Hyperbolic arc-(co)tangent

Percentage Accurate: 8.4% → 99.7%

Time: 5.8s

Alternatives: 5

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: 8.4% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(-2 \cdot x + \left(-0.6666666666666666 \cdot {x}^{3} + -0.4 \cdot {x}^{5}\right)\right) \cdot -0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ (* -2.0 x) (+ (* -0.6666666666666666 (pow x 3.0)) (* -0.4 (pow x 5.0))))
  -0.5))

C

double code(double x) {
	return ((-2.0 * x) + ((-0.6666666666666666 * pow(x, 3.0)) + (-0.4 * pow(x, 5.0)))) * -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-2.0d0) * x) + (((-0.6666666666666666d0) * (x ** 3.0d0)) + ((-0.4d0) * (x ** 5.0d0)))) * (-0.5d0)
end function

Java

public static double code(double x) {
	return ((-2.0 * x) + ((-0.6666666666666666 * Math.pow(x, 3.0)) + (-0.4 * Math.pow(x, 5.0)))) * -0.5;
}

Python

def code(x):
	return ((-2.0 * x) + ((-0.6666666666666666 * math.pow(x, 3.0)) + (-0.4 * math.pow(x, 5.0)))) * -0.5

Julia

function code(x)
	return Float64(Float64(Float64(-2.0 * x) + Float64(Float64(-0.6666666666666666 * (x ^ 3.0)) + Float64(-0.4 * (x ^ 5.0)))) * -0.5)
end

MATLAB

function tmp = code(x)
	tmp = ((-2.0 * x) + ((-0.6666666666666666 * (x ^ 3.0)) + (-0.4 * (x ^ 5.0)))) * -0.5;
end

Wolfram

code[x_] := N[(N[(N[(-2.0 * x), $MachinePrecision] + N[(N[(-0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.4 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

   1. *-commutative 6.7%

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

   2. log-div 6.7%

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

   3. sub-neg 6.7%

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

   4. remove-double-neg 6.7%

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

   5. sub-neg 6.7%

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

   6. +-commutative 6.7%

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

   7. neg-sub0 6.7%

      \[\leadsto \left(\color{blue}{\left(0 - \log \left(1 - x\right)\right)} + \log \left(1 - \left(-x\right)\right)\right) \cdot \frac{1}{2} \]

   8. associate--r- 6.7%

      \[\leadsto \color{blue}{\left(0 - \left(\log \left(1 - x\right) - \log \left(1 - \left(-x\right)\right)\right)\right)} \cdot \frac{1}{2} \]

   9. sub-neg 6.7%

      \[\leadsto \left(0 - \left(\log \color{blue}{\left(1 + \left(-x\right)\right)} - \log \left(1 - \left(-x\right)\right)\right)\right) \cdot \frac{1}{2} \]

   10. log-div 6.7%

       \[\leadsto \left(0 - \color{blue}{\log \left(\frac{1 + \left(-x\right)}{1 - \left(-x\right)}\right)}\right) \cdot \frac{1}{2} \]

   11. neg-sub0 6.7%

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

   12. distribute-lft-neg-in 6.7%

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

   13. distribute-rgt-neg-in 6.7%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\left(-2 \cdot x + \left(-0.6666666666666666 \cdot {x}^{3} + -0.4 \cdot {x}^{5}\right)\right)} \cdot -0.5 \]

5. Final simplification 100.0%

   \[\leadsto \left(-2 \cdot x + \left(-0.6666666666666666 \cdot {x}^{3} + -0.4 \cdot {x}^{5}\right)\right) \cdot -0.5 \]

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(Float64(-x)) - log1p(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 6.7%

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

   1. *-commutative 6.7%

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

   2. log-div 6.7%

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

   3. sub-neg 6.7%

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

   4. remove-double-neg 6.7%

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

   5. sub-neg 6.7%

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

   6. +-commutative 6.7%

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

   7. neg-sub0 6.7%

      \[\leadsto \left(\color{blue}{\left(0 - \log \left(1 - x\right)\right)} + \log \left(1 - \left(-x\right)\right)\right) \cdot \frac{1}{2} \]

   8. associate--r- 6.7%

      \[\leadsto \color{blue}{\left(0 - \left(\log \left(1 - x\right) - \log \left(1 - \left(-x\right)\right)\right)\right)} \cdot \frac{1}{2} \]

   9. sub-neg 6.7%

      \[\leadsto \left(0 - \left(\log \color{blue}{\left(1 + \left(-x\right)\right)} - \log \left(1 - \left(-x\right)\right)\right)\right) \cdot \frac{1}{2} \]

   10. log-div 6.7%

       \[\leadsto \left(0 - \color{blue}{\log \left(\frac{1 + \left(-x\right)}{1 - \left(-x\right)}\right)}\right) \cdot \frac{1}{2} \]

   11. neg-sub0 6.7%

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

   12. distribute-lft-neg-in 6.7%

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

   13. distribute-rgt-neg-in 6.7%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. *-commutative 6.7%

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

   2. log-div 6.7%

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

   3. sub-neg 6.7%

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

   4. remove-double-neg 6.7%

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

   5. sub-neg 6.7%

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

   6. +-commutative 6.7%

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

   7. neg-sub0 6.7%

      \[\leadsto \left(\color{blue}{\left(0 - \log \left(1 - x\right)\right)} + \log \left(1 - \left(-x\right)\right)\right) \cdot \frac{1}{2} \]

   8. associate--r- 6.7%

      \[\leadsto \color{blue}{\left(0 - \left(\log \left(1 - x\right) - \log \left(1 - \left(-x\right)\right)\right)\right)} \cdot \frac{1}{2} \]

   9. sub-neg 6.7%

      \[\leadsto \left(0 - \left(\log \color{blue}{\left(1 + \left(-x\right)\right)} - \log \left(1 - \left(-x\right)\right)\right)\right) \cdot \frac{1}{2} \]

   10. log-div 6.7%

       \[\leadsto \left(0 - \color{blue}{\log \left(\frac{1 + \left(-x\right)}{1 - \left(-x\right)}\right)}\right) \cdot \frac{1}{2} \]

   11. neg-sub0 6.7%

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

   12. distribute-lft-neg-in 6.7%

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

   13. distribute-rgt-neg-in 6.7%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 99.0% accurate, 22.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. *-commutative 6.7%

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

   2. log-div 6.7%

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

   3. sub-neg 6.7%

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

   4. remove-double-neg 6.7%

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

   5. sub-neg 6.7%

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

   6. +-commutative 6.7%

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

   7. neg-sub0 6.7%

      \[\leadsto \left(\color{blue}{\left(0 - \log \left(1 - x\right)\right)} + \log \left(1 - \left(-x\right)\right)\right) \cdot \frac{1}{2} \]

   8. associate--r- 6.7%

      \[\leadsto \color{blue}{\left(0 - \left(\log \left(1 - x\right) - \log \left(1 - \left(-x\right)\right)\right)\right)} \cdot \frac{1}{2} \]

   9. sub-neg 6.7%

      \[\leadsto \left(0 - \left(\log \color{blue}{\left(1 + \left(-x\right)\right)} - \log \left(1 - \left(-x\right)\right)\right)\right) \cdot \frac{1}{2} \]

   10. log-div 6.7%

       \[\leadsto \left(0 - \color{blue}{\log \left(\frac{1 + \left(-x\right)}{1 - \left(-x\right)}\right)}\right) \cdot \frac{1}{2} \]

   11. neg-sub0 6.7%

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

   12. distribute-lft-neg-in 6.7%

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

   13. distribute-rgt-neg-in 6.7%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 5.3% 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 6.7%

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

   1. metadata-eval 6.7%

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

3. Simplified 6.7%

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

   1. log-div 6.7%

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

   2. log1p-udef 19.6%

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

   3. sub-neg 19.6%

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

   4. log1p-udef 100.0%

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

   5. *-un-lft-identity 100.0%

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

   6. cancel-sign-sub-inv 100.0%

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

   7. metadata-eval 100.0%

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

   8. add-sqr-sqrt 52.5%

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

   9. sqrt-unprod 35.0%

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

   10. sqr-neg 35.0%

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

   11. sqrt-unprod 3.1%

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

   12. add-sqr-sqrt 5.6%

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

   13. neg-mul-1 5.6%

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

5. Applied egg-rr 5.6%

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

   1. sub-neg 5.6%

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

   2. +-inverses 5.6%

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

7. Simplified 5.6%

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

8. Final simplification 5.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023321 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))