qlog (example 3.10)

Percentage Accurate: 4.0% → 100.0%

Time: 7.3s

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 4.5%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   3. *-lft-identity N/A

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

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

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

   5. distribute-lft-neg-in N/A

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

   6. *-lft-identity N/A

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

   7. lower-log1p.f64 N/A

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

   8. lower-neg.f64 5.3

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

4. Applied rewrites 5.3%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 100.0%

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

Alternative 2: 99.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3611111111111111, x, 0.3333333333333333\right), x, 0.25\right) \cdot x\right) \cdot x - 1}{\mathsf{fma}\left(\left(-0.25 \cdot x - 0.3333333333333333\right) \cdot x - 0.5, x, 1\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, x, 0.3333333333333333\right) \cdot x - 0.5, x, 1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (/
    (-
     (* (* (fma (fma 0.3611111111111111 x 0.3333333333333333) x 0.25) x) x)
     1.0)
    (fma (- (* (- (* -0.25 x) 0.3333333333333333) x) 0.5) x 1.0))
   x)
  (* (fma (- (* (fma -0.25 x 0.3333333333333333) x) 0.5) x 1.0) x)))

C

double code(double x) {
	return (((((fma(fma(0.3611111111111111, x, 0.3333333333333333), x, 0.25) * x) * x) - 1.0) / fma(((((-0.25 * x) - 0.3333333333333333) * x) - 0.5), x, 1.0)) * x) / (fma(((fma(-0.25, x, 0.3333333333333333) * x) - 0.5), x, 1.0) * x);
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(fma(fma(0.3611111111111111, x, 0.3333333333333333), x, 0.25) * x) * x) - 1.0) / fma(Float64(Float64(Float64(Float64(-0.25 * x) - 0.3333333333333333) * x) - 0.5), x, 1.0)) * x) / Float64(fma(Float64(Float64(fma(-0.25, x, 0.3333333333333333) * x) - 0.5), x, 1.0) * x))
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(N[(N[(0.3611111111111111 * x + 0.3333333333333333), $MachinePrecision] * x + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[(N[(N[(N[(-0.25 * x), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[(N[(N[(-0.25 * x + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 4.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 5.1

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

5. Applied rewrites 5.1%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 99.8

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

8. Applied rewrites 99.8%

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

10. Applied rewrites 99.8%

    \[\leadsto \frac{\frac{{\left(\left(\left(-0.25 \cdot x - 0.3333333333333333\right) \cdot x - 0.5\right) \cdot x\right)}^{2} - 1}{\mathsf{fma}\left(\left(-0.25 \cdot x - 0.3333333333333333\right) \cdot x - 0.5, x, 1\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, x, 0.3333333333333333\right) \cdot x - 0.5, x, 1\right) \cdot x} \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{\frac{{x}^{2} \cdot \left(\frac{1}{4} + x \cdot \left(\frac{1}{3} + \frac{13}{36} \cdot x\right)\right) - 1}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot x - \frac{1}{3}\right) \cdot x - \frac{1}{2}, x, 1\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, x, \frac{1}{3}\right) \cdot x - \frac{1}{2}, x, 1\right) \cdot x} \]
12. Step-by-step derivation

13. Applied rewrites 99.9%

    \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3611111111111111, x, 0.3333333333333333\right), x, 0.25\right) \cdot x\right) \cdot x - 1}{\mathsf{fma}\left(\left(-0.25 \cdot x - 0.3333333333333333\right) \cdot x - 0.5, x, 1\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, x, 0.3333333333333333\right) \cdot x - 0.5, x, 1\right) \cdot x} \]
14. Add Preprocessing

Alternative 3: 99.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 5.1

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

5. Applied rewrites 5.1%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 99.8

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

8. Applied rewrites 99.8%

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

10. Applied rewrites 99.8%

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

Alternative 4: 99.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 5.1

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

5. Applied rewrites 5.1%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 5: 99.5% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((((-0.4166666666666667d0) * x) - 0.5d0) * x) - 1.0d0) * x) - 1.0d0
end function

Java

public static double code(double x) {
	return (((((-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0;
}

Python

def code(x):
	return (((((-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = (((((-0.4166666666666667 * x) - 0.5) * x) - 1.0) * x) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 4.5%

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

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 6: 99.3% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((-0.5d0) * x) - 1.0d0) * x) - 1.0d0
end function

Java

public static double code(double x) {
	return (((-0.5 * x) - 1.0) * x) - 1.0;
}

Python

def code(x):
	return (((-0.5 * x) - 1.0) * x) - 1.0

Julia

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

MATLAB

function tmp = code(x)
	tmp = (((-0.5 * x) - 1.0) * x) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 4.5%

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

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 7: 99.0% accurate, 36.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 4.5%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 99.0

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

5. Applied rewrites 99.0%

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

Alternative 8: 98.0% 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 4.5%

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
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 2024339 
(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))))