Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 4

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. Add Preprocessing
3. Add Preprocessing

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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 8.1%

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

   2. *-un-lft-identity 8.1%

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

   3. log-prod 8.1%

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

   4. metadata-eval 8.1%

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

   5. add-log-exp 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{0 + \frac{2 \cdot x}{1 - x}}\right) \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 99.8%

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

   4. div-sub 99.8%

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

   5. sub-neg 99.8%

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

   6. *-inverses 99.8%

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

   7. metadata-eval 99.8%

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

6. Simplified 99.8%

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

Alternative 3: 99.4% 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. Add Preprocessing

3. Taylor expanded in x around 0 99.6%

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

   1. distribute-lft-in 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r* 99.7%

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

   5. unpow2 99.7%

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

   6. cube-mult 99.7%

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

5. Simplified 99.7%

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

Alternative 4: 98.9% 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. Add Preprocessing

3. Taylor expanded in x around 0 99.4%

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

Reproduce

herbie shell --seed 2024113 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))