Rust f64::atanh

Average Error: 0.0 → 0.0

Time: 2.6s

Precision: binary64

Cost: 6976

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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(((2.0 * x) / (1.0 - x)));
}

↓

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

↓

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

↓

function code(x)
	return Float64(0.5 * log1p(Float64(x * Float64(2.0 / 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]

↓

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   associate-*l/ [<=] 0.0	\[ 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{2}{1 - x} \cdot x}\right) \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	1.4
Cost	6720

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

Alternative 2
Error	64.0
Cost	6592

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

Reproduce

herbie shell --seed 2023039 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))