Rust f32::acosh

Average Accuracy: 49.7% → 96.8%

Time: 5.6s

Precision: binary32

Cost: 6496

\[x \geq 1\]

Math

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

↓

\[\log 2 + \log x \]

FPCore

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

↓

(FPCore (x) :precision binary32 (+ (log 2.0) (log x)))

C

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

↓

float code(float x) {
	return logf(2.0f) + logf(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(2.0e0) + log(x)
end function

Julia

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

↓

function code(x)
	return Float32(log(Float32(2.0)) + log(x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log(single(2.0)) + log(x);
end

Target

Original	49.7%
Target	99.2%
Herbie	96.8%

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

Derivation

1. Initial program 49.7%

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

2. Taylor expanded in x around inf 96.8%

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

3. Simplified 96.8%

   \[\leadsto \color{blue}{\log 2 + \log x} \]
   Proof

   [Start] 96.8	\[ \log 2 + -1 \cdot \log \left(\frac{1}{x}\right) \]
   mul-1-neg [=>] 96.8	\[ \log 2 + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} \]
   log-rec [=>] 96.8	\[ \log 2 + \left(-\color{blue}{\left(-\log x\right)}\right) \]
   remove-double-neg [=>] 96.8	\[ \log 2 + \color{blue}{\log x} \]

4. Final simplification 96.8%

   \[\leadsto \log 2 + \log x \]

Alternatives

Alternative 1
Accuracy	98.2%
Cost	3424

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

Alternative 2
Accuracy	96.9%
Cost	3296

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

Alternative 3
Accuracy	6.1%
Cost	32

\[0 \]

Alternative 4
Accuracy	24.4%
Cost	32

\[6 \]

Alternative 5
Accuracy	29.0%
Cost	32

\[27.333333333333332 \]

Reproduce

herbie shell --seed 2023138 
(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)))))