Rust f32::acosh

Average Error: 16.2 → 0.6

Time: 17.1s

Precision: binary32

Cost: 4128

\[x \geq 1\]

Math

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

↓

\[\log \left(x + \frac{\left(\frac{1}{x} + \left(\frac{1}{x} + \left(x + x\right)\right)\right) + \left(\left(x + \left(\frac{1}{x} - x\right)\right) - 4 \cdot \frac{1}{x}\right)}{2}\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (log
  (+
   x
   (/
    (+
     (+ (/ 1.0 x) (+ (/ 1.0 x) (+ x x)))
     (- (+ x (- (/ 1.0 x) x)) (* 4.0 (/ 1.0 x))))
    2.0))))

C

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

↓

float code(float x) {
	return logf((x + ((((1.0f / x) + ((1.0f / x) + (x + x))) + ((x + ((1.0f / x) - x)) - (4.0f * (1.0f / x)))) / 2.0f)));
}

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 + ((((1.0e0 / x) + ((1.0e0 / x) + (x + x))) + ((x + ((1.0e0 / x) - x)) - (4.0e0 * (1.0e0 / x)))) / 2.0e0)))
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(x + Float32(Float32(Float32(Float32(Float32(1.0) / x) + Float32(Float32(Float32(1.0) / x) + Float32(x + x))) + Float32(Float32(x + Float32(Float32(Float32(1.0) / x) - x)) - Float32(Float32(4.0) * Float32(Float32(1.0) / x)))) / Float32(2.0))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log((x + ((((single(1.0) / x) + ((single(1.0) / x) + (x + x))) + ((x + ((single(1.0) / x) - x)) - (single(4.0) * (single(1.0) / x)))) / single(2.0))));
end

Target

Original	16.2
Target	0.3
Herbie	0.6

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

Derivation

1. Initial program 16.2

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

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. Applied egg-rr 0.6

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

4. Simplified 0.6

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

   [Start] 0.6	\[ \log \left(x + \frac{\left(x + x\right) - \frac{1}{x}}{2}\right) \]
   rational_best.json-simplify-4 [<=] 0.6	\[ \log \left(x + \frac{\left(x + x\right) - \color{blue}{\left(\frac{1}{x} + 0\right)}}{2}\right) \]
   rational_best.json-simplify-1 [<=] 0.6	\[ \log \left(x + \frac{\left(x + x\right) - \color{blue}{\left(0 + \frac{1}{x}\right)}}{2}\right) \]
   rational_best.json-simplify-51 [=>] 0.6	\[ \log \left(x + \frac{\color{blue}{\left(x - 0\right) - \left(\frac{1}{x} - x\right)}}{2}\right) \]
   rational_best.json-simplify-6 [=>] 0.6	\[ \log \left(x + \frac{\color{blue}{x} - \left(\frac{1}{x} - x\right)}{2}\right) \]

5. Applied egg-rr 0.6

   \[\leadsto \log \left(x + \frac{\color{blue}{\left(\left(x + \frac{1}{x}\right) + \left(x + \frac{1}{x}\right)\right) + \left(\left(x + \left(\frac{1}{x} - x\right)\right) - \frac{1}{x} \cdot 4\right)}}{2}\right) \]

6. Simplified 0.6

   \[\leadsto \log \left(x + \frac{\color{blue}{\left(\frac{1}{x} + \left(\frac{1}{x} + \left(x + x\right)\right)\right) + \left(\left(x + \left(\frac{1}{x} - x\right)\right) - 4 \cdot \frac{1}{x}\right)}}{2}\right) \]
   Proof

   [Start] 0.6	\[ \log \left(x + \frac{\left(\left(x + \frac{1}{x}\right) + \left(x + \frac{1}{x}\right)\right) + \left(\left(x + \left(\frac{1}{x} - x\right)\right) - \frac{1}{x} \cdot 4\right)}{2}\right) \]
   rational_best.json-simplify-43 [=>] 0.6	\[ \log \left(x + \frac{\color{blue}{\left(\frac{1}{x} + \left(x + \left(x + \frac{1}{x}\right)\right)\right)} + \left(\left(x + \left(\frac{1}{x} - x\right)\right) - \frac{1}{x} \cdot 4\right)}{2}\right) \]
   rational_best.json-simplify-43 [=>] 0.6	\[ \log \left(x + \frac{\left(\frac{1}{x} + \color{blue}{\left(\frac{1}{x} + \left(x + x\right)\right)}\right) + \left(\left(x + \left(\frac{1}{x} - x\right)\right) - \frac{1}{x} \cdot 4\right)}{2}\right) \]
   rational_best.json-simplify-2 [=>] 0.6	\[ \log \left(x + \frac{\left(\frac{1}{x} + \left(\frac{1}{x} + \left(x + x\right)\right)\right) + \left(\left(x + \left(\frac{1}{x} - x\right)\right) - \color{blue}{4 \cdot \frac{1}{x}}\right)}{2}\right) \]

7. Final simplification 0.6

   \[\leadsto \log \left(x + \frac{\left(\frac{1}{x} + \left(\frac{1}{x} + \left(x + x\right)\right)\right) + \left(\left(x + \left(\frac{1}{x} - x\right)\right) - 4 \cdot \frac{1}{x}\right)}{2}\right) \]

Alternatives

Alternative 1
Error	0.6
Cost	3552

\[\log \left(x + \frac{x - \left(\frac{1}{x} - x\right)}{2}\right) \]

Alternative 2
Error	0.6
Cost	3488

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

Alternative 3
Error	1.0
Cost	3296

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

Alternative 4
Error	17.9
Cost	3232

\[\log x \]

Reproduce

herbie shell --seed 2023090 
(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)))))