Hyperbolic arc-(co)tangent

Average Error: 58.7 → 0.0

Time: 6.0s

Precision: binary64

Cost: 19840

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 58.7

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

2. Simplified 58.7

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]
   Proof

   [Start] 58.7	\[ \frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
   metadata-eval [=>] 58.7	\[ \color{blue}{0.5} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]

3. Applied egg-rr 0.5

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

4. Simplified 0.0

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

   [Start] 0.5	\[ 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) - \log \left(1 - x \cdot x\right)\right) + \mathsf{log1p}\left(x\right)\right) \]
   +-commutative [=>] 0.5	\[ 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) + \left(\mathsf{log1p}\left(x\right) - \log \left(1 - x \cdot x\right)\right)\right)} \]
   associate-+r- [=>] 0.5	\[ 0.5 \cdot \color{blue}{\left(\left(\mathsf{log1p}\left(x\right) + \mathsf{log1p}\left(x\right)\right) - \log \left(1 - x \cdot x\right)\right)} \]
   cancel-sign-sub-inv [=>] 0.5	\[ 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) + \mathsf{log1p}\left(x\right)\right) - \log \color{blue}{\left(1 + \left(-x\right) \cdot x\right)}\right) \]
   *-commutative [<=] 0.5	\[ 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) + \mathsf{log1p}\left(x\right)\right) - \log \left(1 + \color{blue}{x \cdot \left(-x\right)}\right)\right) \]
   log1p-def [=>] 0.0	\[ 0.5 \cdot \left(\left(\mathsf{log1p}\left(x\right) + \mathsf{log1p}\left(x\right)\right) - \color{blue}{\mathsf{log1p}\left(x \cdot \left(-x\right)\right)}\right) \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	13184

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

Alternative 2
Error	0.3
Cost	7040

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

Alternative 3
Error	0.6
Cost	320

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

Alternative 4
Error	60.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023066 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))