Hyperbolic arc-(co)tangent

Percentage Accurate: 8.5% → 100.0%

Time: 11.2s

Alternatives: 8

Speedup: 111.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 8.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. metadata-eval 7.6%

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

3. Simplified 7.6%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. flip-+ 7.4%

      \[\leadsto 0.5 \cdot \log \left(\frac{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}}{1 - x}\right) \]

   2. associate-/l/ 7.4%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left(\frac{1 \cdot 1 - x \cdot x}{\left(1 - x\right) \cdot \left(1 - x\right)}\right)} \]

   3. log-div 7.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\log \left(1 \cdot 1 - x \cdot x\right) - \log \left(\left(1 - x\right) \cdot \left(1 - x\right)\right)\right)} \]

   4. metadata-eval 7.4%

      \[\leadsto 0.5 \cdot \left(\log \left(\color{blue}{1} - x \cdot x\right) - \log \left(\left(1 - x\right) \cdot \left(1 - x\right)\right)\right) \]

   5. sub-neg 7.4%

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

   6. log1p-define 7.8%

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

   7. pow2 7.8%

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

   8. pow2 7.8%

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

   9. log-pow 7.8%

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

   10. sub-neg 7.8%

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

   11. log1p-define 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(-{x}^{2}\right) - 2 \cdot \mathsf{log1p}\left(-x\right)\right)} \]
7. Step-by-step derivation

   1. unpow2 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x \cdot \left(-x\right)\right) - 2 \cdot \mathsf{log1p}\left(-x\right)\right) \]
10. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (- (log1p x) (log1p (- x)))))

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. metadata-eval 7.6%

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

3. Simplified 7.6%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 7.6%

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

   2. *-commutative 7.6%

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

   3. log-prod 7.6%

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

   4. log-div 7.6%

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

   5. log1p-define 20.1%

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

   6. sub-neg 20.1%

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

   7. log1p-define 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) + \color{blue}{0}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) + 0\right)} \]
7. Step-by-step derivation

   1. +-rgt-identity 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

8. Simplified 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + {x}^{3} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x (* (pow x 3.0) 0.3333333333333333)))

C

double code(double x) {
	return x + (pow(x, 3.0) * 0.3333333333333333);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((x ** 3.0d0) * 0.3333333333333333d0)
end function

Java

public static double code(double x) {
	return x + (Math.pow(x, 3.0) * 0.3333333333333333);
}

Python

def code(x):
	return x + (math.pow(x, 3.0) * 0.3333333333333333)

Julia

function code(x)
	return Float64(x + Float64((x ^ 3.0) * 0.3333333333333333))
end

MATLAB

function tmp = code(x)
	tmp = x + ((x ^ 3.0) * 0.3333333333333333);
end

Wolfram

code[x_] := N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. metadata-eval 7.6%

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

3. Simplified 7.6%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in 99.5%

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(0.3333333333333333 \cdot {x}^{2}\right)} \]

   2. *-rgt-identity 99.5%

      \[\leadsto \color{blue}{x} + x \cdot \left(0.3333333333333333 \cdot {x}^{2}\right) \]

   3. *-commutative 99.5%

      \[\leadsto x + x \cdot \color{blue}{\left({x}^{2} \cdot 0.3333333333333333\right)} \]

   4. associate-*r* 99.5%

      \[\leadsto x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot 0.3333333333333333} \]

   5. unpow2 99.5%

      \[\leadsto x + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 0.3333333333333333 \]

   6. cube-mult 99.5%

      \[\leadsto x + \color{blue}{{x}^{3}} \cdot 0.3333333333333333 \]

7. Simplified 99.5%

   \[\leadsto \color{blue}{x + {x}^{3} \cdot 0.3333333333333333} \]
8. Add Preprocessing

Alternative 4: 99.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.3333333333333333 + x \cdot -0.25\right) - 0.5\right)\right) + x \cdot \left(1 - x \cdot \left(x \cdot \left(x \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.5
  (+
   (* x (+ 1.0 (* x (- (* x (+ 0.3333333333333333 (* x -0.25))) 0.5))))
   (* x (- 1.0 (* x (- (* x (- (* x -0.25) 0.3333333333333333)) 0.5)))))))

C

double code(double x) {
	return 0.5 * ((x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)))) + (x * (1.0 - (x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((x * (1.0d0 + (x * ((x * (0.3333333333333333d0 + (x * (-0.25d0)))) - 0.5d0)))) + (x * (1.0d0 - (x * ((x * ((x * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0)))))
end function

Java

public static double code(double x) {
	return 0.5 * ((x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)))) + (x * (1.0 - (x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)))));
}

Python

def code(x):
	return 0.5 * ((x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)))) + (x * (1.0 - (x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)))))

Julia

function code(x)
	return Float64(0.5 * Float64(Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.3333333333333333 + Float64(x * -0.25))) - 0.5)))) + Float64(x * Float64(1.0 - Float64(x * Float64(Float64(x * Float64(Float64(x * -0.25) - 0.3333333333333333)) - 0.5))))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)))) + (x * (1.0 - (x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)))));
end

Wolfram

code[x_] := N[(0.5 * N[(N[(x * N[(1.0 + N[(x * N[(N[(x * N[(0.3333333333333333 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(1.0 - N[(x * N[(N[(x * N[(N[(x * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. metadata-eval 7.6%

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

3. Simplified 7.6%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 7.6%

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

   2. *-commutative 7.6%

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

   3. log-prod 7.6%

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

   4. log-div 7.6%

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

   5. log1p-define 20.1%

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

   6. sub-neg 20.1%

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

   7. log1p-define 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) + \color{blue}{0}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) + 0\right)} \]
7. Step-by-step derivation

   1. +-rgt-identity 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

8. Simplified 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

9. Taylor expanded in x around 0 99.5%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(-0.25 \cdot x - 0.3333333333333333\right) - 0.5\right) - 1\right)}\right) \]

10. Taylor expanded in x around 0 99.5%

    \[\leadsto 0.5 \cdot \left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.3333333333333333 + -0.25 \cdot x\right) - 0.5\right)\right)} - x \cdot \left(x \cdot \left(x \cdot \left(-0.25 \cdot x - 0.3333333333333333\right) - 0.5\right) - 1\right)\right) \]

11. Final simplification 99.5%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.3333333333333333 + x \cdot -0.25\right) - 0.5\right)\right) + x \cdot \left(1 - x \cdot \left(x \cdot \left(x \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) \]
12. Add Preprocessing

Alternative 5: 99.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\right) + x \cdot \left(1 + x \cdot \left(0.5 - x \cdot -0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.5
  (+
   (* x (+ 1.0 (* x (- (* x 0.3333333333333333) 0.5))))
   (* x (+ 1.0 (* x (- 0.5 (* x -0.3333333333333333))))))))

C

double code(double x) {
	return 0.5 * ((x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))) + (x * (1.0 + (x * (0.5 - (x * -0.3333333333333333))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((x * (1.0d0 + (x * ((x * 0.3333333333333333d0) - 0.5d0)))) + (x * (1.0d0 + (x * (0.5d0 - (x * (-0.3333333333333333d0)))))))
end function

Java

public static double code(double x) {
	return 0.5 * ((x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))) + (x * (1.0 + (x * (0.5 - (x * -0.3333333333333333))))));
}

Python

def code(x):
	return 0.5 * ((x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))) + (x * (1.0 + (x * (0.5 - (x * -0.3333333333333333))))))

Julia

function code(x)
	return Float64(0.5 * Float64(Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * 0.3333333333333333) - 0.5)))) + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 - Float64(x * -0.3333333333333333)))))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))) + (x * (1.0 + (x * (0.5 - (x * -0.3333333333333333))))));
end

Wolfram

code[x_] := N[(0.5 * N[(N[(x * N[(1.0 + N[(x * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(1.0 + N[(x * N[(0.5 - N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. metadata-eval 7.6%

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

3. Simplified 7.6%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 7.6%

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

   2. *-commutative 7.6%

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

   3. log-prod 7.6%

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

   4. log-div 7.6%

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

   5. log1p-define 20.1%

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

   6. sub-neg 20.1%

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

   7. log1p-define 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) + \color{blue}{0}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) + 0\right)} \]
7. Step-by-step derivation

   1. +-rgt-identity 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

8. Simplified 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

9. Taylor expanded in x around 0 99.5%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(-0.25 \cdot x - 0.3333333333333333\right) - 0.5\right) - 1\right)}\right) \]

10. Taylor expanded in x around 0 99.4%

    \[\leadsto 0.5 \cdot \left(\color{blue}{x \cdot \left(1 + x \cdot \left(0.3333333333333333 \cdot x - 0.5\right)\right)} - x \cdot \left(x \cdot \left(x \cdot \left(-0.25 \cdot x - 0.3333333333333333\right) - 0.5\right) - 1\right)\right) \]

11. Taylor expanded in x around 0 99.5%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(1 + x \cdot \left(0.3333333333333333 \cdot x - 0.5\right)\right) - x \cdot \left(x \cdot \left(x \cdot \color{blue}{-0.3333333333333333} - 0.5\right) - 1\right)\right) \]

12. Final simplification 99.5%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\right) + x \cdot \left(1 + x \cdot \left(0.5 - x \cdot -0.3333333333333333\right)\right)\right) \]
13. Add Preprocessing

Alternative 6: 99.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \left(1 + x \cdot -0.5\right) + x \cdot \left(1 - x \cdot \left(x \cdot \left(x \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.5
  (+
   (* x (+ 1.0 (* x -0.5)))
   (* x (- 1.0 (* x (- (* x (- (* x -0.25) 0.3333333333333333)) 0.5)))))))

C

double code(double x) {
	return 0.5 * ((x * (1.0 + (x * -0.5))) + (x * (1.0 - (x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((x * (1.0d0 + (x * (-0.5d0)))) + (x * (1.0d0 - (x * ((x * ((x * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0)))))
end function

Java

public static double code(double x) {
	return 0.5 * ((x * (1.0 + (x * -0.5))) + (x * (1.0 - (x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)))));
}

Python

def code(x):
	return 0.5 * ((x * (1.0 + (x * -0.5))) + (x * (1.0 - (x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)))))

Julia

function code(x)
	return Float64(0.5 * Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) + Float64(x * Float64(1.0 - Float64(x * Float64(Float64(x * Float64(Float64(x * -0.25) - 0.3333333333333333)) - 0.5))))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((x * (1.0 + (x * -0.5))) + (x * (1.0 - (x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)))));
end

Wolfram

code[x_] := N[(0.5 * N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(1.0 - N[(x * N[(N[(x * N[(N[(x * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. metadata-eval 7.6%

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

3. Simplified 7.6%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 7.6%

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

   2. *-commutative 7.6%

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

   3. log-prod 7.6%

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

   4. log-div 7.6%

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

   5. log1p-define 20.1%

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

   6. sub-neg 20.1%

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

   7. log1p-define 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) + \color{blue}{0}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) + 0\right)} \]
7. Step-by-step derivation

   1. +-rgt-identity 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

8. Simplified 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

9. Taylor expanded in x around 0 99.5%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(-0.25 \cdot x - 0.3333333333333333\right) - 0.5\right) - 1\right)}\right) \]

10. Taylor expanded in x around 0 99.3%

    \[\leadsto 0.5 \cdot \left(\color{blue}{x \cdot \left(1 + -0.5 \cdot x\right)} - x \cdot \left(x \cdot \left(x \cdot \left(-0.25 \cdot x - 0.3333333333333333\right) - 0.5\right) - 1\right)\right) \]

11. Final simplification 99.3%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(1 + x \cdot -0.5\right) + x \cdot \left(1 - x \cdot \left(x \cdot \left(x \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) \]
12. Add Preprocessing

Alternative 7: 99.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\right) + x \cdot \left(1 - x \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.5
  (+
   (* x (+ 1.0 (* x (- (* x 0.3333333333333333) 0.5))))
   (* x (- 1.0 (* x -0.5))))))

C

double code(double x) {
	return 0.5 * ((x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))) + (x * (1.0 - (x * -0.5))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((x * (1.0d0 + (x * ((x * 0.3333333333333333d0) - 0.5d0)))) + (x * (1.0d0 - (x * (-0.5d0)))))
end function

Java

public static double code(double x) {
	return 0.5 * ((x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))) + (x * (1.0 - (x * -0.5))));
}

Python

def code(x):
	return 0.5 * ((x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))) + (x * (1.0 - (x * -0.5))))

Julia

function code(x)
	return Float64(0.5 * Float64(Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * 0.3333333333333333) - 0.5)))) + Float64(x * Float64(1.0 - Float64(x * -0.5)))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))) + (x * (1.0 - (x * -0.5))));
end

Wolfram

code[x_] := N[(0.5 * N[(N[(x * N[(1.0 + N[(x * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. metadata-eval 7.6%

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

3. Simplified 7.6%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 7.6%

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

   2. *-commutative 7.6%

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

   3. log-prod 7.6%

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

   4. log-div 7.6%

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

   5. log1p-define 20.1%

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

   6. sub-neg 20.1%

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

   7. log1p-define 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) + \color{blue}{0}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) + 0\right)} \]
7. Step-by-step derivation

   1. +-rgt-identity 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

8. Simplified 100.0%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

9. Taylor expanded in x around 0 99.5%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(-0.25 \cdot x - 0.3333333333333333\right) - 0.5\right) - 1\right)}\right) \]

10. Taylor expanded in x around 0 99.4%

    \[\leadsto 0.5 \cdot \left(\color{blue}{x \cdot \left(1 + x \cdot \left(0.3333333333333333 \cdot x - 0.5\right)\right)} - x \cdot \left(x \cdot \left(x \cdot \left(-0.25 \cdot x - 0.3333333333333333\right) - 0.5\right) - 1\right)\right) \]

11. Taylor expanded in x around 0 99.3%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(1 + x \cdot \left(0.3333333333333333 \cdot x - 0.5\right)\right) - \color{blue}{x \cdot \left(-0.5 \cdot x - 1\right)}\right) \]

12. Final simplification 99.3%

    \[\leadsto 0.5 \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\right) + x \cdot \left(1 - x \cdot -0.5\right)\right) \]
13. Add Preprocessing

Alternative 8: 99.0% accurate, 111.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 7.6%

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

   1. metadata-eval 7.6%

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

3. Simplified 7.6%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))