qlog (example 3.10)

Percentage Accurate: 4.0% → 100.0%

Time: 17.9s

Alternatives: 6

Speedup: 207.0×

Specification

\[-1 < x \land x < 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 4.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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 5.0%

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

   1. sub-neg 5.0%

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

   2. log1p-def 5.1%

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

   3. log1p-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ -1 + \left(\frac{x \cdot \left(x \cdot 0.25\right)}{-0.5 - x \cdot -0.4166666666666667} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ -1.0 (- (/ (* x (* x 0.25)) (- -0.5 (* x -0.4166666666666667))) x)))

C

double code(double x) {
	return -1.0 + (((x * (x * 0.25)) / (-0.5 - (x * -0.4166666666666667))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) + (((x * (x * 0.25d0)) / ((-0.5d0) - (x * (-0.4166666666666667d0)))) - x)
end function

Java

public static double code(double x) {
	return -1.0 + (((x * (x * 0.25)) / (-0.5 - (x * -0.4166666666666667))) - x);
}

Python

def code(x):
	return -1.0 + (((x * (x * 0.25)) / (-0.5 - (x * -0.4166666666666667))) - x)

Julia

function code(x)
	return Float64(-1.0 + Float64(Float64(Float64(x * Float64(x * 0.25)) / Float64(-0.5 - Float64(x * -0.4166666666666667))) - x))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + (((x * (x * 0.25)) / (-0.5 - (x * -0.4166666666666667))) - x);
end

Wolfram

code[x_] := N[(-1.0 + N[(N[(N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / N[(-0.5 - N[(x * -0.4166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.0%

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

2. Taylor expanded in x around 0 99.1%

   \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) - 1} \]
3. Step-by-step derivation

   1. sub-neg 99.1%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) + \left(-1\right)} \]

   2. metadata-eval 99.1%

      \[\leadsto \left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) + \color{blue}{-1} \]

   3. +-commutative 99.1%

      \[\leadsto \color{blue}{-1 + \left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right)} \]

   4. +-commutative 99.1%

      \[\leadsto -1 + \color{blue}{\left(\left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right) + -0.5 \cdot {x}^{2}\right)} \]

   5. associate-+l+ 99.1%

      \[\leadsto -1 + \color{blue}{\left(-1 \cdot x + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right)\right)} \]

   6. associate-+r+ 99.1%

      \[\leadsto \color{blue}{\left(-1 + -1 \cdot x\right) + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right)} \]

   7. mul-1-neg 99.1%

      \[\leadsto \left(-1 + \color{blue}{\left(-x\right)}\right) + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right) \]

   8. unsub-neg 99.1%

      \[\leadsto \color{blue}{\left(-1 - x\right)} + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right) \]

   9. *-commutative 99.1%

      \[\leadsto \left(-1 - x\right) + \left(\color{blue}{{x}^{3} \cdot -0.4166666666666667} + -0.5 \cdot {x}^{2}\right) \]

   10. unpow3 99.1%

       \[\leadsto \left(-1 - x\right) + \left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot -0.4166666666666667 + -0.5 \cdot {x}^{2}\right) \]

   11. unpow2 99.1%

       \[\leadsto \left(-1 - x\right) + \left(\left(\color{blue}{{x}^{2}} \cdot x\right) \cdot -0.4166666666666667 + -0.5 \cdot {x}^{2}\right) \]

   12. associate-*l* 99.1%

       \[\leadsto \left(-1 - x\right) + \left(\color{blue}{{x}^{2} \cdot \left(x \cdot -0.4166666666666667\right)} + -0.5 \cdot {x}^{2}\right) \]

   13. *-commutative 99.1%

       \[\leadsto \left(-1 - x\right) + \left({x}^{2} \cdot \left(x \cdot -0.4166666666666667\right) + \color{blue}{{x}^{2} \cdot -0.5}\right) \]

   14. distribute-lft-out 99.1%

       \[\leadsto \left(-1 - x\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot -0.4166666666666667 + -0.5\right)} \]

   15. unpow2 99.1%

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

4. Simplified 99.1%

   \[\leadsto \color{blue}{\left(-1 - x\right) + \left(x \cdot x\right) \cdot \left(x \cdot -0.4166666666666667 + -0.5\right)} \]
5. Step-by-step derivation

   1. +-commutative 99.1%

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

   2. flip-+ 99.1%

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

   3. metadata-eval 99.1%

      \[\leadsto \left(-1 - x\right) + \left(x \cdot x\right) \cdot \frac{\color{blue}{0.25} - \left(x \cdot -0.4166666666666667\right) \cdot \left(x \cdot -0.4166666666666667\right)}{-0.5 - x \cdot -0.4166666666666667} \]

   4. swap-sqr 99.1%

      \[\leadsto \left(-1 - x\right) + \left(x \cdot x\right) \cdot \frac{0.25 - \color{blue}{\left(x \cdot x\right) \cdot \left(-0.4166666666666667 \cdot -0.4166666666666667\right)}}{-0.5 - x \cdot -0.4166666666666667} \]

   5. metadata-eval 99.1%

      \[\leadsto \left(-1 - x\right) + \left(x \cdot x\right) \cdot \frac{0.25 - \left(x \cdot x\right) \cdot \color{blue}{0.1736111111111111}}{-0.5 - x \cdot -0.4166666666666667} \]

6. Applied egg-rr 99.1%

   \[\leadsto \left(-1 - x\right) + \left(x \cdot x\right) \cdot \color{blue}{\frac{0.25 - \left(x \cdot x\right) \cdot 0.1736111111111111}{-0.5 - x \cdot -0.4166666666666667}} \]
7. Step-by-step derivation

   1. associate-+l- 99.1%

      \[\leadsto \color{blue}{-1 - \left(x - \left(x \cdot x\right) \cdot \frac{0.25 - \left(x \cdot x\right) \cdot 0.1736111111111111}{-0.5 - x \cdot -0.4166666666666667}\right)} \]

   2. associate-*r/ 99.1%

      \[\leadsto -1 - \left(x - \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.1736111111111111\right)}{-0.5 - x \cdot -0.4166666666666667}}\right) \]

8. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{-1 - \left(x - \frac{\left(x \cdot x\right) \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.1736111111111111\right)}{-0.5 - x \cdot -0.4166666666666667}\right)} \]

9. Taylor expanded in x around 0 99.2%

   \[\leadsto -1 - \left(x - \frac{\color{blue}{0.25 \cdot {x}^{2}}}{-0.5 - x \cdot -0.4166666666666667}\right) \]
10. Step-by-step derivation

    1. unpow2 99.2%

       \[\leadsto -1 - \left(x - \frac{0.25 \cdot \color{blue}{\left(x \cdot x\right)}}{-0.5 - x \cdot -0.4166666666666667}\right) \]

    2. associate-*r* 99.2%

       \[\leadsto -1 - \left(x - \frac{\color{blue}{\left(0.25 \cdot x\right) \cdot x}}{-0.5 - x \cdot -0.4166666666666667}\right) \]

11. Simplified 99.2%

    \[\leadsto -1 - \left(x - \frac{\color{blue}{\left(0.25 \cdot x\right) \cdot x}}{-0.5 - x \cdot -0.4166666666666667}\right) \]

12. Final simplification 99.2%

    \[\leadsto -1 + \left(\frac{x \cdot \left(x \cdot 0.25\right)}{-0.5 - x \cdot -0.4166666666666667} - x\right) \]

Alternative 3: 99.5% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.0%

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

2. Taylor expanded in x around 0 99.1%

   \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) - 1} \]
3. Step-by-step derivation

   1. sub-neg 99.1%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) + \left(-1\right)} \]

   2. metadata-eval 99.1%

      \[\leadsto \left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) + \color{blue}{-1} \]

   3. +-commutative 99.1%

      \[\leadsto \color{blue}{-1 + \left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right)} \]

   4. +-commutative 99.1%

      \[\leadsto -1 + \color{blue}{\left(\left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right) + -0.5 \cdot {x}^{2}\right)} \]

   5. associate-+l+ 99.1%

      \[\leadsto -1 + \color{blue}{\left(-1 \cdot x + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right)\right)} \]

   6. associate-+r+ 99.1%

      \[\leadsto \color{blue}{\left(-1 + -1 \cdot x\right) + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right)} \]

   7. mul-1-neg 99.1%

      \[\leadsto \left(-1 + \color{blue}{\left(-x\right)}\right) + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right) \]

   8. unsub-neg 99.1%

      \[\leadsto \color{blue}{\left(-1 - x\right)} + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right) \]

   9. *-commutative 99.1%

      \[\leadsto \left(-1 - x\right) + \left(\color{blue}{{x}^{3} \cdot -0.4166666666666667} + -0.5 \cdot {x}^{2}\right) \]

   10. unpow3 99.1%

       \[\leadsto \left(-1 - x\right) + \left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot -0.4166666666666667 + -0.5 \cdot {x}^{2}\right) \]

   11. unpow2 99.1%

       \[\leadsto \left(-1 - x\right) + \left(\left(\color{blue}{{x}^{2}} \cdot x\right) \cdot -0.4166666666666667 + -0.5 \cdot {x}^{2}\right) \]

   12. associate-*l* 99.1%

       \[\leadsto \left(-1 - x\right) + \left(\color{blue}{{x}^{2} \cdot \left(x \cdot -0.4166666666666667\right)} + -0.5 \cdot {x}^{2}\right) \]

   13. *-commutative 99.1%

       \[\leadsto \left(-1 - x\right) + \left({x}^{2} \cdot \left(x \cdot -0.4166666666666667\right) + \color{blue}{{x}^{2} \cdot -0.5}\right) \]

   14. distribute-lft-out 99.1%

       \[\leadsto \left(-1 - x\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot -0.4166666666666667 + -0.5\right)} \]

   15. unpow2 99.1%

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

4. Simplified 99.1%

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

5. Final simplification 99.1%

   \[\leadsto \left(-1 - x\right) + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot -0.4166666666666667\right) \]

Alternative 4: 99.3% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.0%

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

2. Taylor expanded in x around 0 99.1%

   \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) - 1} \]
3. Step-by-step derivation

   1. sub-neg 99.1%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) + \left(-1\right)} \]

   2. metadata-eval 99.1%

      \[\leadsto \left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right) + \color{blue}{-1} \]

   3. +-commutative 99.1%

      \[\leadsto \color{blue}{-1 + \left(-0.5 \cdot {x}^{2} + \left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right)\right)} \]

   4. +-commutative 99.1%

      \[\leadsto -1 + \color{blue}{\left(\left(-1 \cdot x + -0.4166666666666667 \cdot {x}^{3}\right) + -0.5 \cdot {x}^{2}\right)} \]

   5. associate-+l+ 99.1%

      \[\leadsto -1 + \color{blue}{\left(-1 \cdot x + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right)\right)} \]

   6. associate-+r+ 99.1%

      \[\leadsto \color{blue}{\left(-1 + -1 \cdot x\right) + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right)} \]

   7. mul-1-neg 99.1%

      \[\leadsto \left(-1 + \color{blue}{\left(-x\right)}\right) + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right) \]

   8. unsub-neg 99.1%

      \[\leadsto \color{blue}{\left(-1 - x\right)} + \left(-0.4166666666666667 \cdot {x}^{3} + -0.5 \cdot {x}^{2}\right) \]

   9. *-commutative 99.1%

      \[\leadsto \left(-1 - x\right) + \left(\color{blue}{{x}^{3} \cdot -0.4166666666666667} + -0.5 \cdot {x}^{2}\right) \]

   10. unpow3 99.1%

       \[\leadsto \left(-1 - x\right) + \left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot -0.4166666666666667 + -0.5 \cdot {x}^{2}\right) \]

   11. unpow2 99.1%

       \[\leadsto \left(-1 - x\right) + \left(\left(\color{blue}{{x}^{2}} \cdot x\right) \cdot -0.4166666666666667 + -0.5 \cdot {x}^{2}\right) \]

   12. associate-*l* 99.1%

       \[\leadsto \left(-1 - x\right) + \left(\color{blue}{{x}^{2} \cdot \left(x \cdot -0.4166666666666667\right)} + -0.5 \cdot {x}^{2}\right) \]

   13. *-commutative 99.1%

       \[\leadsto \left(-1 - x\right) + \left({x}^{2} \cdot \left(x \cdot -0.4166666666666667\right) + \color{blue}{{x}^{2} \cdot -0.5}\right) \]

   14. distribute-lft-out 99.1%

       \[\leadsto \left(-1 - x\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot -0.4166666666666667 + -0.5\right)} \]

   15. unpow2 99.1%

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

4. Simplified 99.1%

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

5. Taylor expanded in x around 0 99.0%

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

6. Final simplification 99.0%

   \[\leadsto \left(-1 - x\right) + -0.5 \cdot \left(x \cdot x\right) \]

Alternative 5: 99.0% accurate, 69.0× 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 5.0%

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

2. Taylor expanded in x around 0 98.7%

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

   1. sub-neg 98.7%

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

   2. metadata-eval 98.7%

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

   3. +-commutative 98.7%

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

   4. mul-1-neg 98.7%

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

   5. unsub-neg 98.7%

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

4. Simplified 98.7%

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

5. Final simplification 98.7%

   \[\leadsto -1 - x \]

Alternative 6: 97.9% accurate, 207.0× 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 5.0%

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

2. Taylor expanded in x around 0 97.9%

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

3. Final simplification 97.9%

   \[\leadsto -1 \]

Developer target: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ -\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666667 \cdot {x}^{3}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0)))))

C

double code(double x) {
	return -(((1.0 + x) + ((x * x) / 2.0)) + (0.4166666666666667 * pow(x, 3.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -(((1.0d0 + x) + ((x * x) / 2.0d0)) + (0.4166666666666667d0 * (x ** 3.0d0)))
end function

Java

public static double code(double x) {
	return -(((1.0 + x) + ((x * x) / 2.0)) + (0.4166666666666667 * Math.pow(x, 3.0)));
}

Python

def code(x):
	return -(((1.0 + x) + ((x * x) / 2.0)) + (0.4166666666666667 * math.pow(x, 3.0)))

Julia

function code(x)
	return Float64(-Float64(Float64(Float64(1.0 + x) + Float64(Float64(x * x) / 2.0)) + Float64(0.4166666666666667 * (x ^ 3.0))))
end

MATLAB

function tmp = code(x)
	tmp = -(((1.0 + x) + ((x * x) / 2.0)) + (0.4166666666666667 * (x ^ 3.0)));
end

Wolfram

code[x_] := (-N[(N[(N[(1.0 + x), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(0.4166666666666667 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Reproduce

herbie shell --seed 2023279 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (- (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))