Rust f32::acosh

Average Error: 15.8 → 0.6

Time: 8.9s

Precision: binary32

Cost: 6688

\[x \geq 1\]

Math

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

↓

\[\log 2 + \left(\log x + \frac{-0.25}{x \cdot x}\right) \]

FPCore

(FPCore (x) :precision binary32 (log (+ x (sqrt (- (* x x) 1.0)))))

↓

(FPCore (x) :precision binary32 (+ (log 2.0) (+ (log x) (/ -0.25 (* x x)))))

C

float code(float x) {
	return logf((x + sqrtf(((x * x) - 1.0f))));
}

↓

float code(float x) {
	return logf(2.0f) + (logf(x) + (-0.25f / (x * x)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + sqrt(((x * x) - 1.0e0))))
end function

↓

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(2.0e0) + (log(x) + ((-0.25e0) / (x * x)))
end function

Julia

function code(x)
	return log(Float32(x + sqrt(Float32(Float32(x * x) - Float32(1.0)))))
end

↓

function code(x)
	return Float32(log(Float32(2.0)) + Float32(log(x) + Float32(Float32(-0.25) / Float32(x * x))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log(single(2.0)) + (log(x) + (single(-0.25) / (x * x)));
end

Target

Original	15.8
Target	0.2
Herbie	0.6

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

Derivation

1. Initial program 15.8

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

2. Taylor expanded in x around inf 0.6

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

3. Simplified 0.6

   \[\leadsto \color{blue}{\log 2 + \left(\log x + \frac{-0.25}{x \cdot x}\right)} \]
   Proof

   [Start] 0.6	\[ \left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) - 0.25 \cdot \frac{1}{{x}^{2}} \]
   cancel-sign-sub-inv [=>] 0.6	\[ \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) + \left(-0.25\right) \cdot \frac{1}{{x}^{2}}} \]
   +-commutative [=>] 0.6	\[ \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right)} + \left(-0.25\right) \cdot \frac{1}{{x}^{2}} \]
   associate-+l+ [=>] 0.6	\[ \color{blue}{\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-0.25\right) \cdot \frac{1}{{x}^{2}}\right)} \]
   mul-1-neg [=>] 0.6	\[ \log 2 + \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \left(-0.25\right) \cdot \frac{1}{{x}^{2}}\right) \]
   log-rec [=>] 0.6	\[ \log 2 + \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \left(-0.25\right) \cdot \frac{1}{{x}^{2}}\right) \]
   remove-double-neg [=>] 0.6	\[ \log 2 + \left(\color{blue}{\log x} + \left(-0.25\right) \cdot \frac{1}{{x}^{2}}\right) \]
   unpow2 [=>] 0.6	\[ \log 2 + \left(\log x + \left(-0.25\right) \cdot \frac{1}{\color{blue}{x \cdot x}}\right) \]
   associate-*r/ [=>] 0.6	\[ \log 2 + \left(\log x + \color{blue}{\frac{\left(-0.25\right) \cdot 1}{x \cdot x}}\right) \]
   metadata-eval [=>] 0.6	\[ \log 2 + \left(\log x + \frac{\color{blue}{-0.25} \cdot 1}{x \cdot x}\right) \]
   metadata-eval [=>] 0.6	\[ \log 2 + \left(\log x + \frac{\color{blue}{-0.25}}{x \cdot x}\right) \]

4. Final simplification 0.6

   \[\leadsto \log 2 + \left(\log x + \frac{-0.25}{x \cdot x}\right) \]

Alternatives

Alternative 1
Error	0.5
Cost	3808

\[\log \left(\left(\frac{-0.5}{x} + \frac{-0.25}{\left(x \cdot x\right) \cdot \left(x + \frac{0.5}{x}\right)}\right) + 2 \cdot x\right) \]

Alternative 2
Error	0.5
Cost	3424

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

Alternative 3
Error	0.9
Cost	3296

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

Alternative 4
Error	24.2
Cost	3232

\[\mathsf{expm1}\left(1.9453125\right) \]

Alternative 5
Error	24.2
Cost	3232

\[\mathsf{expm1}\left(1.9583333333333333\right) \]

Alternative 6
Error	17.9
Cost	3232

\[\log x \]

Alternative 7
Error	30.0
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

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

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