Rust f32::acosh

Average Error: 15.8 → 0.8

Time: 3.1s

Precision: binary32

Cost: 6496

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary32 (fabs (log (/ 0.5 x))))

C

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

↓

float code(float x) {
	return fabsf(logf((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 = abs(log((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 abs(log(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 = abs(log((single(0.5) / x)));
end

Target

Original	15.8
Target	0.2
Herbie	0.8

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

Derivation

1. Initial program 15.8

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

2. Taylor expanded in x around -inf 31.0

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

3. Applied egg-rr 0.8

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

4. Applied egg-rr 0.8

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

5. Final simplification 0.8

   \[\leadsto \left|\log \left(\frac{0.5}{x}\right)\right| \]

Alternatives

Alternative 1
Error	0.6
Cost	3424

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

Alternative 2
Error	1.0
Cost	3296

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

Alternative 3
Error	30.0
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023045 
(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)))))