?

Average Error: 16.2 → 0.6
Time: 4.9s
Precision: binary32
Cost: 3488

?

\[x \geq 1\]
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(\left(x + x\right) - 0.5 \cdot \frac{1}{x}\right) \]
(FPCore (x) :precision binary32 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x) :precision binary32 (log (- (+ x x) (* 0.5 (/ 1.0 x)))))
float code(float x) {
	return logf((x + sqrtf(((x * x) - 1.0f))));
}
↓
float code(float x) {
	return logf(((x + x) - (0.5f * (1.0f / x))));
}

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 + x) - (0.5e0 * (1.0e0 / x))))
end function

function code(x)
	return log(Float32(x + sqrt(Float32(Float32(x * x) - Float32(1.0)))))
end
↓
function code(x)
	return log(Float32(Float32(x + x) - Float32(Float32(0.5) * Float32(Float32(1.0) / x))))
end

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - single(1.0)))));
end
↓
function tmp = code(x)
	tmp = log(((x + x) - (single(0.5) * (single(1.0) / x))));
end

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\left(x + x\right) - 0.5 \cdot \frac{1}{x}\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.2
Target0.2
Herbie0.6
\[\log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \]

Derivation?

  1. Initial program 16.2

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

  2. Taylor expanded in x around inf 0.6

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

  3. Simplified0.6

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

    [Start]0.6

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

    metadata-eval [<=]0.6

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

    rational.json-simplify-7 [<=]0.6

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

    rational.json-simplify-30 [=>]0.6

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

    rational.json-simplify-7 [=>]0.6

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

  4. Final simplification0.6

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

Alternatives

Alternative 1
Error1.0
Cost3296
\[\log \left(x + x\right) \]

Error

Reproduce?

herbie shell --seed 2023074 
(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)))))