Rust f64::acosh

Average Error: 50.09% → 0.32%

Time: 3.3s

Precision: binary64

Cost: 14080

\[x \geq 1\]

Math

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

↓

\[\log \left(x + \left(x + \left(\frac{-0.5}{x} + \left(\frac{\frac{-0.125}{x}}{x \cdot x} + \frac{-0.0625}{{x}^{5}}\right)\right)\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (log
  (+
   x
   (+ x (+ (/ -0.5 x) (+ (/ (/ -0.125 x) (* x x)) (/ -0.0625 (pow x 5.0))))))))

C

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

↓

double code(double x) {
	return log((x + (x + ((-0.5 / x) + (((-0.125 / x) / (x * x)) + (-0.0625 / pow(x, 5.0)))))));
}

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 + (((-0.5d0) / x) + ((((-0.125d0) / x) / (x * x)) + ((-0.0625d0) / (x ** 5.0d0)))))))
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 + ((-0.5 / x) + (((-0.125 / x) / (x * x)) + (-0.0625 / Math.pow(x, 5.0)))))));
}

Python

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

↓

def code(x):
	return math.log((x + (x + ((-0.5 / x) + (((-0.125 / x) / (x * x)) + (-0.0625 / math.pow(x, 5.0)))))))

Julia

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(-0.5 / x) + Float64(Float64(Float64(-0.125 / x) / Float64(x * x)) + Float64(-0.0625 / (x ^ 5.0)))))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log((x + (x + ((-0.5 / x) + (((-0.125 / x) / (x * x)) + (-0.0625 / (x ^ 5.0)))))));
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[(x + N[(N[(-0.5 / x), $MachinePrecision] + N[(N[(N[(-0.125 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	50.09%
Target	0.17%
Herbie	0.32%

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

Derivation

1. Initial program 50.09

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

2. Taylor expanded in x around inf 0.32

   \[\leadsto \log \left(x + \color{blue}{\left(x - \left(0.5 \cdot \frac{1}{x} + \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)}\right) \]

3. Simplified 0.32

   \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{0.5}{x} + \left(\frac{0.125}{{x}^{3}} + \frac{0.0625}{{x}^{5}}\right)\right)\right)}\right) \]
   Proof

   [Start] 0.32	\[ \log \left(x + \left(x - \left(0.5 \cdot \frac{1}{x} + \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right) \]
   +-rgt-identity [<=] 0.32	\[ \log \left(x + \left(x - \left(0.5 \cdot \frac{1}{x} + \color{blue}{\left(\left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) + 0\right)}\right)\right)\right) \]
   associate-*r/ [=>] 0.32	\[ \log \left(x + \left(x - \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(\left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) + 0\right)\right)\right)\right) \]
   metadata-eval [=>] 0.32	\[ \log \left(x + \left(x - \left(\frac{\color{blue}{0.5}}{x} + \left(\left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) + 0\right)\right)\right)\right) \]
   +-rgt-identity [=>] 0.32	\[ \log \left(x + \left(x - \left(\frac{0.5}{x} + \color{blue}{\left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right)\right)\right) \]
   +-commutative [=>] 0.32	\[ \log \left(x + \left(x - \left(\frac{0.5}{x} + \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right)\right)\right) \]
   associate-*r/ [=>] 0.32	\[ \log \left(x + \left(x - \left(\frac{0.5}{x} + \left(\color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\right)\right) \]
   metadata-eval [=>] 0.32	\[ \log \left(x + \left(x - \left(\frac{0.5}{x} + \left(\frac{\color{blue}{0.125}}{{x}^{3}} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\right)\right) \]
   associate-*r/ [=>] 0.32	\[ \log \left(x + \left(x - \left(\frac{0.5}{x} + \left(\frac{0.125}{{x}^{3}} + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{5}}}\right)\right)\right)\right) \]
   metadata-eval [=>] 0.32	\[ \log \left(x + \left(x - \left(\frac{0.5}{x} + \left(\frac{0.125}{{x}^{3}} + \frac{\color{blue}{0.0625}}{{x}^{5}}\right)\right)\right)\right) \]

4. Applied egg-rr 0.32

   \[\leadsto \log \left(x + \left(x - \left(\frac{0.5}{x} + \left(\color{blue}{\frac{\frac{1}{x}}{x} \cdot \frac{0.125}{x}} + \frac{0.0625}{{x}^{5}}\right)\right)\right)\right) \]

5. Applied egg-rr 0.32

   \[\leadsto \log \left(x + \left(x - \left(\frac{0.5}{x} + \left(\color{blue}{\frac{\frac{0.125}{x}}{x \cdot x}} + \frac{0.0625}{{x}^{5}}\right)\right)\right)\right) \]

6. Final simplification 0.32

   \[\leadsto \log \left(x + \left(x + \left(\frac{-0.5}{x} + \left(\frac{\frac{-0.125}{x}}{x \cdot x} + \frac{-0.0625}{{x}^{5}}\right)\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.49%
Cost	6848

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

Alternative 2
Error	0.91%
Cost	6592

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

Reproduce

herbie shell --seed 2023088 
(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)))))