Rust f64::acosh

Average Error: 31.7 → 0.2

Time: 3.3s

Precision: binary64

Cost: 7360

\[x \geq 1\]

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (log (+ x (+ (+ x (/ (/ (/ -0.125 x) x) x)) (/ -0.5 x)))))

C

double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}

↓

double code(double x) {
	return log((x + ((x + (((-0.125 / x) / x) / x)) + (-0.5 / x))));
}

Fortran

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 + ((((-0.125d0) / x) / x) / x)) + ((-0.5d0) / x))))
end function

Java

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 + (((-0.125 / x) / x) / x)) + (-0.5 / x))));
}

Python

def code(x):
	return math.log((x + math.sqrt(((x * x) - 1.0))))

↓

def code(x):
	return math.log((x + ((x + (((-0.125 / x) / x) / x)) + (-0.5 / x))))

Julia

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end

↓

function code(x)
	return log(Float64(x + Float64(Float64(x + Float64(Float64(Float64(-0.125 / x) / x) / x)) + Float64(-0.5 / x))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log((x + ((x + (((-0.125 / x) / x) / x)) + (-0.5 / x))));
end

Wolfram

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[(N[(x + N[(N[(N[(-0.125 / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	31.7
Target	0.0
Herbie	0.2

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

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. Simplified 0.2

   \[\leadsto \log \left(x + \color{blue}{\left(\left(x - \frac{0.125}{{x}^{3}}\right) - \frac{0.5}{x}\right)}\right) \]
   Proof
   (-.f64 (-.f64 x (/.f64 1/8 (pow.f64 x 3))) (/.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 x (/.f64 (Rewrite<= metadata-eval (*.f64 1/8 1)) (pow.f64 x 3))) (/.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 x (Rewrite<= associate-*r/_binary64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))))) (/.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 x (*.f64 1/8 (/.f64 1 (pow.f64 x 3)))) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) x)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 x (*.f64 1/8 (/.f64 1 (pow.f64 x 3)))) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--r+_binary64 (-.f64 x (+.f64 (*.f64 1/8 (/.f64 1 (pow.f64 x 3))) (*.f64 1/2 (/.f64 1 x))))): 0 points increase in error, 1 points decrease in error
   (-.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 1 x)) (*.f64 1/8 (/.f64 1 (pow.f64 x 3)))))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.2

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

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	6848

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

Alternative 2
Error	0.7
Cost	6592

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

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)))))