neg log

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto -\log \left(\frac{1}{x} + -1\right) \]
4. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (fma x 0.5 1.0) (log x)))

C

double code(double x) {
	return fma(x, fma(x, 0.5, 1.0), log(x));
}

Julia

function code(x)
	return fma(x, fma(x, 0.5, 1.0), log(x))
end

Wolfram

code[x_] := N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

   8. lower-log.f64 99.6

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]
6. Add Preprocessing

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. lower-+.f64 N/A

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

   5. lower-log.f64 99.3

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

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{x + \log x} \]
6. Add Preprocessing

Alternative 4: 98.0% accurate, 1.2× 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 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\log x} \]
4. Step-by-step derivation

   1. lower-log.f64 98.1

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

5. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\log x} \]
6. Add Preprocessing

Alternative 5: 7.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.125, 1\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x, 0.25, -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{1}{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 1.0 x) 5e+106)
   (/
    (* (fma x (* (* x x) 0.125) 1.0) (* x (* x x)))
    (* (* x x) (* x (fma x 0.25 -0.5))))
   (/ 0.5 (/ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if ((1.0 / x) <= 5e+106) {
		tmp = (fma(x, ((x * x) * 0.125), 1.0) * (x * (x * x))) / ((x * x) * (x * fma(x, 0.25, -0.5)));
	} else {
		tmp = 0.5 / (1.0 / (x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 / x) <= 5e+106)
		tmp = Float64(Float64(fma(x, Float64(Float64(x * x) * 0.125), 1.0) * Float64(x * Float64(x * x))) / Float64(Float64(x * x) * Float64(x * fma(x, 0.25, -0.5))));
	else
		tmp = Float64(0.5 / Float64(1.0 / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 / x), $MachinePrecision], 5e+106], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.25 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) x) < 4.9999999999999998e106

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

      8. lower-log.f64 98.9

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{x}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.9%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, x\right) \]

   10. Applied rewrites 1.9%

       \[\leadsto \frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.125, 1\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\mathsf{fma}\left(x, \color{blue}{x}, \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.25 - 0.5\right)\right)\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{1}{8}, 1\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{{x}^{4} \cdot \left(\frac{1}{4} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 14.8%

       \[\leadsto \frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.125, 1\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, 0.25, -0.5\right)}\right)} \]

   if 4.9999999999999998e106 < (/.f64 #s(literal 1 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

      8. lower-log.f64 100.0

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{0.5}, x + \log x\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 3.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 3.0%

       \[\leadsto \frac{0.5}{\frac{1}{\color{blue}{x \cdot x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 2.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{1}{x \cdot x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (/ 1.0 (* x x))))

C

double code(double x) {
	return 0.5 / (1.0 / (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 / (1.0d0 / (x * x))
end function

Java

public static double code(double x) {
	return 0.5 / (1.0 / (x * x));
}

Python

def code(x):
	return 0.5 / (1.0 / (x * x))

Julia

function code(x)
	return Float64(0.5 / Float64(1.0 / Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / (1.0 / (x * x));
end

Wolfram

code[x_] := N[(0.5 / N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

   8. lower-log.f64 99.6

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{0.5}, x + \log x\right) \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
9. Step-by-step derivation

10. Applied rewrites 2.7%

    \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]
11. Step-by-step derivation

12. Applied rewrites 2.7%

    \[\leadsto \frac{0.5}{\frac{1}{\color{blue}{x \cdot x}}} \]
13. Add Preprocessing

Alternative 7: 2.7% accurate, 10.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (* x 0.5)))

C

double code(double x) {
	return x * (x * 0.5);
}

Fortran

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

Java

public static double code(double x) {
	return x * (x * 0.5);
}

Python

def code(x):
	return x * (x * 0.5)

Julia

function code(x)
	return Float64(x * Float64(x * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * 0.5);
end

Wolfram

code[x_] := N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

   8. lower-log.f64 99.6

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
7. Step-by-step derivation

8. Applied rewrites 2.7%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot 0.5\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))