Rust f64::acosh

Average Accuracy: 49.9% → 99.5%

Time: 4.2s

Precision: binary64

Cost: 6976

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 + x) + ((-0.5 / x) + -1.0)));
}

Python

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

↓

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

Julia

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

↓

function code(x)
	return log1p(Float64(Float64(x + x) + Float64(Float64(-0.5 / x) + -1.0)))
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 + x), $MachinePrecision] + N[(N[(-0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	49.9%
Target	99.9%
Herbie	99.5%

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

Derivation

1. Initial program 49.9%

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

2. Taylor expanded in x around inf 99.5%

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

3. Simplified 99.5%

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

   [Start] 99.5	\[ \log \left(x + \left(x - 0.5 \cdot \frac{1}{x}\right)\right) \]
   associate-*r/ [=>] 99.5	\[ \log \left(x + \left(x - \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \]
   metadata-eval [=>] 99.5	\[ \log \left(x + \left(x - \frac{\color{blue}{0.5}}{x}\right)\right) \]

4. Applied egg-rr 99.5%

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

   [Start] 99.5	\[ \log \left(x + \left(x - \frac{0.5}{x}\right)\right) \]
   log1p-expm1-u [=>] 99.5	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \left(x - \frac{0.5}{x}\right)\right)\right)\right)} \]
   expm1-udef [=>] 99.5	\[ \mathsf{log1p}\left(\color{blue}{e^{\log \left(x + \left(x - \frac{0.5}{x}\right)\right)} - 1}\right) \]
   add-exp-log [<=] 99.5	\[ \mathsf{log1p}\left(\color{blue}{\left(x + \left(x - \frac{0.5}{x}\right)\right)} - 1\right) \]
   sub-neg [=>] 99.5	\[ \mathsf{log1p}\left(\left(x + \color{blue}{\left(x + \left(-\frac{0.5}{x}\right)\right)}\right) - 1\right) \]
   associate-+r+ [=>] 99.5	\[ \mathsf{log1p}\left(\color{blue}{\left(\left(x + x\right) + \left(-\frac{0.5}{x}\right)\right)} - 1\right) \]
   count-2 [=>] 99.5	\[ \mathsf{log1p}\left(\left(\color{blue}{2 \cdot x} + \left(-\frac{0.5}{x}\right)\right) - 1\right) \]
   *-commutative [<=] 99.5	\[ \mathsf{log1p}\left(\left(\color{blue}{x \cdot 2} + \left(-\frac{0.5}{x}\right)\right) - 1\right) \]
   metadata-eval [<=] 99.5	\[ \mathsf{log1p}\left(\left(x \cdot \color{blue}{\frac{1}{0.5}} + \left(-\frac{0.5}{x}\right)\right) - 1\right) \]
   div-inv [<=] 99.5	\[ \mathsf{log1p}\left(\left(\color{blue}{\frac{x}{0.5}} + \left(-\frac{0.5}{x}\right)\right) - 1\right) \]
   associate--l+ [=>] 99.5	\[ \mathsf{log1p}\left(\color{blue}{\frac{x}{0.5} + \left(\left(-\frac{0.5}{x}\right) - 1\right)}\right) \]
   div-inv [=>] 99.5	\[ \mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{0.5}} + \left(\left(-\frac{0.5}{x}\right) - 1\right)\right) \]
   metadata-eval [=>] 99.5	\[ \mathsf{log1p}\left(x \cdot \color{blue}{2} + \left(\left(-\frac{0.5}{x}\right) - 1\right)\right) \]
   *-commutative [=>] 99.5	\[ \mathsf{log1p}\left(\color{blue}{2 \cdot x} + \left(\left(-\frac{0.5}{x}\right) - 1\right)\right) \]
   count-2 [<=] 99.5	\[ \mathsf{log1p}\left(\color{blue}{\left(x + x\right)} + \left(\left(-\frac{0.5}{x}\right) - 1\right)\right) \]
   distribute-neg-frac [=>] 99.5	\[ \mathsf{log1p}\left(\left(x + x\right) + \left(\color{blue}{\frac{-0.5}{x}} - 1\right)\right) \]
   metadata-eval [=>] 99.5	\[ \mathsf{log1p}\left(\left(x + x\right) + \left(\frac{\color{blue}{-0.5}}{x} - 1\right)\right) \]

5. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6848

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

Alternative 2
Accuracy	99.0%
Cost	6592

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

Reproduce

herbie shell --seed 2023130 
(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)))))