Rust f64::atanh

Average Accuracy: 100.0% → 99.6%

Time: 2.0s

Precision: binary64

Cost: 7040

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 * ((2.0 * x) + (0.6666666666666666 * Math.pow(x, 3.0)));
}

Python

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

↓

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

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 * Float64(Float64(2.0 * x) + Float64(0.6666666666666666 * (x ^ 3.0))))
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[(N[(2.0 * x), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 100.0%

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

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6976

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

Alternative 2
Accuracy	99.1%
Cost	320

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

Reproduce

herbie shell --seed 2023157 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))