qlog (example 3.10)

Percentage Accurate: 4.1% → 100.0%

Time: 7.8s

Alternatives: 12

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} - \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)}} \]
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. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. *-inverses N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. metadata-eval N/A

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

   13. remove-double-neg N/A

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

   14. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-\mathsf{log1p}\left(\left(-x\right) \cdot x\right), \frac{-1}{\mathsf{log1p}\left(x\right)}, -1\right)} \]
7. 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 4.9%

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} - \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)}} \]
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. sub-div N/A

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

   5. lift-log1p.f64 N/A

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

   6. lift-log1p.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. diff-log N/A

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

   9. lift-*.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

   13. lift-+.f64 N/A

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

   14. flip-- N/A

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

   15. lift--.f64 N/A

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

   16. lift-log.f64 N/A

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

   17. lift-/.f64 6.7

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

6. Applied rewrites 100.0%

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

Alternative 3: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} - \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)}} \]
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. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. *-inverses N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. metadata-eval N/A

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

   13. remove-double-neg N/A

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

   14. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

9. Applied rewrites 99.6%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-neg.f64 N/A

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

    4. distribute-lft-neg-out N/A

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

    5. unsub-neg N/A

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

    6. lower--.f64 N/A

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

    7. lift-/.f64 N/A

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 4: 99.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} - \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)}} \]
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. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. *-inverses N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. metadata-eval N/A

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

   13. remove-double-neg N/A

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

   14. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.5

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

9. Applied rewrites 99.5%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-neg.f64 N/A

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

    4. distribute-lft-neg-out N/A

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

    5. unsub-neg N/A

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

    6. lower--.f64 N/A

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

    7. lift-/.f64 N/A

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

    8. lift-log1p.f64 N/A

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

    9. frac-2neg N/A

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

    10. metadata-eval N/A

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

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

Alternative 5: 99.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x \cdot x, -0.5\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, x, 0.08333333333333333\right), x, -0.5\right), x, -1\right)}{x}, -1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (- (* (* (fma (fma -0.3333333333333333 (* x x) -0.5) (* x x) -1.0) x) x))
  (/
   (fma (fma (fma -0.041666666666666664 x 0.08333333333333333) x -0.5) x -1.0)
   x)
  -1.0))

C

double code(double x) {
	return fma(-((fma(fma(-0.3333333333333333, (x * x), -0.5), (x * x), -1.0) * x) * x), (fma(fma(fma(-0.041666666666666664, x, 0.08333333333333333), x, -0.5), x, -1.0) / x), -1.0);
}

Julia

function code(x)
	return fma(Float64(-Float64(Float64(fma(fma(-0.3333333333333333, Float64(x * x), -0.5), Float64(x * x), -1.0) * x) * x)), Float64(fma(fma(fma(-0.041666666666666664, x, 0.08333333333333333), x, -0.5), x, -1.0) / x), -1.0)
end

Wolfram

code[x_] := N[((-N[(N[(N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]) * N[(N[(N[(N[(-0.041666666666666664 * x + 0.08333333333333333), $MachinePrecision] * x + -0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 4.9%

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} - \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)}} \]
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. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. *-inverses N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. metadata-eval N/A

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

   13. remove-double-neg N/A

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

   14. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.5

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \color{blue}{\frac{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right) - 1}{x}}, -1\right) \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \color{blue}{\frac{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right) - 1}{x}}, -1\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)}}{x}, -1\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right) \cdot x} + \left(\mathsf{neg}\left(1\right)\right)}{x}, -1\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}\right) \cdot x + \color{blue}{-1}}{x}, -1\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1}{12} + \frac{-1}{24} \cdot x\right) - \frac{1}{2}, x, -1\right)}}{x}, -1\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{12} + \frac{-1}{24} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, x, -1\right)}{x}, -1\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{-1}{24} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), x, -1\right)}{x}, -1\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{-1}{24} \cdot x\right) \cdot x + \color{blue}{\frac{-1}{2}}, x, -1\right)}{x}, -1\right) \]

    9. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{-1}{24} \cdot x, x, \frac{-1}{2}\right)}, x, -1\right)}{x}, -1\right) \]

    10. +-commutative N/A

        \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x \cdot x, \frac{-1}{2}\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{24} \cdot x + \frac{1}{12}}, x, \frac{-1}{2}\right), x, -1\right)}{x}, -1\right) \]

    11. lower-fma.f64 99.4

        \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x \cdot x, -0.5\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.041666666666666664, x, 0.08333333333333333\right)}, x, -0.5\right), x, -1\right)}{x}, -1\right) \]

12. Applied rewrites 99.4%

    \[\leadsto \mathsf{fma}\left(-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x \cdot x, -0.5\right), x \cdot x, -1\right) \cdot x\right) \cdot x, \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, x, 0.08333333333333333\right), x, -0.5\right), x, -1\right)}{x}}, -1\right) \]
13. Add Preprocessing

Alternative 6: 99.5% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 5.0

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

5. Applied rewrites 5.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, x, \frac{-1}{3}\right), x, \frac{-1}{2}\right), x, -1\right) \cdot x}{\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 99.3

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

8. Applied rewrites 99.3%

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

10. Applied rewrites 99.3%

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

Alternative 7: 99.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 5.0

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

5. Applied rewrites 5.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, x, \frac{-1}{3}\right), x, \frac{-1}{2}\right), x, -1\right) \cdot x}{\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 99.3

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

8. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, x, -0.3333333333333333\right), x, -0.5\right), x, -1\right) \cdot x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, x, 0.3333333333333333\right), x, -0.5\right), x, 1\right) \cdot x}} \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

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

Alternative 8: 99.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 5.0

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

5. Applied rewrites 5.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, x, \frac{-1}{3}\right), x, \frac{-1}{2}\right), x, -1\right) \cdot x}{\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 99.3

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

8. Applied rewrites 99.3%

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

Alternative 9: 99.4% accurate, 11.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 99.3

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

5. Applied rewrites 99.3%

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

Alternative 10: 99.3% accurate, 16.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.9%

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

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 99.0

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

5. Applied rewrites 99.0%

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

Alternative 11: 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 4.9%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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 98.7

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

5. Applied rewrites 98.7%

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

Alternative 12: 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 4.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 97.4%

   \[\leadsto \color{blue}{-1} \]
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 2024295 
(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))))