arccos

Percentage Accurate: 100.0% → 100.0%

Time: 9.5s

Alternatives: 8

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, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. unpow2 N/A

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

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

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. distribute-rgt-neg-in N/A

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

   17. mul-1-neg N/A

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

   18. *-commutative N/A

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

   19. distribute-rgt1-in N/A

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

   20. +-commutative N/A

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

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

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

   22. mul-1-neg N/A

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

   23. sub-neg N/A

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

   24. --lowering--.f64 N/A

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

   25. *-commutative N/A

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

   26. *-lowering-*.f64 99.7%

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

5. Simplified 99.7%

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

   1. *-commutative N/A

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

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

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

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

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

   4. distribute-rgt-in N/A

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

   5. associate-+r+ N/A

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

   6. neg-mul-1 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt1-in N/A

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

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

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

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

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

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

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

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

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

   16. --lowering--.f64 99.7%

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

7. Applied egg-rr 99.7%

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

   1. remove-double-div N/A

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

   2. un-div-inv N/A

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

   3. flip-+ N/A

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

   4. div-inv N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. times-frac N/A

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

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

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in x around 0

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

    3. +-commutative N/A

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

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

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

    5. unpow2 N/A

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

    6. *-lowering-*.f64 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. unpow2 N/A

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

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

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. distribute-rgt-neg-in N/A

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

   17. mul-1-neg N/A

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

   18. *-commutative N/A

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

   19. distribute-rgt1-in N/A

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

   20. +-commutative N/A

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

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

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

   22. mul-1-neg N/A

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

   23. sub-neg N/A

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

   24. --lowering--.f64 N/A

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

   25. *-commutative N/A

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

   26. *-lowering-*.f64 99.7%

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

5. Simplified 99.7%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(1 + x \cdot \left(-1 + \left(1 - x\right) \cdot \left(x \cdot 0.5\right)\right)\right)} \]
6. 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. 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. distribute-lft-in N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. unpow2 N/A

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

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

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. distribute-rgt-neg-in N/A

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

   17. mul-1-neg N/A

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

   18. *-commutative N/A

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

   19. distribute-rgt1-in N/A

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

   20. +-commutative N/A

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

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

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

   22. mul-1-neg N/A

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

   23. sub-neg N/A

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

   24. --lowering--.f64 N/A

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

   25. *-commutative N/A

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

   26. *-lowering-*.f64 99.7%

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

5. Simplified 99.7%

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

   1. *-commutative N/A

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

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

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

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

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

   4. distribute-rgt-in N/A

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

   5. associate-+r+ N/A

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

   6. neg-mul-1 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt1-in N/A

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

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

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

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

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

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

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

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

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

   16. --lowering--.f64 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

   \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\right) \]
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 99.6%

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

5. Simplified 99.6%

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

Alternative 6: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(1.0 + N[(x * 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. 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 99.2%

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

5. Simplified 99.2%

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

   1. flip3-- N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\left(\frac{{1}^{3} - {x}^{3}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\left(\frac{1}{\frac{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)}{{1}^{3} - {x}^{3}}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)}{{1}^{3} - {x}^{3}}\right)\right)\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{{1}^{3} - {x}^{3}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)}}\right)\right)\right)\right) \]

   5. flip3-- N/A

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

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

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

   7. --lowering--.f64 99.2%

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

7. Applied egg-rr 99.2%

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

8. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in x around 0

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

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

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

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

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. +-lowering-+.f64 99.2%

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

13. Simplified 99.2%

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

Alternative 7: 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 99.2%

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

5. Simplified 99.2%

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

Alternative 8: 97.8% 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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))