neg log

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

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: 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. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fma (* x x) 0.5 (+ (log x) x)))

C

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

Julia

function code(x)
	return fma(Float64(x * x), 0.5, Float64(log(x) + x))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * 0.5 + N[(N[Log[x], $MachinePrecision] + 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. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   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 99.5

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.4% 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 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. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   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 99.5

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

5. Applied rewrites 99.5%

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

Alternative 4: 99.1% 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 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. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-log.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 5: 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 98.3

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\log x} \]
6. 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. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   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 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 2.7%

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

9. Final simplification 2.7%

   \[\leadsto 0.5 \cdot \left(x \cdot x\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024308 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))