Rust f64::atanh

Percentage Accurate: 100.0% → 99.4%

Time: 2.5s

Alternatives: 5

Speedup: 1.0×

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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.5 (+ (* 0.6666666666666666 (pow x 3.0)) (* x 2.0))))

C

double code(double x) {
	return 0.5 * ((0.6666666666666666 * pow(x, 3.0)) + (x * 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((0.6666666666666666d0 * (x ** 3.0d0)) + (x * 2.0d0))
end function

Java

public static double code(double x) {
	return 0.5 * ((0.6666666666666666 * Math.pow(x, 3.0)) + (x * 2.0));
}

Python

def code(x):
	return 0.5 * ((0.6666666666666666 * math.pow(x, 3.0)) + (x * 2.0))

Julia

function code(x)
	return Float64(0.5 * Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(x * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((0.6666666666666666 * (x ^ 3.0)) + (x * 2.0));
end

Wolfram

code[x_] := N[(0.5 * N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $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. associate-/l* 99.7%

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

   2. div-sub 99.7%

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

   3. *-inverses 99.7%

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

   4. sub-neg 99.7%

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

   5. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(2.0 / N[(N[(1.0 / x), $MachinePrecision] + -1.0), $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. associate-/l* 99.7%

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

   2. div-sub 99.7%

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

   3. *-inverses 99.7%

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

   4. sub-neg 99.7%

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

   5. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

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(Float64(x * 2.0) / Float64(1.0 - x))))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 0.5 (log1p (* x 2.0))))

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(x * 2.0), $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. associate-/l* 99.7%

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

   2. div-sub 99.7%

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

   3. *-inverses 99.7%

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

   4. sub-neg 99.7%

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

   5. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 98.9%

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

5. Final simplification 98.9%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot 2\right) \]

Alternative 5: 0.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + -2.0], $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. associate-/l* 99.7%

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

   2. div-sub 99.7%

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

   3. *-inverses 99.7%

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

   4. sub-neg 99.7%

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

   5. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around inf 0.0%

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

5. Final simplification 0.0%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(-2\right) \]

Reproduce

herbie shell --seed 2023305 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))