qlog (example 3.10)

Percentage Accurate: 3.8% → 99.7%

Time: 9.6s

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: 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: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  (/
   (*
    (* x x)
    (fma
     (* x x)
     (fma (* x x) (fma x (* x -0.25) -0.3333333333333333) -0.5)
     -1.0))
   (log1p x))
  1.0))

C

double code(double x) {
	return (((x * x) * fma((x * x), fma((x * x), fma(x, (x * -0.25), -0.3333333333333333), -0.5), -1.0)) / log1p(x)) - 1.0;
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * -0.25), -0.3333333333333333), -0.5), -1.0)) / log1p(x)) - 1.0)
end

Wolfram

code[x_] := N[(N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.25), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[Log[1 + x], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 3.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{log1p}\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}{\mathsf{log1p}\left(x\right)} - \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{\mathsf{log1p}\left(x\right)} - 1 \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{\mathsf{log1p}\left(x\right)} - 1 \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   4. sub-neg N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\mathsf{log1p}\left(x\right)} - 1 \]

   5. metadata-eval N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{\mathsf{log1p}\left(x\right)} - 1 \]

   7. unpow2 N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   9. sub-neg N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   10. metadata-eval N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   12. unpow2 N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}, \frac{-1}{2}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4} \cdot {x}^{2} - \frac{1}{3}, \frac{-1}{2}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   14. sub-neg N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   15. unpow2 N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{4} \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   16. associate-*r* N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{-1}{4} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   17. *-commutative N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{-1}{4} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   18. metadata-eval N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{-1}{4} \cdot x\right) + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   19. lower-fma.f64 N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{-1}{4} \cdot x, \frac{-1}{3}\right)}, \frac{-1}{2}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   20. *-commutative N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{4}}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   21. lower-*.f64 100.0

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -0.25}, -0.3333333333333333\right), -0.5\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

10. Applied rewrites 100.0%

    \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{\mathsf{log1p}\left(x\right)} - 1 \]
11. Add Preprocessing

Alternative 2: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.3333333333333333, -0.5\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  (/
   (* (* x x) (fma x (* x (fma x (* x -0.3333333333333333) -0.5)) -1.0))
   (log1p x))
  1.0))

C

double code(double x) {
	return (((x * x) * fma(x, (x * fma(x, (x * -0.3333333333333333), -0.5)), -1.0)) / log1p(x)) - 1.0;
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(x * x) * fma(x, Float64(x * fma(x, Float64(x * -0.3333333333333333), -0.5)), -1.0)) / log1p(x)) - 1.0)
end

Wolfram

code[x_] := N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[Log[1 + x], $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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{log1p}\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}{\mathsf{log1p}\left(x\right)} - \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{3} \cdot {x}^{2} - \frac{1}{2}\right) - 1\right)}}{\mathsf{log1p}\left(x\right)} - 1 \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{3} \cdot {x}^{2} - \frac{1}{2}\right) - 1\right)}}{\mathsf{log1p}\left(x\right)} - 1 \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{-1}{3} \cdot {x}^{2} - \frac{1}{2}\right) - 1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{-1}{3} \cdot {x}^{2} - \frac{1}{2}\right) - 1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   4. sub-neg N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{3} \cdot {x}^{2} - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\mathsf{log1p}\left(x\right)} - 1 \]

   5. unpow2 N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{3} \cdot {x}^{2} - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   6. associate-*l* N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{3} \cdot {x}^{2} - \frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   7. metadata-eval N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} \cdot {x}^{2} - \frac{1}{2}\right)\right) + \color{blue}{-1}\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{3} \cdot {x}^{2} - \frac{1}{2}\right), -1\right)}}{\mathsf{log1p}\left(x\right)} - 1 \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{-1}{3} \cdot {x}^{2} - \frac{1}{2}\right)}, -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   10. sub-neg N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{3} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   11. *-commutative N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   12. unpow2 N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{3} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   13. associate-*l* N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{3}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   14. *-commutative N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{3} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   15. metadata-eval N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \left(\frac{-1}{3} \cdot x\right) + \color{blue}{\frac{-1}{2}}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{3} \cdot x, \frac{-1}{2}\right)}, -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   17. *-commutative N/A

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

   18. lower-*.f64 99.7

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.3333333333333333}, -0.5\right), -1\right)}{\mathsf{log1p}\left(x\right)} - 1 \]

10. Applied rewrites 99.7%

    \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.3333333333333333, -0.5\right), -1\right)}}{\mathsf{log1p}\left(x\right)} - 1 \]
11. 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 2024228 
(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))))