Rust f64::acosh

Average Error: 31.9 → 0.8

Time: 3.8s

Precision: binary64

Cost: 12992

\[x \geq 1\]

Math

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

↓

\[\log 2 + \log x \]

FPCore

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

↓

(FPCore (x) :precision binary64 (+ (log 2.0) (log x)))

C

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

↓

double code(double x) {
	return log(2.0) + log(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(2.0d0) + log(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(2.0) + Math.log(x);
}

Python

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

↓

def code(x):
	return math.log(2.0) + math.log(x)

Julia

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

↓

function code(x)
	return Float64(log(2.0) + log(x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log(2.0) + log(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[(N[Log[2.0], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]

Target

Original	31.9
Target	0.1
Herbie	0.8

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

Derivation

1. Initial program 31.9

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

2. Taylor expanded in x around inf 0.8

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

3. Simplified 0.8

   \[\leadsto \color{blue}{\log 2 + \log x} \]
   Proof
   (+.f64 (log.f64 2) (log.f64 x)): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 2) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 x))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 2) (neg.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 x))))): 1 points increase in error, 0 points decrease in error
   (+.f64 (log.f64 2) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.8

   \[\leadsto \log 2 + \log x \]

Alternatives

Alternative 1
Error	0.3
Cost	6848

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

Alternative 2
Error	0.7
Cost	6592

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

Alternative 3
Error	62.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022325 
(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)))))