Hyperbolic arc-(co)tangent

Percentage Accurate: 8.3% → 100.0%

Time: 8.5s

Alternatives: 3

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: 8.3% 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%

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

   2. log-div 9.2%

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

   3. log1p-def 21.7%

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

   4. sub-neg 21.7%

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

   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: 99.5% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 0.5 * (((x * x) * (x * 0.6666666666666666)) + (x * 2.0));
}

Python

def code(x):
	return 0.5 * (((x * x) * (x * 0.6666666666666666)) + (x * 2.0))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $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. Taylor expanded in x around 0 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 99.5%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.6666666666666666 \cdot {x}^{3}\right)\right)} + 2 \cdot x\right) \]

   2. expm1-udef 99.2%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.6666666666666666 \cdot {x}^{3}\right)} - 1\right)} + 2 \cdot x\right) \]

   3. log1p-udef 99.2%

      \[\leadsto 0.5 \cdot \left(\left(e^{\color{blue}{\log \left(1 + 0.6666666666666666 \cdot {x}^{3}\right)}} - 1\right) + 2 \cdot x\right) \]

   4. add-exp-log 99.2%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\left(1 + 0.6666666666666666 \cdot {x}^{3}\right)} - 1\right) + 2 \cdot x\right) \]

6. Applied egg-rr 99.2%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\left(1 + 0.6666666666666666 \cdot {x}^{3}\right) - 1\right)} + 2 \cdot x\right) \]
7. Step-by-step derivation

   1. add-exp-log 99.2%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{e^{\log \left(1 + 0.6666666666666666 \cdot {x}^{3}\right)}} - 1\right) + 2 \cdot x\right) \]

   2. log1p-udef 99.2%

      \[\leadsto 0.5 \cdot \left(\left(e^{\color{blue}{\mathsf{log1p}\left(0.6666666666666666 \cdot {x}^{3}\right)}} - 1\right) + 2 \cdot x\right) \]

   3. expm1-udef 99.5%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.6666666666666666 \cdot {x}^{3}\right)\right)} + 2 \cdot x\right) \]

   4. expm1-log1p-u 99.5%

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

   5. *-commutative 99.5%

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

   6. unpow3 99.5%

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

   7. associate-*l* 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

   \[\leadsto 0.5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.6666666666666666\right) + x \cdot 2\right) \]

Alternative 3: 99.0% 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 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. Taylor expanded in x around 0 98.7%

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

5. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2023278 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))