qlog (example 3.10)

Percentage Accurate: 4.1% → 99.5%

Time: 10.3s

Alternatives: 9

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: 4.1% 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, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot -0.4166666666666667, -1\right)\\ t_1 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.4166666666666667, -0.5\right), -1\right)\\ \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot x, 0.5833333333333334, 1\right)}{\frac{t\_0 \cdot t\_0}{t\_1} - \frac{x \cdot \left(x \cdot 0.25\right)}{t\_1}}, -1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (* x -0.4166666666666667) -1.0))
        (t_1 (fma x (fma x -0.4166666666666667 -0.5) -1.0)))
   (fma
    x
    (/
     (fma (* x x) 0.5833333333333334 1.0)
     (- (/ (* t_0 t_0) t_1) (/ (* x (* x 0.25)) t_1)))
    -1.0)))

C

double code(double x) {
	double t_0 = fma(x, (x * -0.4166666666666667), -1.0);
	double t_1 = fma(x, fma(x, -0.4166666666666667, -0.5), -1.0);
	return fma(x, (fma((x * x), 0.5833333333333334, 1.0) / (((t_0 * t_0) / t_1) - ((x * (x * 0.25)) / t_1))), -1.0);
}

Julia

function code(x)
	t_0 = fma(x, Float64(x * -0.4166666666666667), -1.0)
	t_1 = fma(x, fma(x, -0.4166666666666667, -0.5), -1.0)
	return fma(x, Float64(fma(Float64(x * x), 0.5833333333333334, 1.0) / Float64(Float64(Float64(t_0 * t_0) / t_1) - Float64(Float64(x * Float64(x * 0.25)) / t_1))), -1.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * -0.4166666666666667), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * -0.4166666666666667 + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]}, N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.5833333333333334 + 1.0), $MachinePrecision] / N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Initial program 2.8%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 99.7

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.4166666666666667, -0.5\right), -1\right), -1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left(x, \frac{\left(-1 + x \cdot \left(x \cdot -0.4166666666666667\right)\right) \cdot \left(-1 + x \cdot \left(x \cdot -0.4166666666666667\right)\right) - 0.25 \cdot \left(x \cdot x\right)}{\color{blue}{\left(-1 + x \cdot \left(x \cdot -0.4166666666666667\right)\right) - x \cdot -0.5}}, -1\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \frac{1 + \frac{7}{12} \cdot {x}^{2}}{\color{blue}{\left(-1 + x \cdot \left(x \cdot \frac{-5}{12}\right)\right)} - x \cdot \frac{-1}{2}}, -1\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.7%

    \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot x, 0.5833333333333334, 1\right)}{\color{blue}{\left(-1 + x \cdot \left(x \cdot -0.4166666666666667\right)\right)} - x \cdot -0.5}, -1\right) \]

12. Applied rewrites 99.7%

    \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot x, 0.5833333333333334, 1\right)}{\frac{\mathsf{fma}\left(x, x \cdot -0.4166666666666667, -1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.4166666666666667, -1\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.4166666666666667, -0.5\right), -1\right)} - \color{blue}{\frac{x \cdot \left(x \cdot 0.25\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.4166666666666667, -0.5\right), -1\right)}}}, -1\right) \]
13. Add Preprocessing

Alternative 2: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x, -0.4166666666666667, 0.5\right)\\ \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot x, 0.5833333333333334, 1\right)}{\frac{1 - t\_0 \cdot t\_0}{-1 - t\_0}}, -1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fma x -0.4166666666666667 0.5))))
   (fma
    x
    (/
     (fma (* x x) 0.5833333333333334 1.0)
     (/ (- 1.0 (* t_0 t_0)) (- -1.0 t_0)))
    -1.0)))

C

double code(double x) {
	double t_0 = x * fma(x, -0.4166666666666667, 0.5);
	return fma(x, (fma((x * x), 0.5833333333333334, 1.0) / ((1.0 - (t_0 * t_0)) / (-1.0 - t_0))), -1.0);
}

Julia

function code(x)
	t_0 = Float64(x * fma(x, -0.4166666666666667, 0.5))
	return fma(x, Float64(fma(Float64(x * x), 0.5833333333333334, 1.0) / Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(-1.0 - t_0))), -1.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * -0.4166666666666667 + 0.5), $MachinePrecision]), $MachinePrecision]}, N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.5833333333333334 + 1.0), $MachinePrecision] / N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Initial program 4.1%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.4166666666666667, -0.5\right), -1\right), -1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(x, \frac{\left(-1 + x \cdot \left(x \cdot -0.4166666666666667\right)\right) \cdot \left(-1 + x \cdot \left(x \cdot -0.4166666666666667\right)\right) - 0.25 \cdot \left(x \cdot x\right)}{\color{blue}{\left(-1 + x \cdot \left(x \cdot -0.4166666666666667\right)\right) - x \cdot -0.5}}, -1\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \frac{1 + \frac{7}{12} \cdot {x}^{2}}{\color{blue}{\left(-1 + x \cdot \left(x \cdot \frac{-5}{12}\right)\right)} - x \cdot \frac{-1}{2}}, -1\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.5%

    \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot x, 0.5833333333333334, 1\right)}{\color{blue}{\left(-1 + x \cdot \left(x \cdot -0.4166666666666667\right)\right)} - x \cdot -0.5}, -1\right) \]
11. Step-by-step derivation

12. Applied rewrites 99.5%

    \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot x, 0.5833333333333334, 1\right)}{\frac{1 - \left(x \cdot \mathsf{fma}\left(x, -0.4166666666666667, 0.5\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, -0.4166666666666667, 0.5\right)\right)}{\color{blue}{-1 - x \cdot \mathsf{fma}\left(x, -0.4166666666666667, 0.5\right)}}}, -1\right) \]
13. 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 2024229 
(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))))