qlog (example 3.10)

Percentage Accurate: 4.4% → 99.5%

Time: 9.9s

Alternatives: 4

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: 4.4% 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, 1.8× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot \left(-0.5 + x \cdot -0.4166666666666667\right) + \left(-1 - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (* (pow x 2.0) (+ -0.5 (* x -0.4166666666666667))) (- -1.0 x)))

C

double code(double x) {
	return (pow(x, 2.0) * (-0.5 + (x * -0.4166666666666667))) + (-1.0 - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x ** 2.0d0) * ((-0.5d0) + (x * (-0.4166666666666667d0)))) + ((-1.0d0) - x)
end function

Java

public static double code(double x) {
	return (Math.pow(x, 2.0) * (-0.5 + (x * -0.4166666666666667))) + (-1.0 - x);
}

Python

def code(x):
	return (math.pow(x, 2.0) * (-0.5 + (x * -0.4166666666666667))) + (-1.0 - x)

Julia

function code(x)
	return Float64(Float64((x ^ 2.0) * Float64(-0.5 + Float64(x * -0.4166666666666667))) + Float64(-1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = ((x ^ 2.0) * (-0.5 + (x * -0.4166666666666667))) + (-1.0 - x);
end

Wolfram

code[x_] := N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(-0.5 + N[(x * -0.4166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.8%

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

3. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\left(-1 \cdot x + \left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right)\right) - 1} \]
4. Step-by-step derivation

   1. associate--l+ 100.0%

      \[\leadsto \color{blue}{-1 \cdot x + \left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) - 1\right)} \]

   2. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) - 1\right) + -1 \cdot x} \]

   3. mul-1-neg 100.0%

      \[\leadsto \left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) - 1\right) + \color{blue}{\left(-x\right)} \]

   4. unsub-neg 100.0%

      \[\leadsto \color{blue}{\left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) - 1\right) - x} \]

   5. sub-neg 100.0%

      \[\leadsto \color{blue}{\left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) + \left(-1\right)\right)} - x \]

   6. metadata-eval 100.0%

      \[\leadsto \left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) + \color{blue}{-1}\right) - x \]

   7. associate-+r- 100.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) + \left(-1 - x\right)} \]

   8. *-commutative 100.0%

      \[\leadsto \left(\color{blue}{{x}^{2} \cdot -0.5} + -0.4166666666666667 \cdot {x}^{3}\right) + \left(-1 - x\right) \]

   9. *-commutative 100.0%

      \[\leadsto \left({x}^{2} \cdot -0.5 + \color{blue}{{x}^{3} \cdot -0.4166666666666667}\right) + \left(-1 - x\right) \]

   10. unpow3 100.0%

       \[\leadsto \left({x}^{2} \cdot -0.5 + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot -0.4166666666666667\right) + \left(-1 - x\right) \]

   11. unpow2 100.0%

       \[\leadsto \left({x}^{2} \cdot -0.5 + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot -0.4166666666666667\right) + \left(-1 - x\right) \]

   12. associate-*l* 100.0%

       \[\leadsto \left({x}^{2} \cdot -0.5 + \color{blue}{{x}^{2} \cdot \left(x \cdot -0.4166666666666667\right)}\right) + \left(-1 - x\right) \]

   13. distribute-lft-out 100.0%

       \[\leadsto \color{blue}{{x}^{2} \cdot \left(-0.5 + x \cdot -0.4166666666666667\right)} + \left(-1 - x\right) \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(-0.5 + x \cdot -0.4166666666666667\right) + \left(-1 - x\right)} \]

6. Final simplification 100.0%

   \[\leadsto {x}^{2} \cdot \left(-0.5 + x \cdot -0.4166666666666667\right) + \left(-1 - x\right) \]
7. Add Preprocessing

Alternative 2: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.8%

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

3. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\left(-1 \cdot x + \left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right)\right) - 1} \]
4. Step-by-step derivation

   1. associate--l+ 100.0%

      \[\leadsto \color{blue}{-1 \cdot x + \left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) - 1\right)} \]

   2. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) - 1\right) + -1 \cdot x} \]

   3. mul-1-neg 100.0%

      \[\leadsto \left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) - 1\right) + \color{blue}{\left(-x\right)} \]

   4. unsub-neg 100.0%

      \[\leadsto \color{blue}{\left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) - 1\right) - x} \]

   5. sub-neg 100.0%

      \[\leadsto \color{blue}{\left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) + \left(-1\right)\right)} - x \]

   6. metadata-eval 100.0%

      \[\leadsto \left(\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) + \color{blue}{-1}\right) - x \]

   7. associate-+r- 100.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) + \left(-1 - x\right)} \]

   8. *-commutative 100.0%

      \[\leadsto \left(\color{blue}{{x}^{2} \cdot -0.5} + -0.4166666666666667 \cdot {x}^{3}\right) + \left(-1 - x\right) \]

   9. *-commutative 100.0%

      \[\leadsto \left({x}^{2} \cdot -0.5 + \color{blue}{{x}^{3} \cdot -0.4166666666666667}\right) + \left(-1 - x\right) \]

   10. unpow3 100.0%

       \[\leadsto \left({x}^{2} \cdot -0.5 + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot -0.4166666666666667\right) + \left(-1 - x\right) \]

   11. unpow2 100.0%

       \[\leadsto \left({x}^{2} \cdot -0.5 + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot -0.4166666666666667\right) + \left(-1 - x\right) \]

   12. associate-*l* 100.0%

       \[\leadsto \left({x}^{2} \cdot -0.5 + \color{blue}{{x}^{2} \cdot \left(x \cdot -0.4166666666666667\right)}\right) + \left(-1 - x\right) \]

   13. distribute-lft-out 100.0%

       \[\leadsto \color{blue}{{x}^{2} \cdot \left(-0.5 + x \cdot -0.4166666666666667\right)} + \left(-1 - x\right) \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(-0.5 + x \cdot -0.4166666666666667\right) + \left(-1 - x\right)} \]

6. Taylor expanded in x around 0 99.9%

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

   1. unpow2 99.9%

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

   2. associate-*r* 99.9%

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

   3. fma-define 99.9%

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in x around 0 99.9%

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

    1. unpow2 99.9%

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

    2. associate-*r* 99.9%

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

    3. distribute-rgt-out 99.9%

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

    4. *-commutative 99.9%

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

    5. fma-neg 99.9%

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

    6. metadata-eval 99.9%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

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

Alternative 3: 98.9% 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 3.8%

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

3. Taylor expanded in x around 0 99.4%

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

   1. sub-neg 99.4%

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

   2. metadata-eval 99.4%

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

   3. +-commutative 99.4%

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

   4. mul-1-neg 99.4%

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

   5. unsub-neg 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 97.7% 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 3.8%

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

3. Taylor expanded in x around 0 98.3%

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

4. Final simplification 98.3%

   \[\leadsto -1 \]
5. 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 2024046 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (<= (fabs x) 1.0)

  :alt
  (/ (log1p (- x)) (log1p x))

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