Average Error: 16.0 → 0.7

Time: 6.4s

Precision: binary32

Cost: 3424

\[x \geq 1\]

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

↓

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

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

↓

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

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

↓

float code(float x) {
	return logf((x + (x + (-0.5f / x))));
}

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 + (x + ((-0.5e0) / x))))
end function

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

↓

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

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

↓

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

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

↓

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

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	16.0
Target	0.3
Herbie	0.7

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

Derivation

Initial program 16.0
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
Taylor expanded in x around inf 0.7
\[\leadsto \log \left(x + \color{blue}{\left(x - 0.5 \cdot \frac{1}{x}\right)}\right) \]
Simplified0.7
\[\leadsto \log \left(x + \color{blue}{\left(x - \frac{0.5}{x}\right)}\right) \]
Proof
(-.f32 x (/.f32 1/2 x)): 0 points increase in error, 0 points decrease in error
(-.f32 x (/.f32 (Rewrite<= metadata-eval (*.f32 1/2 1)) x)): 0 points increase in error, 0 points decrease in error
(-.f32 x (Rewrite<= associate-*r/_binary32 (*.f32 1/2 (/.f32 1 x)))): 0 points increase in error, 0 points decrease in error
Final simplification0.7
\[\leadsto \log \left(x + \left(x + \frac{-0.5}{x}\right)\right) \]

Alternatives

Alternative 1
Error	1.1
Cost	3296

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

Alternative 2
Error	30.0
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2022332 
(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)))))