Rust f64::acosh

Average Error: 32.3 → 0.2

Time: 5.4s

Precision: binary64

Cost: 7488

\[x \geq 1\]

Math

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

↓

\[\mathsf{log1p}\left(x \cdot 2 + \left(\frac{-0.5}{x} - \left(1 + \frac{\frac{0.125}{x \cdot x}}{x}\right)\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (log1p (+ (* x 2.0) (- (/ -0.5 x) (+ 1.0 (/ (/ 0.125 (* x x)) x))))))

C

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

↓

double code(double x) {
	return log1p(((x * 2.0) + ((-0.5 / x) - (1.0 + ((0.125 / (x * x)) / x)))));
}

Java

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

↓

public static double code(double x) {
	return Math.log1p(((x * 2.0) + ((-0.5 / x) - (1.0 + ((0.125 / (x * x)) / x)))));
}

Python

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

↓

def code(x):
	return math.log1p(((x * 2.0) + ((-0.5 / x) - (1.0 + ((0.125 / (x * x)) / x)))))

Julia

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

↓

function code(x)
	return log1p(Float64(Float64(x * 2.0) + Float64(Float64(-0.5 / x) - Float64(1.0 + Float64(Float64(0.125 / Float64(x * x)) / x)))))
end

Wolfram

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

↓

code[x_] := N[Log[1 + N[(N[(x * 2.0), $MachinePrecision] + N[(N[(-0.5 / x), $MachinePrecision] - N[(1.0 + N[(N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	32.3
Target	0.1
Herbie	0.2

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

Derivation

1. Initial program 32.3

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

2. Applied egg-rr 32.3

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} - 1\right)\right)} \]

3. Simplified 32.3

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + \left(-1 + x\right)\right)} \]
   Proof

   [Start] 32.3	\[ \mathsf{log1p}\left(x + \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} - 1\right)\right) \]
   +-commutative [=>] 32.3	\[ \mathsf{log1p}\left(\color{blue}{\left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} - 1\right) + x}\right) \]
   sub-neg [=>] 32.3	\[ \mathsf{log1p}\left(\color{blue}{\left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + \left(-1\right)\right)} + x\right) \]
   metadata-eval [=>] 32.3	\[ \mathsf{log1p}\left(\left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + \color{blue}{-1}\right) + x\right) \]
   associate-+l+ [=>] 32.3	\[ \mathsf{log1p}\left(\color{blue}{\sqrt{\mathsf{fma}\left(x, x, -1\right)} + \left(-1 + x\right)}\right) \]

4. Taylor expanded in x around inf 0.2

   \[\leadsto \mathsf{log1p}\left(\color{blue}{2 \cdot x - \left(0.5 \cdot \frac{1}{x} + \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}\right) \]

5. Simplified 0.2

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot 2 - \left(\frac{0.5}{x} + \left(1 + \frac{0.125}{{x}^{3}}\right)\right)}\right) \]
   Proof

   [Start] 0.2	\[ \mathsf{log1p}\left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]
   *-commutative [=>] 0.2	\[ \mathsf{log1p}\left(\color{blue}{x \cdot 2} - \left(0.5 \cdot \frac{1}{x} + \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]
   associate-*r/ [=>] 0.2	\[ \mathsf{log1p}\left(x \cdot 2 - \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]
   metadata-eval [=>] 0.2	\[ \mathsf{log1p}\left(x \cdot 2 - \left(\frac{\color{blue}{0.5}}{x} + \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]
   associate-*r/ [=>] 0.2	\[ \mathsf{log1p}\left(x \cdot 2 - \left(\frac{0.5}{x} + \left(1 + \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right)\right)\right) \]
   metadata-eval [=>] 0.2	\[ \mathsf{log1p}\left(x \cdot 2 - \left(\frac{0.5}{x} + \left(1 + \frac{\color{blue}{0.125}}{{x}^{3}}\right)\right)\right) \]

6. Applied egg-rr 0.2

   \[\leadsto \mathsf{log1p}\left(x \cdot 2 - \left(\frac{0.5}{x} + \left(1 + \color{blue}{\frac{\frac{0.25}{x \cdot x}}{x} \cdot 0.5}\right)\right)\right) \]

7. Applied egg-rr 0.2

   \[\leadsto \mathsf{log1p}\left(x \cdot 2 - \left(\frac{0.5}{x} + \left(1 + \color{blue}{\frac{\frac{0.125}{x \cdot x}}{x}}\right)\right)\right) \]

8. Final simplification 0.2

   \[\leadsto \mathsf{log1p}\left(x \cdot 2 + \left(\frac{-0.5}{x} - \left(1 + \frac{\frac{0.125}{x \cdot x}}{x}\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.2
Cost	7360

\[\log \left(\left(x \cdot 2 + \frac{-0.5}{x}\right) + \frac{\frac{-0.125}{x \cdot x}}{x}\right) \]

Alternative 2
Error	0.3
Cost	6848

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

Alternative 3
Error	0.6
Cost	6592

\[\log \left(x + x\right) \]

Alternative 4
Error	64.0
Cost	6464

\[\mathsf{log1p}\left(-1\right) \]

Reproduce

herbie shell --seed 2023045 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

  (log (+ x (sqrt (- (* x x) 1.0)))))