Rust f32::acosh

Average Error: 15.9 → 0.6

Time: 11.8s

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	15.9
Target	0.3
Herbie	0.6

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

Derivation

1. Initial program 15.9

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

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

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

   [Start] 0.6	\[ \log \left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right) \]
   rational.json-simplify-2 [=>] 0.6	\[ \log \left(\color{blue}{x \cdot 2} - 0.5 \cdot \frac{1}{x}\right) \]

4. Taylor expanded in x around 0 0.6

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

5. Final simplification 0.6

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

Alternatives

Alternative 1
Error	1.1
Cost	3296

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

Reproduce

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