Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Precision: binary64

Cost: 8000

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

code[x_] := N[(0.5 * N[Log[1 + N[(N[(N[(2.0 * x), $MachinePrecision] / N[(N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x + 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. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   flip-- [=>] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 + x}}}\right) \]
   associate-/r/ [=>] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{2 \cdot x}{1 \cdot 1 - x \cdot x} \cdot \left(1 + x\right)}\right) \]
   metadata-eval [=>] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\color{blue}{1} - x \cdot x} \cdot \left(1 + x\right)\right) \]
   +-commutative [=>] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x \cdot x} \cdot \color{blue}{\left(x + 1\right)}\right) \]

3. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x \cdot x} \cdot \left(x + 1\right)\right) \]
   sub-neg [=>] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\color{blue}{1 + \left(-x \cdot x\right)}} \cdot \left(x + 1\right)\right) \]
   flip-+ [=>] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\color{blue}{\frac{1 \cdot 1 - \left(-x \cdot x\right) \cdot \left(-x \cdot x\right)}{1 - \left(-x \cdot x\right)}}} \cdot \left(x + 1\right)\right) \]
   metadata-eval [=>] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\frac{\color{blue}{1} - \left(-x \cdot x\right) \cdot \left(-x \cdot x\right)}{1 - \left(-x \cdot x\right)}} \cdot \left(x + 1\right)\right) \]
   distribute-rgt-neg-in [=>] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\frac{1 - \color{blue}{\left(x \cdot \left(-x\right)\right)} \cdot \left(-x \cdot x\right)}{1 - \left(-x \cdot x\right)}} \cdot \left(x + 1\right)\right) \]
   distribute-rgt-neg-in [=>] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\frac{1 - \left(x \cdot \left(-x\right)\right) \cdot \color{blue}{\left(x \cdot \left(-x\right)\right)}}{1 - \left(-x \cdot x\right)}} \cdot \left(x + 1\right)\right) \]
   distribute-rgt-neg-in [=>] 100.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\frac{1 - \left(x \cdot \left(-x\right)\right) \cdot \left(x \cdot \left(-x\right)\right)}{1 - \color{blue}{x \cdot \left(-x\right)}}} \cdot \left(x + 1\right)\right) \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6976

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

Alternative 2
Accuracy	98.9%
Cost	320

\[0.5 \cdot \left(2 \cdot x\right) \]

Reproduce

herbie shell --seed 2023160 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))