qlog (example 3.10)

Percentage Accurate: 3.9% → 100.0%

Time: 11.2s

Alternatives: 9

Speedup: 218.0×

Specification

\[\left|x\right| \leq 1\]

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (log (- 1.0 x)) (log (+ 1.0 x))))

C

double code(double x) {
	return log((1.0 - x)) / log((1.0 + x));
}

Fortran

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

Java

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

Python

def code(x):
	return math.log((1.0 - x)) / math.log((1.0 + x))

Julia

function code(x)
	return Float64(log(Float64(1.0 - x)) / log(Float64(1.0 + x)))
end

MATLAB

function tmp = code(x)
	tmp = log((1.0 - x)) / log((1.0 + x));
end

Wolfram

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

Initial Program: 3.9% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(-x\right)\\ \frac{t\_0}{\mathsf{log1p}\left(x \cdot \left(-x\right)\right) - t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log1p (- x)))) (/ t_0 (- (log1p (* x (- x))) t_0))))

C

double code(double x) {
	double t_0 = log1p(-x);
	return t_0 / (log1p((x * -x)) - t_0);
}

Java

public static double code(double x) {
	double t_0 = Math.log1p(-x);
	return t_0 / (Math.log1p((x * -x)) - t_0);
}

Python

def code(x):
	t_0 = math.log1p(-x)
	return t_0 / (math.log1p((x * -x)) - t_0)

Julia

function code(x)
	t_0 = log1p(Float64(-x))
	return Float64(t_0 / Float64(log1p(Float64(x * Float64(-x))) - t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Log[1 + (-x)], $MachinePrecision]}, N[(t$95$0 / N[(N[Log[1 + N[(x * (-x)), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 2.7%

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

   1. lower-log1p.f64 5.4

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

4. Applied rewrites 5.4%

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

   1. flip-+ N/A

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

   2. lift--.f64 N/A

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

   3. log-div N/A

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

   4. lift-log.f64 N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. sub-neg N/A

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

   9. lower-log1p.f64 N/A

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

   10. lift-*.f64 N/A

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

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

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

   12. neg-mul-1 N/A

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

   13. lower-*.f64 N/A

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

   14. neg-mul-1 N/A

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

   15. lower-neg.f64 4.8

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

   16. lift-log.f64 N/A

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

   17. lift--.f64 N/A

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

   18. sub-neg N/A

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

   19. neg-mul-1 N/A

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

   20. lower-log1p.f64 N/A

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

   21. neg-mul-1 N/A

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

   22. lower-neg.f64 5.4

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

6. Applied rewrites 5.4%

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

   1. sub-neg N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-log1p.f64 100.0

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

8. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-x\right)}}{\mathsf{log1p}\left(x \cdot \left(-x\right)\right) - \mathsf{log1p}\left(-x\right)} \]
9. 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 2.7%

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

   1. lower-log1p.f64 5.4

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

4. Applied rewrites 5.4%

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

   1. sub-neg N/A

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

   2. neg-mul-1 N/A

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

   3. lower-log1p.f64 N/A

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

   4. neg-mul-1 N/A

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

   5. lower-neg.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 3: 99.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.16666666666666666, 0.1111111111111111\right), -0.25\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, -0.3333333333333333\right), 0.5\right)}, -1\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   x
   (fma
    x
    (*
     (fma (* x x) (fma x 0.16666666666666666 0.1111111111111111) -0.25)
     (/ 1.0 (fma x (fma x -0.25 -0.3333333333333333) 0.5)))
    -1.0))
  (fma (* x x) (fma x (fma x -0.25 0.3333333333333333) -0.5) x)))

C

double code(double x) {
	return (x * fma(x, (fma((x * x), fma(x, 0.16666666666666666, 0.1111111111111111), -0.25) * (1.0 / fma(x, fma(x, -0.25, -0.3333333333333333), 0.5))), -1.0)) / fma((x * x), fma(x, fma(x, -0.25, 0.3333333333333333), -0.5), x);
}

Julia

function code(x)
	return Float64(Float64(x * fma(x, Float64(fma(Float64(x * x), fma(x, 0.16666666666666666, 0.1111111111111111), -0.25) * Float64(1.0 / fma(x, fma(x, -0.25, -0.3333333333333333), 0.5))), -1.0)) / fma(Float64(x * x), fma(x, fma(x, -0.25, 0.3333333333333333), -0.5), x))
end

Wolfram

code[x_] := N[(N[(x * N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666 + 0.1111111111111111), $MachinePrecision] + -0.25), $MachinePrecision] * N[(1.0 / N[(x * N[(x * -0.25 + -0.3333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.25 + 0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.7%

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 3.9

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

5. Applied rewrites 3.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. flip-+ N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

10. Applied rewrites 99.7%

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

11. Taylor expanded in x around 0

    \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{9} + \frac{1}{6} \cdot x}, \frac{-1}{4}\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{4}, \frac{-1}{3}\right), \frac{1}{2}\right)}, -1\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{4}, \frac{1}{3}\right), \frac{-1}{2}\right), x\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{9}}, \frac{-1}{4}\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{4}, \frac{-1}{3}\right), \frac{1}{2}\right)}, -1\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{4}, \frac{1}{3}\right), \frac{-1}{2}\right), x\right)} \]

    2. *-commutative N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{9}, \frac{-1}{4}\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{4}, \frac{-1}{3}\right), \frac{1}{2}\right)}, -1\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{4}, \frac{1}{3}\right), \frac{-1}{2}\right), x\right)} \]

    3. lower-fma.f64 99.7

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.1111111111111111\right)}, -0.25\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, -0.3333333333333333\right), 0.5\right)}, -1\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), x\right)} \]

13. Applied rewrites 99.7%

    \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.1111111111111111\right)}, -0.25\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, -0.3333333333333333\right), 0.5\right)}, -1\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), x\right)} \]
14. Add Preprocessing

Alternative 4: 99.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.7%

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 3.9

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

5. Applied rewrites 3.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 5: 99.5% accurate, 11.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.7%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 6: 99.3% accurate, 14.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.7%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 7: 99.3% accurate, 16.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.7%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 8: 99.0% accurate, 54.5× speedup

Math

\[\begin{array}{l} \\ -1 - x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 - x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 2.7%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 9: 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 2.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 98.4%

   \[\leadsto \color{blue}{-1} \]
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 2024216 
(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))))