Hyperbolic arc-cosine

Percentage Accurate: 51.6% → 99.4%

Time: 5.1s

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: 51.6% 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: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around inf 98.7%

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

   1. associate-*r/ 98.7%

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

   2. metadata-eval 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

   \[\leadsto \log \left(x + \left(x - \frac{0.5}{x}\right)\right) \]
7. Add Preprocessing

Alternative 2: 31.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + -2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ x -2.0)))

C

double code(double x) {
	return log((x + -2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + (-2.0d0)))
end function

Java

public static double code(double x) {
	return Math.log((x + -2.0));
}

Python

def code(x):
	return math.log((x + -2.0))

Julia

function code(x)
	return log(Float64(x + -2.0))
end

MATLAB

function tmp = code(x)
	tmp = log((x + -2.0));
end

Wolfram

code[x_] := N[Log[N[(x + -2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around inf 98.7%

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

   1. associate-*r/ 98.7%

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

   2. metadata-eval 98.7%

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

5. Simplified 98.7%

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

   1. +-commutative 98.7%

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

   2. associate-+l- 98.7%

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

   3. clear-num 98.7%

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

   4. div-inv 98.7%

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

   5. metadata-eval 98.7%

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

   6. *-commutative 98.7%

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

   7. count-2 98.7%

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

   8. flip-+ 0.0%

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

   9. +-inverses 0.0%

      \[\leadsto \log \left(x - \left(\frac{1}{\frac{\color{blue}{0}}{x - x}} - x\right)\right) \]

   10. +-inverses 0.0%

       \[\leadsto \log \left(x - \left(\frac{1}{\frac{0}{\color{blue}{0}}} - x\right)\right) \]

   11. +-inverses 0.0%

       \[\leadsto \log \left(x - \left(\frac{1}{\frac{\color{blue}{x - x}}{0}} - x\right)\right) \]

   12. +-inverses 0.0%

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

   13. clear-num 0.0%

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

   14. +-inverses 0.0%

       \[\leadsto \log \left(x - \left(\frac{\color{blue}{0}}{x - x} - x\right)\right) \]

   15. +-inverses 0.0%

       \[\leadsto \log \left(x - \left(\frac{0}{\color{blue}{0}} - x\right)\right) \]

7. Applied egg-rr 0.0%

   \[\leadsto \log \color{blue}{\left(x - \left(\frac{0}{0} - x\right)\right)} \]

9. Simplified 31.2%

   \[\leadsto \log \color{blue}{\left(x + -2\right)} \]

10. Final simplification 31.2%

    \[\leadsto \log \left(x + -2\right) \]
11. Add Preprocessing

Alternative 3: 98.9% 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 56.5%

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

3. Taylor expanded in x around inf 98.0%

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

4. Final simplification 98.0%

   \[\leadsto \log \left(x + x\right) \]
5. Add Preprocessing

Alternative 4: 1.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around inf 98.0%

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

4. Taylor expanded in x around inf 98.4%

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

6. Simplified 1.6%

   \[\leadsto \color{blue}{-2} \]

7. Final simplification 1.6%

   \[\leadsto -2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))