Hyperbolic arc-(co)tangent

Percentage Accurate: 8.3% → 100.0%

Time: 9.1s

Alternatives: 4

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: 8.3% 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 9.0%

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

   1. metadata-eval 9.0%

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

3. Simplified 9.0%

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

   1. log-pow 8.9%

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

   2. pow1/2 8.9%

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

   3. *-un-lft-identity 8.9%

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

   4. log-prod 8.9%

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

   5. metadata-eval 8.9%

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

   6. pow1/2 8.9%

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

   7. log-pow 9.0%

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

   8. log-div 9.0%

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

   9. log1p-define 21.4%

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

   10. sub-neg 21.4%

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

   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: 99.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot 0.2\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* (* x x) (+ 0.3333333333333333 (* (* x x) 0.2))))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * 0.2))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * (0.3333333333333333d0 + ((x * x) * 0.2d0))))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * 0.2))));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * 0.2))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.3333333333333333 + Float64(Float64(x * x) * 0.2)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * 0.2))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

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

   1. metadata-eval 9.0%

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

3. Simplified 9.0%

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

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + 0.2 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.8%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{x}^{2} \cdot 0.2}\right)\right) \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot 0.2\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 99.8%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) \]

9. Applied egg-rr 99.8%

   \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) \]
10. Step-by-step derivation

    1. unpow2 99.8%

       \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) \]

11. Applied egg-rr 99.8%

    \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot 0.2\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.6% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.0%

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

   1. metadata-eval 9.0%

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

3. Simplified 9.0%

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

5. Taylor expanded in x around 0 99.7%

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

   1. unpow2 99.8%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) \]

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

   \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.3333333333333333\right) \]
9. Add Preprocessing

Alternative 4: 99.0% 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 9.0%

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

   1. metadata-eval 9.0%

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

3. Simplified 9.0%

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

5. Taylor expanded in x around 0 99.1%

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

Reproduce

herbie shell --seed 2024151 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))