \cdot 2} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024242 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))