Rust f32::atanh

Average Accuracy: 99.8% → 99.2%

Time: 6.3s

Precision: binary32

Cost: 10240

Math

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

↓

\[0.5 \cdot \left(0.2857142857142857 \cdot {x}^{7} + \left(x \cdot 2 + \left(0.6666666666666666 \cdot {x}^{3} + 0.4 \cdot {x}^{5}\right)\right)\right) \]

FPCore

(FPCore (x) :precision binary32 (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

↓

(FPCore (x)
 :precision binary32
 (*
  0.5
  (+
   (* 0.2857142857142857 (pow x 7.0))
   (+ (* x 2.0) (+ (* 0.6666666666666666 (pow x 3.0)) (* 0.4 (pow x 5.0)))))))

C

float code(float x) {
	return 0.5f * log1pf(((2.0f * x) / (1.0f - x)));
}

↓

float code(float x) {
	return 0.5f * ((0.2857142857142857f * powf(x, 7.0f)) + ((x * 2.0f) + ((0.6666666666666666f * powf(x, 3.0f)) + (0.4f * powf(x, 5.0f)))));
}

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(2.0) * x) / Float32(Float32(1.0) - x))))
end

↓

function code(x)
	return Float32(Float32(0.5) * Float32(Float32(Float32(0.2857142857142857) * (x ^ Float32(7.0))) + Float32(Float32(x * Float32(2.0)) + Float32(Float32(Float32(0.6666666666666666) * (x ^ Float32(3.0))) + Float32(Float32(0.4) * (x ^ Float32(5.0)))))))
end

Derivation

1. Initial program 99.8%

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

2. Simplified 99.7%

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

   [Start] 99.8	\[ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   associate-*l/ [<=] 99.7	\[ 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{2}{1 - x} \cdot x}\right) \]

3. Taylor expanded in x around 0 99.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(0.2857142857142857 \cdot {x}^{7} + \left(2 \cdot x + \left(0.6666666666666666 \cdot {x}^{3} + 0.4 \cdot {x}^{5}\right)\right)\right)} \]

4. Final simplification 99.2%

   \[\leadsto 0.5 \cdot \left(0.2857142857142857 \cdot {x}^{7} + \left(x \cdot 2 + \left(0.6666666666666666 \cdot {x}^{3} + 0.4 \cdot {x}^{5}\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	3488

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

Alternative 2
Accuracy	99.8%
Cost	3488

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

Alternative 3
Accuracy	96.8%
Cost	160

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

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))