qlog (example 3.10)

Percentage Accurate: 4.2% → 100.0%

Time: 10.5s

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: 4.2% 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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (log1p((x * -x)) / log1p(x)) - 1.0;
}

Java

public static double code(double x) {
	return (Math.log1p((x * -x)) / Math.log1p(x)) - 1.0;
}

Python

def code(x):
	return (math.log1p((x * -x)) / math.log1p(x)) - 1.0

Julia

function code(x)
	return Float64(Float64(log1p(Float64(x * Float64(-x))) / log1p(x)) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 3.6%

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

Alternative 2: 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 3.6%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 4.6

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

4. Applied rewrites 4.6%

   \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-x\right)}}{\log \left(1 + x\right)} \]
5. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-log1p.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 3: 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 3.6%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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 4: 99.6% accurate, 1.5× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[Log[1 + x], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 3.6%

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

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.9

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

10. Applied rewrites 99.9%

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

Alternative 5: 99.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(x * fma(x, fma(x, fma(x, -0.2916666666666667, -0.4166666666666667), -0.5), -1.0)) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 3.6%

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

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(\frac{-7}{24} \cdot x - \frac{5}{12}\right) - \frac{1}{2}\right) - 1\right)} - 1 \]
9. 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 99.8

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

10. Applied rewrites 99.8%

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

Alternative 6: 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 3.6%

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

Alternative 7: 99.2% 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 3.6%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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 99.6

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

5. Applied rewrites 99.6%

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

Alternative 8: 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 3.6%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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 99.3

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

5. Applied rewrites 99.3%

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

Alternative 9: 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 3.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 98.0%

   \[\leadsto \color{blue}{-1} \]
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 2024233 
(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))))