Rust f64::acosh

Average Error: 31.7 → 0.3

Time: 1.7s

Precision: binary64

Cost: 6976

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) - 1.0d0))))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((x + x) - (0.5d0 * (1.0d0 / x))))
end function

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.log(((x + x) - (0.5 * (1.0 / x))));
}

Python

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

↓

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

Julia

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

↓

function code(x)
	return log(Float64(Float64(x + x) - Float64(0.5 * Float64(1.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - 1.0))));
end

↓

function tmp = code(x)
	tmp = log(((x + x) - (0.5 * (1.0 / 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[N[(N[(x + x), $MachinePrecision] - N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	31.7
Target	0.0
Herbie	0.3

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

Derivation

1. Initial program 31.7

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

2. Taylor expanded in x around inf 0.3

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

3. Simplified 0.3

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

   [Start] 0.3	\[ \log \left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right) \]
   metadata-eval [<=] 0.3	\[ \log \left(\color{blue}{\left(1 + 1\right)} \cdot x - 0.5 \cdot \frac{1}{x}\right) \]
   rational.json-simplify-7 [<=] 0.3	\[ \log \left(\left(1 + 1\right) \cdot \color{blue}{\frac{x}{1}} - 0.5 \cdot \frac{1}{x}\right) \]
   rational.json-simplify-30 [=>] 0.3	\[ \log \left(\color{blue}{\left(x + \frac{x}{1}\right)} - 0.5 \cdot \frac{1}{x}\right) \]
   rational.json-simplify-7 [=>] 0.3	\[ \log \left(\left(x + \color{blue}{x}\right) - 0.5 \cdot \frac{1}{x}\right) \]

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.6
Cost	6592

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

Reproduce

herbie shell --seed 2023064 
(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)))))