Rust f32::acosh

Average Error: 16.1 → 0.5

Time: 8.3s

Precision: binary32

Cost: 3680

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Target

Original	16.1
Target	0.3
Herbie	0.5

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

Derivation

1. Initial program 16.1

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

2. Taylor expanded in x around inf 0.5

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

3. Simplified 0.5

   \[\leadsto \log \color{blue}{\left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)} \]
   Proof
   (log.f32 (-.f32 (*.f32 x 2) (+.f32 (/.f32 1/2 x) (/.f32 1/8 (pow.f32 x 3))))): 0 points increase in error, 0 points decrease in error
   (log.f32 (-.f32 (Rewrite<= *-commutative_binary32 (*.f32 2 x)) (+.f32 (/.f32 1/2 x) (/.f32 1/8 (pow.f32 x 3))))): 0 points increase in error, 0 points decrease in error
   (log.f32 (-.f32 (*.f32 2 x) (+.f32 (/.f32 (Rewrite<= metadata-eval (*.f32 1/2 1)) x) (/.f32 1/8 (pow.f32 x 3))))): 0 points increase in error, 0 points decrease in error
   (log.f32 (-.f32 (*.f32 2 x) (+.f32 (Rewrite<= associate-*r/_binary32 (*.f32 1/2 (/.f32 1 x))) (/.f32 1/8 (pow.f32 x 3))))): 0 points increase in error, 0 points decrease in error
   (log.f32 (-.f32 (*.f32 2 x) (+.f32 (*.f32 1/2 (/.f32 1 x)) (/.f32 (Rewrite<= metadata-eval (*.f32 1/8 1)) (pow.f32 x 3))))): 0 points increase in error, 0 points decrease in error
   (log.f32 (-.f32 (*.f32 2 x) (+.f32 (*.f32 1/2 (/.f32 1 x)) (Rewrite<= associate-*r/_binary32 (*.f32 1/8 (/.f32 1 (pow.f32 x 3))))))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.5

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

5. Applied egg-rr 0.5

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

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.7
Cost	3552

\[\mathsf{log1p}\left(\left(x \cdot 2 + 1\right) + \left(-2 + \frac{-0.5}{x}\right)\right) \]

Alternative 2
Error	0.7
Cost	3424

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

Alternative 3
Error	1.1
Cost	3360

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

Alternative 4
Error	1.1
Cost	3296

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

Alternative 5
Error	30.0
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2022343 
(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)))))