Average Error: 31.7 → 0.2
Time: 2.9s
Precision: binary64
Cost: 7360
\[x \geq 1\]
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(x + \left(x + \frac{1}{x} \cdot \left(-0.5 + \frac{-0.125}{x \cdot x}\right)\right)\right) \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (log (+ x (+ x (* (/ 1.0 x) (+ -0.5 (/ -0.125 (* x x))))))))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log((x + (x + ((1.0 / x) * (-0.5 + (-0.125 / (x * x)))))));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) - 1.0d0))))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + (x + ((1.0d0 / x) * ((-0.5d0) + ((-0.125d0) / (x * x)))))))
end function

public static double code(double x) {
	return Math.log((x + Math.sqrt(((x * x) - 1.0))));
}
↓
public static double code(double x) {
	return Math.log((x + (x + ((1.0 / x) * (-0.5 + (-0.125 / (x * x)))))));
}

def code(x):
	return math.log((x + math.sqrt(((x * x) - 1.0))))
↓
def code(x):
	return math.log((x + (x + ((1.0 / x) * (-0.5 + (-0.125 / (x * x)))))))

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end
↓
function code(x)
	return log(Float64(x + Float64(x + Float64(Float64(1.0 / x) * Float64(-0.5 + Float64(-0.125 / Float64(x * x)))))))
end

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - 1.0))));
end
↓
function tmp = code(x)
	tmp = log((x + (x + ((1.0 / x) * (-0.5 + (-0.125 / (x * x)))))));
end

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := N[Log[N[(x + N[(x + N[(N[(1.0 / x), $MachinePrecision] * N[(-0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(x + \frac{1}{x} \cdot \left(-0.5 + \frac{-0.125}{x \cdot x}\right)\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.7
Target0.0
Herbie0.2
\[\log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \]

Derivation

  1. Initial program 31.7

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

  2. Taylor expanded in x around inf 0.2

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

  3. Simplified0.2

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)}\right) \]
    Proof
    (log.f64 (+.f64 x (-.f64 x (+.f64 (/.f64 1/2 x) (/.f64 1/8 (pow.f64 x 3)))))): 0 points increase in error, 0 points decrease in error
    (log.f64 (+.f64 x (-.f64 x (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) x) (/.f64 1/8 (pow.f64 x 3)))))): 0 points increase in error, 0 points decrease in error
    (log.f64 (+.f64 x (-.f64 x (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 x))) (/.f64 1/8 (pow.f64 x 3)))))): 0 points increase in error, 0 points decrease in error
    (log.f64 (+.f64 x (-.f64 x (+.f64 (*.f64 1/2 (/.f64 1 x)) (/.f64 (Rewrite<= metadata-eval (*.f64 1/8 1)) (pow.f64 x 3)))))): 0 points increase in error, 0 points decrease in error
    (log.f64 (+.f64 x (-.f64 x (+.f64 (*.f64 1/2 (/.f64 1 x)) (Rewrite<= associate-*r/_binary64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3)))))))): 0 points increase in error, 0 points decrease in error

  4. Applied egg-rr0.2

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

  5. Applied egg-rr0.2

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

  6. Applied egg-rr0.2

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

  7. Final simplification0.2

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

Alternatives

Alternative 1
Error0.3
Cost6848
\[\log \left(x + \left(x + \frac{-0.5}{x}\right)\right) \]
Alternative 2
Error0.7
Cost6592
\[\log \left(x + x\right) \]

Error

Reproduce

herbie shell --seed 2022329 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

  (log (+ x (sqrt (- (* x x) 1.0)))))