qlog (example 3.10)

Percentage Accurate: 4.1% → 100.0%

Time: 9.1s

Alternatives: 8

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: 100.0% accurate, 1.0× speedup

Math

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

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(Float64(-x) * 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 3.5%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. lift-+.f64 N/A

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

   6. log-div N/A

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

   7. lift-log.f64 N/A

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

   8. div-sub N/A

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

   9. lower--.f64 N/A

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

4. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-sub N/A

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

   6. frac-times N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. pow-div N/A

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

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} \cdot \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)} - {\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{0}} \]

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

6. Applied rewrites 50.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

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

   7. unpow1 N/A

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

   8. unpow1 N/A

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

   9. pow-div N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} \cdot {\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{0}} - 1 \]

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} - 1 \]
10. Add Preprocessing

Alternative 2: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (((((((((-0.25 * x) * x) - 0.3333333333333333) * (x * x)) - 0.5) * (x * x)) - 1.0) * (x * x)) / Math.log1p(x)) - 1.0;
}

Python

def code(x):
	return (((((((((-0.25 * x) * x) - 0.3333333333333333) * (x * x)) - 0.5) * (x * x)) - 1.0) * (x * x)) / math.log1p(x)) - 1.0

Julia

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

Wolfram

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

Derivation

1. Initial program 3.5%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. lift-+.f64 N/A

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

   6. log-div N/A

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

   7. lift-log.f64 N/A

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

   8. div-sub N/A

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

   9. lower--.f64 N/A

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

4. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-sub N/A

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

   6. frac-times N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. pow-div N/A

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

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} \cdot \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)} - {\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{0}} \]

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

6. Applied rewrites 50.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

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

   7. unpow1 N/A

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

   8. unpow1 N/A

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

   9. pow-div N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} \cdot {\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{0}} - 1 \]

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

    \[\leadsto \frac{\left(\left(\left(\left(-0.25 \cdot x\right) \cdot x - 0.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.5\right) \cdot \left(x \cdot x\right) - 1\right) \cdot \left(x \cdot x\right)}{\mathsf{log1p}\left(x\right)} - 1 \]
13. Add Preprocessing

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (((((((-0.3333333333333333 * (x * x)) - 0.5) * x) * x) - 1.0) * (x * x)) / Math.log1p(x)) - 1.0;
}

Python

def code(x):
	return (((((((-0.3333333333333333 * (x * x)) - 0.5) * x) * x) - 1.0) * (x * x)) / math.log1p(x)) - 1.0

Julia

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

Wolfram

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

Derivation

1. Initial program 3.5%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. lift-+.f64 N/A

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

   6. log-div N/A

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

   7. lift-log.f64 N/A

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

   8. div-sub N/A

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

   9. lower--.f64 N/A

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

4. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-sub N/A

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

   6. frac-times N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. pow-div N/A

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

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} \cdot \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)} - {\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{0}} \]

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

6. Applied rewrites 50.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

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

   7. unpow1 N/A

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

   8. unpow1 N/A

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

   9. pow-div N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} \cdot {\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{0}} - 1 \]

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower--.f64 N/A

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

    4. *-commutative N/A

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

    5. unpow2 N/A

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

    6. associate-*r* N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lower--.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. unpow2 N/A

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

    14. lower-*.f64 99.7

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

    \[\leadsto \frac{\left(\left(\left(-0.3333333333333333 \cdot \left(x \cdot x\right) - 0.5\right) \cdot x\right) \cdot x - 1\right) \cdot \left(x \cdot x\right)}{\mathsf{log1p}\left(x\right)} - 1 \]
13. Add Preprocessing

Alternative 4: 99.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((((((-0.2916666666666667d0) * x) - 0.4166666666666667d0) * x) - 0.5d0) * x) - 1.0d0) * x) - 1.0d0
end function

Java

public static double code(double x) {
	return (((((((-0.2916666666666667 * x) - 0.4166666666666667) * x) - 0.5) * x) - 1.0) * x) - 1.0;
}

Python

def code(x):
	return (((((((-0.2916666666666667 * x) - 0.4166666666666667) * x) - 0.5) * x) - 1.0) * x) - 1.0

Julia

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

MATLAB

function tmp = code(x)
	tmp = (((((((-0.2916666666666667 * x) - 0.4166666666666667) * x) - 0.5) * x) - 1.0) * x) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 3.5%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. lift-+.f64 N/A

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

   6. log-div N/A

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

   7. lift-log.f64 N/A

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

   8. div-sub N/A

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

   9. lower--.f64 N/A

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

4. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-sub N/A

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

   6. frac-times N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. pow-div N/A

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

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} \cdot \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)} - {\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{0}} \]

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

6. Applied rewrites 50.4%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 5: 99.5% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (((((-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0;
}

Python

def code(x):
	return (((((-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.5%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 6: 99.4% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (((-0.5 * x) - 1.0) * x) - 1.0;
}

Python

def code(x):
	return (((-0.5 * x) - 1.0) * x) - 1.0

Julia

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

MATLAB

function tmp = code(x)
	tmp = (((-0.5 * x) - 1.0) * x) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 3.5%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 7: 99.0% accurate, 36.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	return -x - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -x - 1.0d0
end function

Java

public static double code(double x) {
	return -x - 1.0;
}

Python

def code(x):
	return -x - 1.0

Julia

function code(x)
	return Float64(Float64(-x) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = -x - 1.0;
end

Wolfram

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

Derivation

1. Initial program 3.5%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Final simplification 99.3%

   \[\leadsto \left(-x\right) - 1 \]
7. Add Preprocessing

Alternative 8: 97.9% 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 3.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{-1} \]
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 2024313 
(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))))