Hyperbolic arc-cosine

Percentage Accurate: 51.5% → 99.1%

Time: 6.9s

Alternatives: 4

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{0.5}{x}\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -log((0.5d0 / x))
end function

Java

public static double code(double x) {
	return -Math.log((0.5 / x));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := (-N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 51.2%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Step-by-step derivation

   1. flip-+ 1.5%

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

   2. div-inv 1.5%

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

   3. log-prod 1.5%

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

   4. add-sqr-sqrt 1.5%

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

   5. fma-neg 1.5%

      \[\leadsto \log \left(x \cdot x - \color{blue}{\mathsf{fma}\left(x, x, -1\right)}\right) + \log \left(\frac{1}{x - \sqrt{x \cdot x - 1}}\right) \]

   6. metadata-eval 1.5%

      \[\leadsto \log \left(x \cdot x - \mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right) + \log \left(\frac{1}{x - \sqrt{x \cdot x - 1}}\right) \]

   7. fma-neg 1.5%

      \[\leadsto \log \left(x \cdot x - \mathsf{fma}\left(x, x, -1\right)\right) + \log \left(\frac{1}{x - \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}}\right) \]

   8. metadata-eval 1.5%

      \[\leadsto \log \left(x \cdot x - \mathsf{fma}\left(x, x, -1\right)\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}}\right) \]

3. Applied egg-rr 1.5%

   \[\leadsto \color{blue}{\log \left(x \cdot x - \mathsf{fma}\left(x, x, -1\right)\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right)} \]
4. Step-by-step derivation

   1. fma-def 1.5%

      \[\leadsto \log \left(x \cdot x - \color{blue}{\left(x \cdot x + -1\right)}\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   2. associate--r+ 3.0%

      \[\leadsto \log \color{blue}{\left(\left(x \cdot x - x \cdot x\right) - -1\right)} + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   3. +-inverses 3.3%

      \[\leadsto \log \left(\color{blue}{0} - -1\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   4. metadata-eval 3.3%

      \[\leadsto \log \color{blue}{1} + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   5. metadata-eval 3.3%

      \[\leadsto \color{blue}{0} + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   6. +-lft-identity 3.3%

      \[\leadsto \color{blue}{\log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right)} \]

   7. log-rec 3.0%

      \[\leadsto \color{blue}{-\log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)} \]

5. Simplified 3.0%

   \[\leadsto \color{blue}{-\log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)} \]

6. Taylor expanded in x around inf 99.2%

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

7. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ x x)))

C

double code(double x) {
	return log((x + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + x))
end function

Java

public static double code(double x) {
	return Math.log((x + x));
}

Python

def code(x):
	return math.log((x + x))

Julia

function code(x)
	return log(Float64(x + x))
end

MATLAB

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

Wolfram

code[x_] := N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.2%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]

3. Final simplification 98.8%

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

Alternative 3: 31.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log x))

C

double code(double x) {
	return log(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(x)
end function

Java

public static double code(double x) {
	return Math.log(x);
}

Python

def code(x):
	return math.log(x)

Julia

function code(x)
	return log(x)
end

MATLAB

function tmp = code(x)
	tmp = log(x);
end

Wolfram

code[x_] := N[Log[x], $MachinePrecision]

Derivation

1. Initial program 51.2%

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

2. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{\left(x - 0.5 \cdot \frac{1}{x}\right)}\right) \]
3. Step-by-step derivation

   1. associate-*r/ 99.0%

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

   2. metadata-eval 99.0%

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

4. Simplified 99.0%

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

5. Taylor expanded in x around 0 0.0%

   \[\leadsto \color{blue}{\log -0.5 + -1 \cdot \log x} \]

7. Simplified 31.4%

   \[\leadsto \color{blue}{\log x} \]

8. Final simplification 31.4%

   \[\leadsto \log x \]

Alternative 4: 1.1% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ -1 + \frac{x}{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -1.0 (/ x -1.0)))

C

double code(double x) {
	return -1.0 + (x / -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) + (x / (-1.0d0))
end function

Java

public static double code(double x) {
	return -1.0 + (x / -1.0);
}

Python

def code(x):
	return -1.0 + (x / -1.0)

Julia

function code(x)
	return Float64(-1.0 + Float64(x / -1.0))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + (x / -1.0);
end

Wolfram

code[x_] := N[(-1.0 + N[(x / -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Step-by-step derivation

   1. difference-of-sqr-1 51.2%

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

   2. sub-neg 51.2%

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

   3. metadata-eval 51.2%

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

3. Applied egg-rr 51.2%

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

4. Taylor expanded in x around 0 0.0%

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

6. Simplified 1.1%

   \[\leadsto \color{blue}{-1 + \frac{x}{-1}} \]

7. Final simplification 1.1%

   \[\leadsto -1 + \frac{x}{-1} \]

Reproduce

herbie shell --seed 2023278 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))