Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 3

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \tanh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (atanh x))

C

double code(double x) {
	return atanh(x);
}

Python

def code(x):
	return math.atanh(x)

Julia

function code(x)
	return atanh(x)
end

MATLAB

function tmp = code(x)
	tmp = atanh(x);
end

Wolfram

code[x_] := N[ArcTanh[x], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

C

double code(double x) {
	return 0.5 * log1p(((2.0 * x) / (1.0 - x)));
}

Java

public static double code(double x) {
	return 0.5 * Math.log1p(((2.0 * x) / (1.0 - x)));
}

Python

def code(x):
	return 0.5 * math.log1p(((2.0 * x) / (1.0 - x)))

Julia

function code(x)
	return Float64(0.5 * log1p(Float64(Float64(2.0 * x) / Float64(1.0 - x))))
end

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(N[(2.0 * x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{1 - x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (log1p (* x (/ 2.0 (- 1.0 x))))))

C

double code(double x) {
	return 0.5 * log1p((x * (2.0 / (1.0 - x))));
}

Java

public static double code(double x) {
	return 0.5 * Math.log1p((x * (2.0 / (1.0 - x))));
}

Python

def code(x):
	return 0.5 * math.log1p((x * (2.0 / (1.0 - x))))

Julia

function code(x)
	return Float64(0.5 * log1p(Float64(x * Float64(2.0 / Float64(1.0 - x)))))
end

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(x * N[(2.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x \cdot 2}}{1 - x}\right) \]

   2. associate-/l* 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{x \cdot \frac{2}{1 - x}}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{1 - x}\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x \cdot 2}}{1 - x}\right) \]

   2. associate-/l* 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{x \cdot \frac{2}{1 - x}}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.5%

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

   1. distribute-lft-in 99.5%

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

   2. *-rgt-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

   5. unpow2 99.5%

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

   6. cube-mult 99.5%

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

7. Simplified 99.5%

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

Alternative 3: 99.0% accurate, 109.0× 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 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x \cdot 2}}{1 - x}\right) \]

   2. associate-/l* 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{x \cdot \frac{2}{1 - x}}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{log1p}\left(x \cdot \frac{2}{1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.3%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))