Jmat.Real.lambertw, estimator

Percentage Accurate: 99.6% → 100.0%

Time: 3.4s

Alternatives: 2

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (* x (/ 1.0 (log x)))))

C

double code(double x) {
	return log((x * (1.0 / log(x))));
}

Fortran

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

Java

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

Python

def code(x):
	return math.log((x * (1.0 / math.log(x))))

Julia

function code(x)
	return log(Float64(x * Float64(1.0 / log(x))))
end

MATLAB

function tmp = code(x)
	tmp = log((x * (1.0 / log(x))));
end

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\log x - \log \log x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 98.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x}} - \log \log x \]

   2. *-un-lft-identity 98.1%

      \[\leadsto \left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} - \color{blue}{1 \cdot \log \log x} \]

   3. prod-diff 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\log \log x \cdot 1\right) + \mathsf{fma}\left(-\log \log x, 1, \log \log x \cdot 1\right)} \]

   4. *-commutative 98.1%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\color{blue}{1 \cdot \log \log x}\right) + \mathsf{fma}\left(-\log \log x, 1, \log \log x \cdot 1\right) \]

   5. *-un-lft-identity 98.1%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\color{blue}{\log \log x}\right) + \mathsf{fma}\left(-\log \log x, 1, \log \log x \cdot 1\right) \]

   6. fma-neg 98.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} - \log \log x\right)} + \mathsf{fma}\left(-\log \log x, 1, \log \log x \cdot 1\right) \]

   7. add-cube-cbrt 99.6%

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

   8. diff-log 100.0%

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

   9. *-commutative 100.0%

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

   10. *-un-lft-identity 100.0%

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

4. Applied egg-rr 100.0%

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

   1. fma-undefine 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. +-commutative 100.0%

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

   4. unsub-neg 100.0%

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

   5. +-inverses 100.0%

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

   6. +-rgt-identity 100.0%

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

6. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \log \left(x \cdot \frac{1}{\log x}\right) \]
10. Add Preprocessing

Alternative 2: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\log x - \log \log x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 98.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x}} - \log \log x \]

   2. *-un-lft-identity 98.1%

      \[\leadsto \left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} - \color{blue}{1 \cdot \log \log x} \]

   3. prod-diff 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\log \log x \cdot 1\right) + \mathsf{fma}\left(-\log \log x, 1, \log \log x \cdot 1\right)} \]

   4. *-commutative 98.1%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\color{blue}{1 \cdot \log \log x}\right) + \mathsf{fma}\left(-\log \log x, 1, \log \log x \cdot 1\right) \]

   5. *-un-lft-identity 98.1%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\color{blue}{\log \log x}\right) + \mathsf{fma}\left(-\log \log x, 1, \log \log x \cdot 1\right) \]

   6. fma-neg 98.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} - \log \log x\right)} + \mathsf{fma}\left(-\log \log x, 1, \log \log x \cdot 1\right) \]

   7. add-cube-cbrt 99.6%

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

   8. diff-log 100.0%

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

   9. *-commutative 100.0%

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

   10. *-un-lft-identity 100.0%

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

4. Applied egg-rr 100.0%

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

   1. fma-undefine 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. +-commutative 100.0%

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

   4. unsub-neg 100.0%

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

   5. +-inverses 100.0%

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

   6. +-rgt-identity 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \log \left(\frac{x}{\log x}\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024072 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))