Rust f32::acosh

Average Error: 16.1 → 0.6

Time: 3.0s

Precision: binary32

Cost: 3424

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary32 (log (+ (* x 2.0) (/ -0.5 x))))

C

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

↓

float code(float x) {
	return logf(((x * 2.0f) + (-0.5f / 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(((x * 2.0e0) + ((-0.5e0) / x)))
end function

Julia

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

↓

function code(x)
	return log(Float32(Float32(x * Float32(2.0)) + Float32(Float32(-0.5) / x)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log(((x * single(2.0)) + (single(-0.5) / x)));
end

Target

Original	16.1
Target	0.3
Herbie	0.6

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

Derivation

1. Initial program 16.1

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

2. Taylor expanded in x around inf 0.6

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

3. Simplified 0.6

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

   [Start] 0.6	\[ \log \left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right) \]
   *-commutative [=>] 0.6	\[ \log \left(\color{blue}{x \cdot 2} - 0.5 \cdot \frac{1}{x}\right) \]
   associate-*r/ [=>] 0.6	\[ \log \left(x \cdot 2 - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) \]
   metadata-eval [=>] 0.6	\[ \log \left(x \cdot 2 - \frac{\color{blue}{0.5}}{x}\right) \]

4. Final simplification 0.6

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

Alternatives

Alternative 1
Error	1.0
Cost	3296

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

Reproduce

herbie shell --seed 2023187 
(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)))))