Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 1.9s

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: 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]

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(2.0 * N[(x + N[(x * 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. Taylor expanded in x around 0 98.3%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{2 \cdot {x}^{2} + 2 \cdot x}\right) \]
3. Step-by-step derivation

   1. distribute-lft-out 98.3%

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

   2. unpow2 98.3%

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

4. Simplified 98.3%

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

5. Final simplification 98.3%

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

Alternative 3: 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-inv 99.7%

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

   3. div-sub 99.7%

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

   4. pow1 99.7%

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

   5. pow1 99.7%

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

   6. pow-div 99.7%

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

   7. metadata-eval 99.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(2 \cdot \frac{1}{\frac{1}{x} - {x}^{\color{blue}{0}}}\right) \]

   8. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. metadata-eval 99.7%

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

   3. sub-neg 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 97.2%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{2 \cdot x}\right) \]
3. Step-by-step derivation

   1. count-2 97.2%

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

4. Simplified 97.2%

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

5. Final simplification 97.2%

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

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. Taylor expanded in x around inf 0.0%

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

3. Final simplification 0.0%

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

Reproduce

herbie shell --seed 2023189 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))