qlog (example 3.10)

Percentage Accurate: 4.0% → 100.0%

Time: 10.6s

Alternatives: 9

Speedup: 218.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 4.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.8%

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

   1. flip-- N/A

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

   2. lift-+.f64 N/A

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

   3. frac-2neg N/A

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

   4. frac-2neg N/A

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

   5. lift-+.f64 N/A

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

   6. flip3-+ N/A

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

   7. distribute-neg-frac N/A

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

   8. distribute-frac-neg2 N/A

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

   9. frac-2neg N/A

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

   10. flip3-+ N/A

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

   11. lift-+.f64 N/A

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

   12. log-div N/A

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

   13. lift-log.f64 N/A

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

   14. lower--.f64 N/A

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

4. Applied rewrites 5.6%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-log1p.f64 N/A

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

   6. lift-log1p.f64 N/A

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

   7. div-sub N/A

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

   8. *-inverses N/A

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

   9. lower--.f64 N/A

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.8%

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

   1. lower-log1p.f64 6.7

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

4. Applied rewrites 6.7%

   \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\mathsf{log1p}\left(x\right)}} \]
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. lift-neg.f64 N/A

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

   3. lower-log1p.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 3: 99.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * N[(N[(-1.0 + N[(x * N[(x * N[(x * -0.25 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $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 4.8%

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

   1. lower-log1p.f64 6.7

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

4. Applied rewrites 6.7%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right)}{\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)}} \]
9. 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 99.6

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

10. Applied rewrites 99.6%

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

    1. lift-fma.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. lift-fma.f64 N/A

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

    5. distribute-rgt-in N/A

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

    6. associate-+r+ N/A

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

    7. lower-+.f64 N/A

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

    8. lower-+.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 N/A

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

    11. lift-fma.f64 N/A

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

    12. metadata-eval N/A

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

    13. sub-neg N/A

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

    14. lower-*.f64 N/A

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

    15. sub-neg N/A

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

    16. metadata-eval N/A

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

    17. lift-fma.f64 N/A

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

    18. *-commutative N/A

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

    19. lower-*.f64 99.6

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

12. Applied rewrites 99.6%

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

Alternative 4: 99.5% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* x (fma (fma x -0.25 -0.3333333333333333) (* x x) (fma x -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(fma(x, -0.25, -0.3333333333333333), (x * x), fma(x, -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(fma(x, -0.25, -0.3333333333333333), Float64(x * x), fma(x, -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[(N[(x * -0.25 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * -0.5 + -1.0), $MachinePrecision]), $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 4.8%

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

   1. lower-log1p.f64 6.7

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

4. Applied rewrites 6.7%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right)}{\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)}} \]
9. 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 99.6

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

10. Applied rewrites 99.6%

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

    1. lift-fma.f64 N/A

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

    2. distribute-rgt-in N/A

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

    3. associate-+l+ N/A

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

    4. lift-fma.f64 N/A

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

    5. metadata-eval N/A

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

    6. sub-neg N/A

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

    7. *-commutative N/A

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

    8. associate-*l* N/A

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

    9. lift-*.f64 N/A

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

    10. lower-fma.f64 N/A

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

    11. sub-neg N/A

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

    12. metadata-eval N/A

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

    13. lift-fma.f64 N/A

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

    14. *-commutative N/A

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

    15. lower-fma.f64 99.6

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

12. Applied rewrites 99.6%

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

Alternative 5: 99.5% 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 4.8%

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

   1. lower-log1p.f64 6.7

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

4. Applied rewrites 6.7%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right)}{\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)}} \]
9. 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 99.6

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

10. Applied rewrites 99.6%

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

Alternative 6: 99.4% accurate, 11.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.8%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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 99.5

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

5. Applied rewrites 99.5%

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

Alternative 7: 99.2% accurate, 16.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.8%

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

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

5. Applied rewrites 99.3%

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

Alternative 8: 98.9% accurate, 54.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 - x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 4.8%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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 98.9

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

5. Applied rewrites 98.9%

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

Alternative 9: 97.8% accurate, 218.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 4.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 97.7%

   \[\leadsto \color{blue}{-1} \]
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 2024214 
(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))))