Hyperbolic arc-(co)tangent

Percentage Accurate: 8.5% → 100.0%

Time: 9.3s

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: 8.5% 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 10.9%

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

   1. metadata-eval 10.9%

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

3. Simplified 10.9%

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

   1. *-un-lft-identity 10.9%

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

   2. *-commutative 10.9%

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

   3. log-prod 10.9%

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

   4. log-div 11.0%

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

   5. log1p-define 23.2%

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

   6. sub-neg 23.2%

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

   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 2: 99.5% 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 10.9%

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

   1. metadata-eval 10.9%

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

3. Simplified 10.9%

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

5. Taylor expanded in x around 0 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. *-commutative 98.8%

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

   3. pow2 98.8%

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

   4. associate-*l* 98.8%

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

   5. pow3 98.8%

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

7. Applied egg-rr 98.8%

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

Alternative 3: 99.5% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.9%

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

   1. metadata-eval 10.9%

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

3. Simplified 10.9%

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

5. Taylor expanded in x around 0 98.8%

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

   1. unpow2 98.8%

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

7. Applied egg-rr 98.8%

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

Alternative 4: 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 10.9%

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

   1. metadata-eval 10.9%

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

3. Simplified 10.9%

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

5. Taylor expanded in x around 0 97.5%

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

6. Final simplification 97.5%

   \[\leadsto 0.5 \cdot \left(x \cdot 2\right) \]
7. Add Preprocessing

Alternative 5: 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 10.9%

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

   1. metadata-eval 10.9%

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

3. Simplified 10.9%

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

   1. *-un-lft-identity 10.9%

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

   2. *-commutative 10.9%

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

   3. log-prod 10.9%

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

   4. log-div 11.0%

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

   5. log1p-define 23.2%

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

   6. sub-neg 23.2%

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

   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. Step-by-step derivation

   1. sub-neg 100.0%

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

   2. add-sqr-sqrt 56.0%

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

   3. sqrt-unprod 39.1%

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

   4. sqr-neg 39.1%

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

   5. sqrt-prod 2.6%

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

   6. add-sqr-sqrt 5.4%

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

10. Applied egg-rr 5.4%

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

    1. sub-neg 5.4%

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

    2. +-inverses 5.4%

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

12. Simplified 5.4%

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

13. Final simplification 5.4%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))