Rust f64::acosh

Average Error: 31.5 → 0.2

Time: 4.8s

Precision: binary64

Cost: 13824

\[x \geq 1\]

Math

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

↓

\[\log \left(x \cdot 2 - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (log (- (* x 2.0) (+ (* 0.5 (/ 1.0 x)) (* 0.125 (/ 1.0 (pow x 3.0)))))))

C

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

↓

double code(double x) {
	return log(((x * 2.0) - ((0.5 * (1.0 / x)) + (0.125 * (1.0 / pow(x, 3.0))))));
}

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 * 2.0d0) - ((0.5d0 * (1.0d0 / x)) + (0.125d0 * (1.0d0 / (x ** 3.0d0))))))
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 * 2.0) - ((0.5 * (1.0 / x)) + (0.125 * (1.0 / Math.pow(x, 3.0))))));
}

Python

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

↓

def code(x):
	return math.log(((x * 2.0) - ((0.5 * (1.0 / x)) + (0.125 * (1.0 / math.pow(x, 3.0))))))

Julia

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

↓

function code(x)
	return log(Float64(Float64(x * 2.0) - Float64(Float64(0.5 * Float64(1.0 / x)) + Float64(0.125 * Float64(1.0 / (x ^ 3.0))))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log(((x * 2.0) - ((0.5 * (1.0 / x)) + (0.125 * (1.0 / (x ^ 3.0))))));
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[N[(N[(x * 2.0), $MachinePrecision] - N[(N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	31.5
Target	0.0
Herbie	0.2

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

Derivation

1. Initial program 31.5

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

2. Taylor expanded in x around inf 0.2

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

3. Simplified 0.2

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

   [Start] 0.2	\[ \log \left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   rational.json-simplify-2 [=>] 0.2	\[ \log \left(\color{blue}{x \cdot 2} - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	6848

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

Alternative 2
Error	0.5
Cost	6592

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

Reproduce

herbie shell --seed 2023077 
(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)))))