Hyperbolic arc-(co)tangent

Percentage Accurate: 8.5% → 100.0%

Time: 10.1s

Alternatives: 5

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

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

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 9.3%

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

   1. metadata-eval 9.3%

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

3. Simplified 9.3%

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

   1. log-pow 9.3%

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

   2. pow1/2 9.3%

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

   3. *-un-lft-identity 9.3%

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

   4. log-prod 9.3%

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

   5. metadata-eval 9.3%

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

   6. pow1/2 9.3%

      \[\leadsto 0 + \log \color{blue}{\left({\left(\frac{1 + x}{1 - x}\right)}^{0.5}\right)} \]

   7. log-pow 9.3%

      \[\leadsto 0 + \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]

   8. log-div 9.4%

      \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\log \left(1 + x\right) - \log \left(1 - x\right)\right)} \]

   9. log1p-define 22.1%

      \[\leadsto 0 + 0.5 \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 - x\right)\right) \]

   10. sub-neg 22.1%

       \[\leadsto 0 + 0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 + \left(-x\right)\right)}\right) \]

   11. log1p-define 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + x \cdot \left(1 + x \cdot \left(0.5 - x \cdot \left(x \cdot -0.25 - 0.3333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.5
  (+
   (log1p x)
   (* x (+ 1.0 (* x (- 0.5 (* x (- (* x -0.25) 0.3333333333333333)))))))))

C

double code(double x) {
	return 0.5 * (log1p(x) + (x * (1.0 + (x * (0.5 - (x * ((x * -0.25) - 0.3333333333333333)))))));
}

Java

public static double code(double x) {
	return 0.5 * (Math.log1p(x) + (x * (1.0 + (x * (0.5 - (x * ((x * -0.25) - 0.3333333333333333)))))));
}

Python

def code(x):
	return 0.5 * (math.log1p(x) + (x * (1.0 + (x * (0.5 - (x * ((x * -0.25) - 0.3333333333333333)))))))

Julia

function code(x)
	return Float64(0.5 * Float64(log1p(x) + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 - Float64(x * Float64(Float64(x * -0.25) - 0.3333333333333333))))))))
end

Wolfram

code[x_] := N[(0.5 * N[(N[Log[1 + x], $MachinePrecision] + N[(x * N[(1.0 + N[(x * N[(0.5 - N[(x * N[(N[(x * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

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

   1. metadata-eval 9.3%

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

3. Simplified 9.3%

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

   1. log-pow 9.3%

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

   2. pow1/2 9.3%

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

   3. *-un-lft-identity 9.3%

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

   4. log-prod 9.3%

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

   5. metadata-eval 9.3%

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

   6. pow1/2 9.3%

      \[\leadsto 0 + \log \color{blue}{\left({\left(\frac{1 + x}{1 - x}\right)}^{0.5}\right)} \]

   7. log-pow 9.3%

      \[\leadsto 0 + \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]

   8. log-div 9.4%

      \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\log \left(1 + x\right) - \log \left(1 - x\right)\right)} \]

   9. log1p-define 22.1%

      \[\leadsto 0 + 0.5 \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 - x\right)\right) \]

   10. sub-neg 22.1%

       \[\leadsto 0 + 0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \log \color{blue}{\left(1 + \left(-x\right)\right)}\right) \]

   11. log1p-define 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 98.8%

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

10. Final simplification 98.8%

    \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(x\right) + x \cdot \left(1 + x \cdot \left(0.5 - x \cdot \left(x \cdot -0.25 - 0.3333333333333333\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.3%

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

   1. metadata-eval 9.3%

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

3. Simplified 9.3%

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

5. Taylor expanded in x around 0 98.7%

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

   1. distribute-rgt-in 98.7%

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

   2. *-lft-identity 98.7%

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

   3. associate-*l* 98.7%

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

   4. unpow2 98.7%

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

   5. unpow3 98.7%

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

7. Simplified 98.7%

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

Alternative 4: 99.6% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.3%

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

   1. metadata-eval 9.3%

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

3. Simplified 9.3%

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

5. Taylor expanded in x around 0 98.7%

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

   1. unpow2 98.7%

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

7. Applied egg-rr 98.7%

   \[\leadsto x \cdot \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
8. Add Preprocessing

Alternative 5: 99.1% 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 9.3%

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

   1. metadata-eval 9.3%

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

3. Simplified 9.3%

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

5. Taylor expanded in x around 0 98.1%

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

Reproduce

herbie shell --seed 2024148 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))