neg log

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 6

Speedup: 0.6×

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Final simplification 99.2%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[-\log \left(\frac{1}{x} - 1\right) \]
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. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-log.f64 99.1

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

5. Applied rewrites 99.1%

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

Alternative 3: 99.2% accurate, 1.1× speedup

Math

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

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 Float64(log(x) + x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

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

      \[\leadsto \color{blue}{x + 1 \cdot \log x} \]

   3. *-lft-identity N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-log.f64 98.8

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

5. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\log x + x} \]
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 99.2%

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

3. Taylor expanded in x around 0

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

   1. lower-log.f64 98.2

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

5. Applied rewrites 98.2%

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

Alternative 5: 5.3% accurate, 9.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (fma -0.5 x -1.0) x))

C

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

Julia

function code(x)
	return Float64(fma(-0.5, x, -1.0) * x)
end

Wolfram

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

Derivation

1. Initial program 99.2%

   \[-\log \left(\frac{1}{x} - 1\right) \]
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. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-log.f64 99.1

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

5. Applied rewrites 99.1%

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

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 2.3%

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

10. Applied rewrites 5.3%

    \[\leadsto \mathsf{fma}\left(-0.5, x, -1\right) \cdot x \]
11. Add Preprocessing

Alternative 6: 5.3% accurate, 39.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 99.2%

   \[-\log \left(\frac{1}{x} - 1\right) \]
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. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-log.f64 99.1

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

5. Applied rewrites 99.1%

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

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 2.3%

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

10. Applied rewrites 5.3%

    \[\leadsto \mathsf{fma}\left(-0.5, x, -1\right) \cdot x \]

11. Taylor expanded in x around 0

    \[\leadsto -1 \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 5.3%

    \[\leadsto -x \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))