Hyperbolic arc-(co)tangent

Percentage Accurate: 8.8% → 100.0%

Time: 8.2s

Alternatives: 4

Speedup: 22.2×

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: 8.8% 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

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

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 8.9%

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

   1. metadata-eval 8.9%

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

   2. log-div 9.0%

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

   3. log1p-def 21.3%

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

   4. sub-neg 21.3%

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

   5. log1p-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.5 (+ (* x 2.0) (* 0.6666666666666666 (pow x 3.0)))))

C

double code(double x) {
	return 0.5 * ((x * 2.0) + (0.6666666666666666 * pow(x, 3.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((x * 2.0d0) + (0.6666666666666666d0 * (x ** 3.0d0)))
end function

Java

public static double code(double x) {
	return 0.5 * ((x * 2.0) + (0.6666666666666666 * Math.pow(x, 3.0)));
}

Python

def code(x):
	return 0.5 * ((x * 2.0) + (0.6666666666666666 * math.pow(x, 3.0)))

Julia

function code(x)
	return Float64(0.5 * Float64(Float64(x * 2.0) + Float64(0.6666666666666666 * (x ^ 3.0))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((x * 2.0) + (0.6666666666666666 * (x ^ 3.0)));
end

Wolfram

code[x_] := N[(0.5 * N[(N[(x * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.9%

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

   1. metadata-eval 8.9%

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

3. Simplified 8.9%

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

4. Taylor expanded in x around 0 99.4%

   \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} \]

5. Final simplification 99.4%

   \[\leadsto 0.5 \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \]

Alternative 3: 98.8% accurate, 22.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 0.5 * (x * 2.0);
}

Python

def code(x):
	return 0.5 * (x * 2.0)

Julia

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

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x * 2.0);
end

Wolfram

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

Derivation

1. Initial program 8.9%

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

   1. metadata-eval 8.9%

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

3. Simplified 8.9%

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

4. Taylor expanded in x around 0 99.1%

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

5. Final simplification 99.1%

   \[\leadsto 0.5 \cdot \left(x \cdot 2\right) \]

Alternative 4: 5.4% accurate, 111.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 8.9%

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

   1. metadata-eval 8.9%

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

3. Simplified 8.9%

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

   1. div-inv 9.0%

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

   2. add-cube-cbrt 8.8%

      \[\leadsto 0.5 \cdot \log \left(\color{blue}{\left(\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right) \cdot \sqrt[3]{1 + x}\right)} \cdot \frac{1}{1 - x}\right) \]

   3. associate-*l* 8.8%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left(\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right) \cdot \left(\sqrt[3]{1 + x} \cdot \frac{1}{1 - x}\right)\right)} \]

   4. pow2 8.8%

      \[\leadsto 0.5 \cdot \log \left(\color{blue}{{\left(\sqrt[3]{1 + x}\right)}^{2}} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{1}{1 - x}\right)\right) \]

   5. frac-2neg 8.8%

      \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \color{blue}{\frac{-1}{-\left(1 - x\right)}}\right)\right) \]

   6. metadata-eval 8.8%

      \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{\color{blue}{-1}}{-\left(1 - x\right)}\right)\right) \]

   7. sub-neg 8.8%

      \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{-\color{blue}{\left(1 + \left(-x\right)\right)}}\right)\right) \]

   8. add-sqr-sqrt 4.9%

      \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{-\left(1 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}\right)\right) \]

   9. sqrt-unprod 7.8%

      \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{-\left(1 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}\right)\right) \]

   10. sqr-neg 7.8%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{-\left(1 + \sqrt{\color{blue}{x \cdot x}}\right)}\right)\right) \]

   11. sqrt-unprod 2.9%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{-\left(1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}\right)\right) \]

   12. add-sqr-sqrt 5.6%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{-\left(1 + \color{blue}{x}\right)}\right)\right) \]

   13. *-un-lft-identity 5.6%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{-\color{blue}{1 \cdot \left(1 + x\right)}}\right)\right) \]

   14. *-un-lft-identity 5.6%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{-\color{blue}{\left(1 + x\right)}}\right)\right) \]

   15. +-commutative 5.6%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{-\color{blue}{\left(x + 1\right)}}\right)\right) \]

   16. distribute-neg-in 5.6%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{\color{blue}{\left(-x\right) + \left(-1\right)}}\right)\right) \]

   17. add-sqr-sqrt 2.6%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + \left(-1\right)}\right)\right) \]

   18. sqrt-unprod 6.5%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + \left(-1\right)}\right)\right) \]

   19. sqr-neg 6.5%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{\sqrt{\color{blue}{x \cdot x}} + \left(-1\right)}\right)\right) \]

   20. sqrt-unprod 3.9%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(-1\right)}\right)\right) \]

   21. add-sqr-sqrt 8.8%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{\color{blue}{x} + \left(-1\right)}\right)\right) \]

   22. metadata-eval 8.8%

       \[\leadsto 0.5 \cdot \log \left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{x + \color{blue}{-1}}\right)\right) \]

5. Applied egg-rr 8.8%

   \[\leadsto 0.5 \cdot \log \color{blue}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \left(\sqrt[3]{1 + x} \cdot \frac{-1}{x + -1}\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* 8.8%

      \[\leadsto 0.5 \cdot \log \color{blue}{\left(\left({\left(\sqrt[3]{1 + x}\right)}^{2} \cdot \sqrt[3]{1 + x}\right) \cdot \frac{-1}{x + -1}\right)} \]

   2. unpow2 8.8%

      \[\leadsto 0.5 \cdot \log \left(\left(\color{blue}{\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)} \cdot \sqrt[3]{1 + x}\right) \cdot \frac{-1}{x + -1}\right) \]

   3. add-cube-cbrt 9.0%

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

   4. add-sqr-sqrt 8.9%

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

   5. associate-*r* 8.9%

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

   6. log-prod 8.9%

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

7. Applied egg-rr 8.8%

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

   1. log-prod 8.8%

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

   2. log1p-def 21.1%

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

   3. associate-+l+ 21.1%

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

   4. log-rec 21.1%

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

   5. neg-mul-1 21.1%

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

   6. log-rec 21.1%

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

   7. neg-mul-1 21.1%

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

   8. distribute-rgt-out 21.1%

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

   9. metadata-eval 21.1%

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

9. Simplified 21.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) + \log \left(\sqrt{1 - x}\right) \cdot -2\right)} \]
10. Step-by-step derivation

    1. add-cbrt-cube 21.1%

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \log \color{blue}{\left(\sqrt[3]{\left(\sqrt{1 - x} \cdot \sqrt{1 - x}\right) \cdot \sqrt{1 - x}}\right)} \cdot -2\right) \]

    2. pow1/3 21.1%

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \log \color{blue}{\left({\left(\left(\sqrt{1 - x} \cdot \sqrt{1 - x}\right) \cdot \sqrt{1 - x}\right)}^{0.3333333333333333}\right)} \cdot -2\right) \]

    3. log-pow 21.2%

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \color{blue}{\left(0.3333333333333333 \cdot \log \left(\left(\sqrt{1 - x} \cdot \sqrt{1 - x}\right) \cdot \sqrt{1 - x}\right)\right)} \cdot -2\right) \]

    4. add-sqr-sqrt 21.3%

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \left(0.3333333333333333 \cdot \log \left(\color{blue}{\left(1 - x\right)} \cdot \sqrt{1 - x}\right)\right) \cdot -2\right) \]

    5. pow1 21.3%

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \left(0.3333333333333333 \cdot \log \left(\color{blue}{{\left(1 - x\right)}^{1}} \cdot \sqrt{1 - x}\right)\right) \cdot -2\right) \]

    6. pow1/2 21.3%

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \left(0.3333333333333333 \cdot \log \left({\left(1 - x\right)}^{1} \cdot \color{blue}{{\left(1 - x\right)}^{0.5}}\right)\right) \cdot -2\right) \]

    7. pow-prod-up 21.2%

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(1 - x\right)}^{\left(1 + 0.5\right)}\right)}\right) \cdot -2\right) \]

    8. log-pow 21.3%

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \left(0.3333333333333333 \cdot \color{blue}{\left(\left(1 + 0.5\right) \cdot \log \left(1 - x\right)\right)}\right) \cdot -2\right) \]

    9. metadata-eval 21.3%

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \left(0.3333333333333333 \cdot \left(\color{blue}{1.5} \cdot \log \left(1 - x\right)\right)\right) \cdot -2\right) \]

    10. sub-neg 21.3%

        \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \left(0.3333333333333333 \cdot \left(1.5 \cdot \log \color{blue}{\left(1 + \left(-x\right)\right)}\right)\right) \cdot -2\right) \]

    11. log1p-def 99.6%

        \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \left(0.3333333333333333 \cdot \left(1.5 \cdot \color{blue}{\mathsf{log1p}\left(-x\right)}\right)\right) \cdot -2\right) \]

11. Applied egg-rr 99.6%

    \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + \color{blue}{\left(0.3333333333333333 \cdot \left(1.5 \cdot \mathsf{log1p}\left(-x\right)\right)\right)} \cdot -2\right) \]
12. Step-by-step derivation

    1. expm1-log1p-u 99.2%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) + \left(0.3333333333333333 \cdot \left(1.5 \cdot \mathsf{log1p}\left(-x\right)\right)\right) \cdot -2\right)\right)} \]

    2. expm1-udef 8.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) + \left(0.3333333333333333 \cdot \left(1.5 \cdot \mathsf{log1p}\left(-x\right)\right)\right) \cdot -2\right)} - 1\right)} \]

13. Applied egg-rr 5.2%

    \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right), \mathsf{log1p}\left(x\right)\right)\right)} - 1\right)} \]
14. Step-by-step derivation

    1. expm1-def 5.2%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right), \mathsf{log1p}\left(x\right)\right)\right)\right)} \]

    2. expm1-log1p 5.2%

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

    3. fma-udef 5.2%

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

    4. distribute-lft1-in 5.2%

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

    5. metadata-eval 5.2%

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

    6. mul0-lft 5.2%

       \[\leadsto 0.5 \cdot \color{blue}{0} \]

15. Simplified 5.2%

    \[\leadsto 0.5 \cdot \color{blue}{0} \]

16. Final simplification 5.2%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))