neg log

Percentage Accurate: 99.9% → 99.9%

Time: 6.0s

Alternatives: 6

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: 99.9% 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: 99.9% 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: 99.4% 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 99.3

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

5. Applied rewrites 99.3%

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

Alternative 3: 99.1% 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 99.0

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

5. Applied rewrites 99.0%

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

Alternative 4: 98.2% 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 98.0

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

5. Applied rewrites 98.0%

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

Alternative 5: 2.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{2}{x \cdot x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 2.0 (* x x))))

C

double code(double x) {
	return 1.0 / (2.0 / (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (2.0d0 / (x * x))
end function

Java

public static double code(double x) {
	return 1.0 / (2.0 / (x * x));
}

Python

def code(x):
	return 1.0 / (2.0 / (x * x))

Julia

function code(x)
	return Float64(1.0 / Float64(2.0 / Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (2.0 / (x * x));
end

Wolfram

code[x_] := N[(1.0 / N[(2.0 / 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 99.3

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

5. Applied rewrites 99.3%

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

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%

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

10. Applied rewrites 2.7%

    \[\leadsto 0.5 \cdot \frac{1}{\frac{1}{\color{blue}{x \cdot x}}} \]
11. Step-by-step derivation

12. Applied rewrites 2.7%

    \[\leadsto \frac{1}{\frac{2}{\color{blue}{x \cdot x}}} \]
13. Add Preprocessing

Alternative 6: 2.7% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 * N[(x * 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 99.3

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

5. Applied rewrites 99.3%

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

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%

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

Reproduce

herbie shell --seed 2024237 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))