qlog (example 3.10)

Percentage Accurate: 4.1% → 100.0%

Time: 10.8s

Alternatives: 8

Speedup: 218.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 4.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.8%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{\mathsf{log1p}\left(x \cdot \left(-x\right)\right)}{\mathsf{log1p}\left(x\right)} - \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)} \]
6. 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

4. Applied egg-rr 100.0%

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

   1. sub-div N/A

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

   2. diff-log N/A

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

   3. unsub-neg N/A

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

   4. metadata-eval N/A

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

   5. flip-- N/A

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

   6. /-lowering-/.f64 N/A

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

   7. sub-neg N/A

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

   8. accelerator-lowering-log1p.f64 N/A

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

   9. neg-lowering-neg.f64 N/A

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

   10. accelerator-lowering-log1p.f64 100.0

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

6. Applied egg-rr 100.0%

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

Alternative 3: 99.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(x * fma(x, fma(x, fma(x, -0.25, -0.3333333333333333), -0.5), -1.0)) / fma(fma(x, fma(x, -0.25, 0.3333333333333333), -0.5), Float64(x * x), 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 * N[(x * -0.25 + 0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 5.7

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

5. Simplified 5.7%

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

6. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f64 99.4

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

8. Simplified 99.4%

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

Alternative 4: 99.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 5.7

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

5. Simplified 5.7%

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

6. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f64 99.4

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

8. Simplified 99.4%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft1-in N/A

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

   4. times-frac N/A

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

   5. *-lowering-*.f64 N/A

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

10. Applied egg-rr 99.3%

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

    1. *-inverses N/A

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

    2. *-rgt-identity N/A

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

    3. /-lowering-/.f64 N/A

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

    4. accelerator-lowering-fma.f64 N/A

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

    5. metadata-eval N/A

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

    6. sub-neg N/A

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

    7. accelerator-lowering-fma.f64 N/A

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

    8. sub-neg N/A

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

    9. metadata-eval N/A

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

    10. accelerator-lowering-fma.f64 N/A

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

    11. accelerator-lowering-fma.f64 N/A

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

12. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), 1\right)}} \]
13. 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 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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 99.3

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

5. Simplified 99.3%

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

Alternative 6: 99.3% accurate, 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. accelerator-lowering-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. accelerator-lowering-fma.f64 99.1

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

5. Simplified 99.1%

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

Alternative 7: 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 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. --lowering--.f64 99.0

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

5. Simplified 99.0%

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

Alternative 8: 97.9% accurate, 218.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 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. Simplified 97.9%

   \[\leadsto \color{blue}{-1} \]
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 2024198 
(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))))