qlog (example 3.10)

Percentage Accurate: 4.0% → 100.0%

Time: 9.2s

Alternatives: 5

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 4.5%

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

   1. sub-neg 4.5%

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

   2. log1p-def 5.0%

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

   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: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(-1 - x\right) + {x}^{2} \cdot \left(-0.5 + x \cdot -0.4166666666666667\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- -1.0 x) (* (pow x 2.0) (+ -0.5 (* x -0.4166666666666667)))))

C

double code(double x) {
	return (-1.0 - x) + (pow(x, 2.0) * (-0.5 + (x * -0.4166666666666667)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-1.0d0) - x) + ((x ** 2.0d0) * ((-0.5d0) + (x * (-0.4166666666666667d0))))
end function

Java

public static double code(double x) {
	return (-1.0 - x) + (Math.pow(x, 2.0) * (-0.5 + (x * -0.4166666666666667)));
}

Python

def code(x):
	return (-1.0 - x) + (math.pow(x, 2.0) * (-0.5 + (x * -0.4166666666666667)))

Julia

function code(x)
	return Float64(Float64(-1.0 - x) + Float64((x ^ 2.0) * Float64(-0.5 + Float64(x * -0.4166666666666667))))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 - x) + ((x ^ 2.0) * (-0.5 + (x * -0.4166666666666667)));
end

Wolfram

code[x_] := N[(N[(-1.0 - x), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] * N[(-0.5 + N[(x * -0.4166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 4.5%

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

2. Taylor expanded in x around 0 99.6%

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

   1. sub-neg 99.6%

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

   2. metadata-eval 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r+ 99.6%

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

   5. mul-1-neg 99.6%

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

   6. neg-sub0 99.6%

      \[\leadsto \left(-1 + \color{blue}{\left(0 - x\right)}\right) + \left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) \]

   7. associate-+r- 99.6%

      \[\leadsto \color{blue}{\left(\left(-1 + 0\right) - x\right)} + \left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) \]

   8. metadata-eval 99.6%

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

   9. *-commutative 99.6%

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

   10. *-commutative 99.6%

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

   11. unpow3 99.6%

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

   12. unpow2 99.6%

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

   13. associate-*l* 99.6%

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

   14. distribute-lft-out 99.6%

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

4. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 99.3% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 4.5%

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

2. Taylor expanded in x around 0 99.6%

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

   1. sub-neg 99.6%

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

   2. metadata-eval 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r+ 99.6%

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

   5. mul-1-neg 99.6%

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

   6. neg-sub0 99.6%

      \[\leadsto \left(-1 + \color{blue}{\left(0 - x\right)}\right) + \left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) \]

   7. associate-+r- 99.6%

      \[\leadsto \color{blue}{\left(\left(-1 + 0\right) - x\right)} + \left(-0.5 \cdot {x}^{2} + -0.4166666666666667 \cdot {x}^{3}\right) \]

   8. metadata-eval 99.6%

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

   9. *-commutative 99.6%

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

   10. *-commutative 99.6%

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

   11. unpow3 99.6%

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

   12. unpow2 99.6%

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

   13. associate-*l* 99.6%

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

   14. distribute-lft-out 99.6%

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

4. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. expm1-log1p-u 96.4%

      \[\leadsto -0.5 \cdot {x}^{2} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 - x\right)\right)} \]

   3. expm1-udef 96.4%

      \[\leadsto -0.5 \cdot {x}^{2} + \color{blue}{\left(e^{\mathsf{log1p}\left(-1 - x\right)} - 1\right)} \]

   4. metadata-eval 96.4%

      \[\leadsto -0.5 \cdot {x}^{2} + \left(e^{\mathsf{log1p}\left(-1 - x\right)} - \color{blue}{\frac{2}{2}}\right) \]

   5. associate-+r- 96.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2} + e^{\mathsf{log1p}\left(-1 - x\right)}\right) - \frac{2}{2}} \]

   6. log1p-udef 96.4%

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

   7. metadata-eval 96.4%

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

   8. associate-+r- 46.8%

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

   9. metadata-eval 46.8%

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

   10. metadata-eval 46.8%

       \[\leadsto \left(-0.5 \cdot {x}^{2} + e^{\log \left(\color{blue}{0} - x\right)}\right) - \frac{2}{2} \]

   11. neg-sub0 46.8%

       \[\leadsto \left(-0.5 \cdot {x}^{2} + e^{\log \color{blue}{\left(-x\right)}}\right) - \frac{2}{2} \]

   12. mul-1-neg 46.8%

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

   13. add-exp-log 99.4%

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

   14. mul-1-neg 99.4%

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

   15. +-commutative 99.4%

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

   16. mul-1-neg 99.4%

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

   17. unpow2 99.4%

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

   18. associate-*r* 99.4%

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

   19. distribute-rgt-out 99.4%

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

   20. *-commutative 99.4%

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

   21. metadata-eval 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 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 4.5%

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

2. Taylor expanded in x around 0 99.0%

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

   1. sub-neg 99.0%

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

   2. metadata-eval 99.0%

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

   3. +-commutative 99.0%

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

   4. mul-1-neg 99.0%

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

   5. neg-sub0 99.0%

      \[\leadsto -1 + \color{blue}{\left(0 - x\right)} \]

   6. associate-+r- 99.0%

      \[\leadsto \color{blue}{\left(-1 + 0\right) - x} \]

   7. metadata-eval 99.0%

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

4. Simplified 99.0%

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

5. Final simplification 99.0%

   \[\leadsto -1 - x \]

Alternative 5: 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 4.5%

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

2. Taylor expanded in x around 0 98.0%

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

3. Final simplification 98.0%

   \[\leadsto -1 \]

Developer target: 99.4% 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 2023301 
(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))))