qlog (example 3.10)

Percentage Accurate: 3.8% → 100.0%

Time: 11.6s

Alternatives: 5

Speedup: 207.0×

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: 3.8% 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: 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]

Derivation

1. Initial program 4.9%

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

   1. sub-neg 4.9%

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

   2. log1p-define 6.6%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-x\right)}}{\log \left(1 + x\right)} \]

   3. log1p-define 100.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-x\right)}{\color{blue}{\mathsf{log1p}\left(x\right)}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)} \]
6. Add Preprocessing

Alternative 2: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

   1. sub-neg 4.9%

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

   2. log1p-define 6.6%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-x\right)}}{\log \left(1 + x\right)} \]

   3. log1p-define 100.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-x\right)}{\color{blue}{\mathsf{log1p}\left(x\right)}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{\left(-1 \cdot x + -0.5 \cdot {x}^{2}\right) - 1} \]
6. Step-by-step derivation

   1. +-commutative 98.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + -1 \cdot x\right)} - 1 \]

   2. neg-mul-1 98.4%

      \[\leadsto \left(-0.5 \cdot {x}^{2} + \color{blue}{\left(-x\right)}\right) - 1 \]

   3. unsub-neg 98.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} - x\right)} - 1 \]

7. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} - x\right)} - 1 \]
8. Step-by-step derivation

   1. unpow2 98.4%

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

   2. associate-*r* 98.4%

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

   3. fma-neg 98.4%

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

9. Applied egg-rr 98.4%

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

10. Final simplification 98.4%

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

Alternative 3: 99.3% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

   1. sub-neg 4.9%

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

   2. log1p-define 6.6%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-x\right)}}{\log \left(1 + x\right)} \]

   3. log1p-define 100.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-x\right)}{\color{blue}{\mathsf{log1p}\left(x\right)}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{\left(-1 \cdot x + -0.5 \cdot {x}^{2}\right) - 1} \]
6. Step-by-step derivation

   1. +-commutative 98.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + -1 \cdot x\right)} - 1 \]

   2. neg-mul-1 98.4%

      \[\leadsto \left(-0.5 \cdot {x}^{2} + \color{blue}{\left(-x\right)}\right) - 1 \]

   3. unsub-neg 98.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} - x\right)} - 1 \]

7. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} - x\right)} - 1 \]

8. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{\left(-1 \cdot x + -0.5 \cdot {x}^{2}\right)} - 1 \]
9. Step-by-step derivation

   1. *-commutative 98.4%

      \[\leadsto \left(\color{blue}{x \cdot -1} + -0.5 \cdot {x}^{2}\right) - 1 \]

   2. unpow2 98.4%

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

   3. associate-*r* 98.4%

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

   4. *-commutative 98.4%

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

   5. distribute-lft-out 98.4%

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

   6. +-commutative 98.4%

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

   7. *-commutative 98.4%

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

   8. fma-define 98.4%

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

10. Simplified 98.4%

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

    1. fma-undefine 98.4%

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

    2. distribute-rgt-in 98.4%

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

    3. mul-1-neg 98.4%

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

12. Applied egg-rr 98.4%

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

13. Final simplification 98.4%

    \[\leadsto \left(x \cdot \left(x \cdot -0.5\right) - x\right) + -1 \]
14. Add Preprocessing

Alternative 4: 99.0% accurate, 69.0× 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 4.9%

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

   1. sub-neg 4.9%

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

   2. log1p-define 6.6%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-x\right)}}{\log \left(1 + x\right)} \]

   3. log1p-define 100.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-x\right)}{\color{blue}{\mathsf{log1p}\left(x\right)}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 98.0%

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

   1. sub-neg 98.0%

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

   2. metadata-eval 98.0%

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

   3. +-commutative 98.0%

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

   4. mul-1-neg 98.0%

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

   5. unsub-neg 98.0%

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

   \[\leadsto -1 - x \]
9. Add Preprocessing

Alternative 5: 98.0% accurate, 207.0× 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 4.9%

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

   1. sub-neg 4.9%

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

   2. log1p-define 6.6%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-x\right)}}{\log \left(1 + x\right)} \]

   3. log1p-define 100.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-x\right)}{\color{blue}{\mathsf{log1p}\left(x\right)}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 96.8%

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

6. Final simplification 96.8%

   \[\leadsto -1 \]
7. Add Preprocessing

Developer target: 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 2024040 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (<= (fabs x) 1.0)

  :herbie-target
  (/ (log1p (- x)) (log1p x))

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