qlog (example 3.10)

Percentage Accurate: 3.7% → 99.5%

Time: 5.8s

Alternatives: 4

Speedup: 218.0×

Specification

\[\left|x\right| \leq 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(-0.4166666666666667 \cdot x - 0.5\right) \cdot x - 1\right) \cdot x - 1 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* (- (* (- (* -0.4166666666666667 x) 0.5) x) 1.0) x) 1.0))

C

double code(double x) {
	return (((((-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((((-0.4166666666666667d0) * x) - 0.5d0) * x) - 1.0d0) * x) - 1.0d0
end function

Java

public static double code(double x) {
	return (((((-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0;
}

Python

def code(x):
	return (((((-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = (((((-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0;
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(-0.4166666666666667 * x), $MachinePrecision] - 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 3.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{-5}{12} \cdot x - \frac{1}{2}\right) - 1\right) - 1} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{-5}{12} \cdot x - \frac{1}{2}\right) - 1\right) - 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-5}{12} \cdot x - \frac{1}{2}\right) - 1\right) \cdot x} - 1 \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-5}{12} \cdot x - \frac{1}{2}\right) - 1\right) \cdot x} - 1 \]

   4. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-5}{12} \cdot x - \frac{1}{2}\right) - 1\right)} \cdot x - 1 \]

   5. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-5}{12} \cdot x - \frac{1}{2}\right) \cdot x} - 1\right) \cdot x - 1 \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-5}{12} \cdot x - \frac{1}{2}\right) \cdot x} - 1\right) \cdot x - 1 \]

   7. lower--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-5}{12} \cdot x - \frac{1}{2}\right)} \cdot x - 1\right) \cdot x - 1 \]

   8. lower-*.f64 100.0

      \[\leadsto \left(\left(\color{blue}{-0.4166666666666667 \cdot x} - 0.5\right) \cdot x - 1\right) \cdot x - 1 \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\left(-0.4166666666666667 \cdot x - 0.5\right) \cdot x - 1\right) \cdot x - 1} \]
6. Add Preprocessing

Alternative 2: 99.4% accurate, 14.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 98.6%

   \[\leadsto \color{blue}{-1} \]

6. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-neg-in N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. rgt-mult-inverse N/A

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

   10. fp-cancel-sub-sign N/A

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

   11. mul-1-neg N/A

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

   12. mul-1-neg N/A

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

   13. distribute-lft-neg-in N/A

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

   14. *-commutative N/A

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

   15. distribute-lft-neg-in N/A

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

   16. metadata-eval N/A

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

   17. mul-1-neg N/A

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

   18. distribute-lft-neg-out N/A

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

   19. rgt-mult-inverse N/A

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

   20. metadata-eval N/A

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

   21. lower-fma.f64 99.9

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

8. Applied rewrites 99.9%

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

Alternative 3: 99.1% accurate, 54.5× speedup

Math

\[\begin{array}{l} \\ -1 - x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return -1.0 - x;
}

Python

def code(x):
	return -1.0 - x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 98.6%

   \[\leadsto \color{blue}{-1} \]

6. Taylor expanded in x around 0

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

   1. *-rgt-identity N/A

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

   2. rgt-mult-inverse N/A

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

   3. fp-cancel-sub-sign N/A

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

   4. mul-1-neg N/A

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

   5. distribute-lft-in N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. mul-1-neg N/A

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

   9. mul-1-neg N/A

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

   10. fp-cancel-sub-sign N/A

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

   11. distribute-lft-neg-out N/A

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

   12. rgt-mult-inverse N/A

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

   13. metadata-eval N/A

       \[\leadsto \color{blue}{-1} - x \cdot 1 \]

   14. *-rgt-identity N/A

       \[\leadsto -1 - \color{blue}{x} \]

   15. lower--.f64 99.6

       \[\leadsto \color{blue}{-1 - x} \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{-1 - x} \]
9. Add Preprocessing

Alternative 4: 98.1% accurate, 218.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 3.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 98.6%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (log1p (- x)) (log1p x)))

C

double code(double x) {
	return log1p(-x) / log1p(x);
}

Java

public static double code(double x) {
	return Math.log1p(-x) / Math.log1p(x);
}

Python

def code(x):
	return math.log1p(-x) / math.log1p(x)

Julia

function code(x)
	return Float64(log1p(Float64(-x)) / log1p(x))
end

Wolfram

code[x_] := N[(N[Log[1 + (-x)], $MachinePrecision] / N[Log[1 + x], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024342 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (<= (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ (log1p (- x)) (log1p x)))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))