qlog (example 3.10)

Percentage Accurate: 4.0% → 100.0%

Time: 7.5s

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: 4.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{\mathsf{log1p}\left(\left(-x\right) \cdot x\right)}{\mathsf{log1p}\left(x\right)} - 1 \]
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 6.6%

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

4. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-div N/A

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

   5. lift-log1p.f64 N/A

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

   6. lift-log1p.f64 N/A

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

   7. diff-log N/A

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

   8. lift-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. flip-- N/A

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

   13. lower-/.f64 N/A

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

   14. sub-neg N/A

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

   15. lift-neg.f64 N/A

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

   16. lower-log1p.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 3: 99.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 7.9

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

5. Applied rewrites 7.9%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 99.5

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.5%

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

Alternative 4: 99.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 7.9

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

5. Applied rewrites 7.9%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 11.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 6: 99.3% accurate, 16.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 99.1

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

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, -1\right), x, -1\right)} \]
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 6.6%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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 98.4

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

5. Applied rewrites 98.4%

   \[\leadsto \color{blue}{-1 - x} \]
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 6.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 96.3%

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

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024302 
(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))))