qlog (example 3.10)

Percentage Accurate: 4.2% → 99.6%

Time: 10.5s

Alternatives: 6

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: 4.2% 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: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.9%

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

4. Applied egg-rr 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-log1p.f64 N/A

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

   4. lift-log1p.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-log1p.f64 N/A

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

   7. lift-log1p.f64 N/A

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

   8. *-inverses N/A

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

   9. lower--.f64 100.0

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

   10. lift-neg.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. neg-mul-1 N/A

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

   14. lower-*.f64 N/A

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

   15. neg-mul-1 N/A

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

   16. lower-neg.f64 100.0

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

9. Simplified 100.0%

   \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.3333333333333333, -0.5\right), -1\right) \cdot x\right) \cdot x}}{\mathsf{log1p}\left(x\right)} - 1 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-log1p.f64 N/A

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

    8. lift-/.f64 N/A

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

    9. sub-neg N/A

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

11. Applied egg-rr 100.0%

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

Alternative 2: 99.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.2916666666666667, -0.4166666666666667\right), -0.5\right), -1\right), x, -1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (fma x (fma x (fma x -0.2916666666666667 -0.4166666666666667) -0.5) -1.0)
  x
  -1.0))

C

double code(double x) {
	return fma(fma(x, fma(x, fma(x, -0.2916666666666667, -0.4166666666666667), -0.5), -1.0), x, -1.0);
}

Julia

function code(x)
	return fma(fma(x, fma(x, fma(x, -0.2916666666666667, -0.4166666666666667), -0.5), -1.0), x, -1.0)
end

Wolfram

code[x_] := N[(N[(x * N[(x * N[(x * -0.2916666666666667 + -0.4166666666666667), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision] * x + -1.0), $MachinePrecision]

Derivation

1. Initial program 2.9%

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

4. Applied egg-rr 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-log1p.f64 N/A

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

   4. lift-log1p.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-log1p.f64 N/A

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

   7. lift-log1p.f64 N/A

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

   8. *-inverses N/A

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

   9. lower--.f64 100.0

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

   10. lift-neg.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. neg-mul-1 N/A

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

   14. lower-*.f64 N/A

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

   15. neg-mul-1 N/A

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

   16. lower-neg.f64 100.0

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(\frac{-7}{24} \cdot x - \frac{5}{12}\right) - \frac{1}{2}\right) - 1\right)} - 1 \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(\frac{-7}{24} \cdot x - \frac{5}{12}\right) - \frac{1}{2}\right) - 1\right)} - 1 \]

   2. sub-neg N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{-7}{24} \cdot x - \frac{5}{12}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} - 1 \]

   3. metadata-eval N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \left(\frac{-7}{24} \cdot x - \frac{5}{12}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right) - 1 \]

   4. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-7}{24} \cdot x - \frac{5}{12}\right) - \frac{1}{2}, -1\right)} - 1 \]

   5. sub-neg N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{-7}{24} \cdot x - \frac{5}{12}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right) - 1 \]

   6. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot \left(\frac{-7}{24} \cdot x - \frac{5}{12}\right) + \color{blue}{\frac{-1}{2}}, -1\right) - 1 \]

   7. lower-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{-7}{24} \cdot x - \frac{5}{12}, \frac{-1}{2}\right)}, -1\right) - 1 \]

   8. sub-neg N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-7}{24} \cdot x + \left(\mathsf{neg}\left(\frac{5}{12}\right)\right)}, \frac{-1}{2}\right), -1\right) - 1 \]

   9. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-7}{24}} + \left(\mathsf{neg}\left(\frac{5}{12}\right)\right), \frac{-1}{2}\right), -1\right) - 1 \]

   10. metadata-eval N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{-7}{24} + \color{blue}{\frac{-5}{12}}, \frac{-1}{2}\right), -1\right) - 1 \]

   11. lower-fma.f64 100.0

       \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.2916666666666667, -0.4166666666666667\right)}, -0.5\right), -1\right) - 1 \]

9. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.2916666666666667, -0.4166666666666667\right), -0.5\right), -1\right)} - 1 \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-7}{24}, \frac{-5}{12}\right)} + \frac{-1}{2}\right) + -1\right) - 1 \]

    2. lift-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-7}{24}, \frac{-5}{12}\right), \frac{-1}{2}\right)} + -1\right) - 1 \]

    3. lift-fma.f64 N/A

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-7}{24}, \frac{-5}{12}\right), \frac{-1}{2}\right), -1\right)} - 1 \]

    4. lift-*.f64 N/A

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-7}{24}, \frac{-5}{12}\right), \frac{-1}{2}\right), -1\right)} - 1 \]

    5. sub-neg N/A

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-7}{24}, \frac{-5}{12}\right), \frac{-1}{2}\right), -1\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

    6. lift-*.f64 N/A

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-7}{24}, \frac{-5}{12}\right), \frac{-1}{2}\right), -1\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-7}{24}, \frac{-5}{12}\right), \frac{-1}{2}\right), -1\right) \cdot x} + \left(\mathsf{neg}\left(1\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-7}{24}, \frac{-5}{12}\right), \frac{-1}{2}\right), -1\right) \cdot x + \color{blue}{-1} \]

    9. lower-fma.f64 100.0

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.2916666666666667, -0.4166666666666667\right), -0.5\right), -1\right), x, -1\right)} \]

11. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.2916666666666667, -0.4166666666666667\right), -0.5\right), -1\right), x, -1\right)} \]
12. Add Preprocessing

Alternative 3: 99.4% accurate, 11.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(x, fma(x, fma(x, -0.4166666666666667, -0.5), -1.0), -1.0)
end

Wolfram

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

Derivation

1. Initial program 2.9%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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 99.9

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

5. Simplified 99.9%

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

Alternative 4: 99.3% accurate, 16.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.9%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 99.9

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

5. Simplified 99.9%

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

Alternative 5: 98.9% 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 2.9%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 99.7

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

5. Simplified 99.7%

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

Alternative 6: 97.8% 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 2.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 98.9%

   \[\leadsto \color{blue}{-1} \]
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 2024207 
(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))))