arccos

Percentage Accurate: 100.0% → 100.0%

Time: 21.3s

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, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot \left(x + -1\right)\right) - -1\\ 2 \cdot \tan^{-1} \left(\left(1 + x \cdot \left(\left(x \cdot \left(-1 + x \cdot 0.5\right)\right) \cdot t\_0\right)\right) \cdot \frac{1}{1 + x \cdot t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (* x (+ x -1.0))) -1.0)))
   (*
    2.0
    (atan
     (*
      (+ 1.0 (* x (* (* x (+ -1.0 (* x 0.5))) t_0)))
      (/ 1.0 (+ 1.0 (* x t_0))))))))

C

double code(double x) {
	double t_0 = (0.5 * (x * (x + -1.0))) - -1.0;
	return 2.0 * atan(((1.0 + (x * ((x * (-1.0 + (x * 0.5))) * t_0))) * (1.0 / (1.0 + (x * t_0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (0.5d0 * (x * (x + (-1.0d0)))) - (-1.0d0)
    code = 2.0d0 * atan(((1.0d0 + (x * ((x * ((-1.0d0) + (x * 0.5d0))) * t_0))) * (1.0d0 / (1.0d0 + (x * t_0)))))
end function

Java

public static double code(double x) {
	double t_0 = (0.5 * (x * (x + -1.0))) - -1.0;
	return 2.0 * Math.atan(((1.0 + (x * ((x * (-1.0 + (x * 0.5))) * t_0))) * (1.0 / (1.0 + (x * t_0)))));
}

Python

def code(x):
	t_0 = (0.5 * (x * (x + -1.0))) - -1.0
	return 2.0 * math.atan(((1.0 + (x * ((x * (-1.0 + (x * 0.5))) * t_0))) * (1.0 / (1.0 + (x * t_0)))))

Julia

function code(x)
	t_0 = Float64(Float64(0.5 * Float64(x * Float64(x + -1.0))) - -1.0)
	return Float64(2.0 * atan(Float64(Float64(1.0 + Float64(x * Float64(Float64(x * Float64(-1.0 + Float64(x * 0.5))) * t_0))) * Float64(1.0 / Float64(1.0 + Float64(x * t_0))))))
end

MATLAB

function tmp = code(x)
	t_0 = (0.5 * (x * (x + -1.0))) - -1.0;
	tmp = 2.0 * atan(((1.0 + (x * ((x * (-1.0 + (x * 0.5))) * t_0))) * (1.0 / (1.0 + (x * t_0)))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]}, N[(2.0 * N[ArcTan[N[(N[(1.0 + N[(x * N[(N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(x * t$95$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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right) - 1\right)\right)}\right)\right) \]
4. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right) - 1\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right) - 1\right)\right)\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right) + -1\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   9. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \left(\mathsf{neg}\left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \left(-1 \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

   13. distribute-rgt1-in N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\left(-1 \cdot x + 1\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\left(1 + -1 \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(1 + -1 \cdot x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

   16. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(1 - x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

   18. --lowering--.f64 99.5%

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.5%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\left(\frac{1 \cdot 1 - \left(x \cdot \left(-1 + x \cdot \left(\left(1 - x\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(x \cdot \left(-1 + x \cdot \left(\left(1 - x\right) \cdot \frac{1}{2}\right)\right)\right)}{1 - x \cdot \left(-1 + x \cdot \left(\left(1 - x\right) \cdot \frac{1}{2}\right)\right)}\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\left(\left(1 \cdot 1 - \left(x \cdot \left(-1 + x \cdot \left(\left(1 - x\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(x \cdot \left(-1 + x \cdot \left(\left(1 - x\right) \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{1}{1 - x \cdot \left(-1 + x \cdot \left(\left(1 - x\right) \cdot \frac{1}{2}\right)\right)}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(1 \cdot 1 - \left(x \cdot \left(-1 + x \cdot \left(\left(1 - x\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(x \cdot \left(-1 + x \cdot \left(\left(1 - x\right) \cdot \frac{1}{2}\right)\right)\right)\right), \left(\frac{1}{1 - x \cdot \left(-1 + x \cdot \left(\left(1 - x\right) \cdot \frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

7. Applied egg-rr 99.5%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right), \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot x - 1\right)\right)}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x - 1\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x + -1\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, \left(-1 + \frac{1}{2} \cdot x\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.6%

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

11. Final simplification 99.6%

    \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 + x \cdot \left(\left(x \cdot \left(-1 + x \cdot 0.5\right)\right) \cdot \left(0.5 \cdot \left(x \cdot \left(x + -1\right)\right) - -1\right)\right)\right) \cdot \frac{1}{1 + x \cdot \left(0.5 \cdot \left(x \cdot \left(x + -1\right)\right) - -1\right)}\right) \]
12. Add Preprocessing

Alternative 3: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(N[(1.0 - x), $MachinePrecision] / N[(1.0 - N[(0.5 * N[(x * 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. Add Preprocessing
3. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{1 + x} - \frac{x}{1 + x}\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{1 + x} - \frac{1}{\frac{1 + x}{x}}\right)\right)\right)\right) \]

   3. frac-sub N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 \cdot \frac{1 + x}{x} - \left(1 + x\right) \cdot 1}{\left(1 + x\right) \cdot \frac{1 + x}{x}}\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot \frac{1 + x}{x} - \left(1 + x\right) \cdot 1\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(1 + x\right)}{x} - \left(1 + x\right) \cdot 1\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   6. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + x}{x} - \left(1 + x\right) \cdot 1\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + x}{x} - 1 \cdot \left(1 + x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + x}{x} - \left(1 + x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1 + x}{x}\right), \left(1 + x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), x\right), \left(1 + x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \left(1 + x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(1, x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\left(1 + x\right), \left(\frac{1 + x}{x}\right)\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(\frac{1 + x}{x}\right)\right)\right)\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{/.f64}\left(\left(1 + x\right), x\right)\right)\right)\right)\right)\right) \]

   16. +-lowering-+.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right) - 1\right)\right)}\right)\right) \]

7. Simplified 99.5%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\left(\frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)}{1 - x \cdot \left(x \cdot \frac{1}{2}\right)} \cdot \left(1 - x\right)\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\left(\frac{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right) \cdot \left(1 - x\right)}{1 - x \cdot \left(x \cdot \frac{1}{2}\right)}\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right) \cdot \left(1 - x\right)\right), \left(1 - x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 99.5%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\frac{\left(1 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.25\right) \cdot \left(1 - x\right)}{1 - 0.5 \cdot \left(x \cdot x\right)}\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + -1 \cdot x\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(1 - x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

    3. --lowering--.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

12. Simplified 99.6%

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

Alternative 4: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 + N[(x * N[(x * 0.5), $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. Add Preprocessing
3. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{1 + x} - \frac{x}{1 + x}\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{1 + x} - \frac{1}{\frac{1 + x}{x}}\right)\right)\right)\right) \]

   3. frac-sub N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 \cdot \frac{1 + x}{x} - \left(1 + x\right) \cdot 1}{\left(1 + x\right) \cdot \frac{1 + x}{x}}\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot \frac{1 + x}{x} - \left(1 + x\right) \cdot 1\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \left(1 + x\right)}{x} - \left(1 + x\right) \cdot 1\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   6. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + x}{x} - \left(1 + x\right) \cdot 1\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + x}{x} - 1 \cdot \left(1 + x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + x}{x} - \left(1 + x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1 + x}{x}\right), \left(1 + x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), x\right), \left(1 + x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \left(1 + x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(1, x\right)\right), \left(\left(1 + x\right) \cdot \frac{1 + x}{x}\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\left(1 + x\right), \left(\frac{1 + x}{x}\right)\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(\frac{1 + x}{x}\right)\right)\right)\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{/.f64}\left(\left(1 + x\right), x\right)\right)\right)\right)\right)\right) \]

   16. +-lowering-+.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right) - 1\right)\right)}\right)\right) \]

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 5: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} \cdot x - 1\right)\right)}\right)\right) \]
4. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} \cdot x - 1\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x - 1\right)\right)\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x + -1\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + \frac{1}{2} \cdot x\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot x\right)\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 99.4%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.4%

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

Alternative 6: 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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\color{blue}{\left(1 + -1 \cdot x\right)}\right)\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\left(1 - x\right)\right)\right) \]

   3. --lowering--.f64 99.1%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{\_.f64}\left(1, x\right)\right)\right) \]

5. Simplified 99.1%

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

Alternative 7: 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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\color{blue}{1}\right)\right) \]
4. Step-by-step derivation

5. Simplified 98.2%

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

Reproduce

herbie shell --seed 2024141 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))