Rust f32::acosh

Average Error: 16.0 → 0.4

Time: 6.0s

Precision: binary32

Cost: 3680

\[x \geq 1\]

Math

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

↓

\[\log \left(x \cdot 2 + \frac{\frac{-0.125}{x} + x \cdot -0.5}{x \cdot x}\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (log (+ (* x 2.0) (/ (+ (/ -0.125 x) (* x -0.5)) (* x x)))))

C

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

↓

float code(float x) {
	return logf(((x * 2.0f) + (((-0.125f / x) + (x * -0.5f)) / (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 * 2.0e0) + ((((-0.125e0) / x) + (x * (-0.5e0))) / (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(2.0)) + Float32(Float32(Float32(Float32(-0.125) / x) + Float32(x * Float32(-0.5))) / 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(2.0)) + (((single(-0.125) / x) + (x * single(-0.5))) / (x * x))));
end

Target

Original	16.0
Target	0.3
Herbie	0.4

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

Derivation

1. Initial program 16.0

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

2. Taylor expanded in x around inf 0.4

   \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

3. Simplified 0.4

   \[\leadsto \log \color{blue}{\left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)} \]
   Proof

   [Start] 0.4	\[ \log \left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   *-commutative [=>] 0.4	\[ \log \left(\color{blue}{x \cdot 2} - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   associate-*r/ [=>] 0.4	\[ \log \left(x \cdot 2 - \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   metadata-eval [=>] 0.4	\[ \log \left(x \cdot 2 - \left(\frac{\color{blue}{0.5}}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   associate-*r/ [=>] 0.4	\[ \log \left(x \cdot 2 - \left(\frac{0.5}{x} + \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right)\right) \]
   metadata-eval [=>] 0.4	\[ \log \left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{\color{blue}{0.125}}{{x}^{3}}\right)\right) \]

4. Applied egg-rr 0.4

   \[\leadsto \log \left(x \cdot 2 - \left(\frac{0.5}{x} + \color{blue}{\frac{\frac{0.25}{x \cdot x}}{x} \cdot 0.5}\right)\right) \]

5. Applied egg-rr 0.4

   \[\leadsto \log \left(x \cdot 2 - \color{blue}{\frac{0.5 \cdot x + x \cdot \frac{0.125}{x \cdot x}}{x \cdot x}}\right) \]

6. Simplified 0.4

   \[\leadsto \log \left(x \cdot 2 - \color{blue}{\frac{x \cdot \left(0.5 + \frac{0.125}{x \cdot x}\right)}{x \cdot x}}\right) \]
   Proof

   [Start] 0.4	\[ \log \left(x \cdot 2 - \frac{0.5 \cdot x + x \cdot \frac{0.125}{x \cdot x}}{x \cdot x}\right) \]
   *-commutative [=>] 0.4	\[ \log \left(x \cdot 2 - \frac{\color{blue}{x \cdot 0.5} + x \cdot \frac{0.125}{x \cdot x}}{x \cdot x}\right) \]
   distribute-lft-out [=>] 0.4	\[ \log \left(x \cdot 2 - \frac{\color{blue}{x \cdot \left(0.5 + \frac{0.125}{x \cdot x}\right)}}{x \cdot x}\right) \]

7. Applied egg-rr 0.4

   \[\leadsto \log \left(x \cdot 2 - \frac{\color{blue}{\frac{0.125}{x} + x \cdot 0.5}}{x \cdot x}\right) \]

8. Final simplification 0.4

   \[\leadsto \log \left(x \cdot 2 + \frac{\frac{-0.125}{x} + x \cdot -0.5}{x \cdot x}\right) \]

Alternatives

Alternative 1
Error	0.4
Cost	3680

\[\log \left(x \cdot 2 + \left(\frac{-0.5}{x} + \frac{\frac{-0.125}{x \cdot x}}{x}\right)\right) \]

Alternative 2
Error	0.6
Cost	3424

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

Alternative 3
Error	1.0
Cost	3296

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

Reproduce

herbie shell --seed 2023016 
(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)))))