neg log

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := (-N[Log[N[(N[(1.0 - 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

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 100.0

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

5. Applied rewrites 100.0%

   \[\leadsto -\log \color{blue}{\left(\frac{1 - x}{x}\right)} \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. neg-log N/A

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

   4. lift--.f64 N/A

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

   5. flip3-- N/A

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

   6. clear-num N/A

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

   7. lower-log.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\log \left({\left(\mathsf{expm1}\left(-\log x\right)\right)}^{-1}\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \log \color{blue}{\left({\left(\mathsf{expm1}\left(-\log x\right)\right)}^{-1}\right)} \]

   2. unpow-1 N/A

      \[\leadsto \log \color{blue}{\left(\frac{1}{\mathsf{expm1}\left(-\log x\right)}\right)} \]

   3. lift-expm1.f64 N/A

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

   4. flip-- N/A

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

   5. associate-/r/ N/A

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

   6. pow2 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. neg-log N/A

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

   10. rem-exp-log N/A

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

   11. pow2 N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. neg-log N/A

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

   15. rem-exp-log N/A

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

6. Applied rewrites 100.0%

   \[\leadsto \log \color{blue}{\left(\frac{x}{1 - x}\right)} \]
7. Add Preprocessing

Alternative 3: 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 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 99.2

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

5. Applied rewrites 99.2%

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

Alternative 4: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. neg-log N/A

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

   4. lift--.f64 N/A

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

   5. flip3-- N/A

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

   6. clear-num N/A

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

   7. lower-log.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. inv-pow N/A

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

   12. lower-pow.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\log \left({\left(\mathsf{expm1}\left(-\log x\right)\right)}^{-1}\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \log \color{blue}{\left({\left(\mathsf{expm1}\left(-\log x\right)\right)}^{-1}\right)} \]

   2. unpow-1 N/A

      \[\leadsto \log \color{blue}{\left(\frac{1}{\mathsf{expm1}\left(-\log x\right)}\right)} \]

   3. lift-expm1.f64 N/A

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

   4. flip-- N/A

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

   5. associate-/r/ N/A

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

   6. pow2 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. neg-log N/A

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

   10. rem-exp-log N/A

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

   11. pow2 N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. neg-log N/A

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

   15. rem-exp-log N/A

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f64 99.2

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

9. Applied rewrites 99.2%

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

Alternative 5: 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 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 6: 98.4% 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 98.7

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

5. Applied rewrites 98.7%

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

Alternative 7: 2.7% accurate, 10.6× speedup

Math

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

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(Float64(0.5 * x) * x)
end

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
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 99.2

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

5. Applied rewrites 99.2%

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

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. Step-by-step derivation

10. Applied rewrites 2.7%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))