Hyperbolic arc-cosine

Percentage Accurate: 52.1% → 99.5%

Time: 4.7s

Alternatives: 5

Speedup: 2.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 52.1% 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: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \log \left(2 \cdot x - \frac{0.5}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (- (* 2.0 x) (/ 0.5 x))))

C

double code(double x) {
	return log(((2.0 * x) - (0.5 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((2.0d0 * x) - (0.5d0 / x)))
end function

Java

public static double code(double x) {
	return Math.log(((2.0 * x) - (0.5 / x)));
}

Python

def code(x):
	return math.log(((2.0 * x) - (0.5 / x)))

Julia

function code(x)
	return log(Float64(Float64(2.0 * x) - Float64(0.5 / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((2.0 * x) - (0.5 / x)));
end

Wolfram

code[x_] := N[Log[N[(N[(2.0 * x), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.2%

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

2. Taylor expanded in x around inf 99.4%

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

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

4. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.4%

   \[\leadsto \log \color{blue}{\left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right)} \]

6. Taylor expanded in x around 0 99.4%

   \[\leadsto \log \left(2 \cdot x - \color{blue}{\frac{0.5}{x}}\right) \]

7. Final simplification 99.4%

   \[\leadsto \log \left(2 \cdot x - \frac{0.5}{x}\right) \]

Alternative 2: 99.5% 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 48.2%

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

2. Taylor expanded in x around inf 99.4%

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

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

4. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 31.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.2%

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

2. Taylor expanded in x around inf 99.4%

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

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

4. Simplified 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-+l- 99.4%

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

   3. clear-num 99.4%

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

   4. div-inv 99.4%

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

   5. metadata-eval 99.4%

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

   6. *-commutative 99.4%

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

   7. count-2 99.4%

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

   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%

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

   11. +-inverses 0.0%

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

   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%

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

6. Applied egg-rr 0.0%

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

8. Simplified 31.5%

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

9. Final simplification 31.5%

   \[\leadsto \log \left(x + -1\right) \]

Alternative 4: 99.1% 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 48.2%

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

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 5: 1.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 48.2%

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

2. Taylor expanded in x around inf 98.8%

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

3. Taylor expanded in x around inf 98.9%

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

5. Simplified 1.6%

   \[\leadsto \color{blue}{-1} \]

6. Final simplification 1.6%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))