Rust f32::acosh

Average Error: 16.0 → 0.6

Time: 8.7s

Precision: binary32

Cost: 3808

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (log (+ (+ x (/ -0.5 x)) (+ x (/ -0.25 (* (- x (/ -0.5 x)) (* x x)))))))

C

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

↓

float code(float x) {
	return logf(((x + (-0.5f / x)) + (x + (-0.25f / ((x - (-0.5f / x)) * (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(((x + ((-0.5e0) / x)) + (x + ((-0.25e0) / ((x - ((-0.5e0) / x)) * (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 log(Float32(Float32(x + Float32(Float32(-0.5) / x)) + Float32(x + Float32(Float32(-0.25) / Float32(Float32(x - Float32(Float32(-0.5) / x)) * 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(((x + (single(-0.5) / x)) + (x + (single(-0.25) / ((x - (single(-0.5) / x)) * (x * x))))));
end

Target

Original	16.0
Target	0.3
Herbie	0.6

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

Derivation

1. Initial program 16.0

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

2. Taylor expanded in x around inf 0.6

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

3. Simplified 0.6

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

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

4. Applied egg-rr 0.6

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

5. Taylor expanded in x around inf 0.6

   \[\leadsto \log \left(\color{blue}{\left(x - 0.5 \cdot \frac{1}{x}\right)} - \left(\frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)} - x\right)\right) \]

6. Simplified 0.6

   \[\leadsto \log \left(\color{blue}{\left(x - \frac{0.5}{x}\right)} - \left(\frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)} - x\right)\right) \]
   Proof

   [Start] 0.6	\[ \log \left(\left(x - 0.5 \cdot \frac{1}{x}\right) - \left(\frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)} - x\right)\right) \]
   associate-*r/ [=>] 0.6	\[ \log \left(\left(x - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) - \left(\frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)} - x\right)\right) \]
   metadata-eval [=>] 0.6	\[ \log \left(\left(x - \frac{\color{blue}{0.5}}{x}\right) - \left(\frac{0.25}{\left(x + \frac{0.5}{x}\right) \cdot \left(x \cdot x\right)} - x\right)\right) \]

7. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	3488

\[\mathsf{log1p}\left(\left(x + x\right) + \left(\frac{-0.5}{x} + -1\right)\right) \]

Alternative 2
Error	0.6
Cost	3424

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

Alternative 3
Error	1.0
Cost	3296

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

Alternative 4
Error	17.9
Cost	3232

\[\log x \]

Alternative 5
Error	30.0
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023066 
(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)))))