qlog (example 3.10)

Percentage Accurate: 4.2% → 100.0%

Time: 7.3s

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: 4.2% 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 5.3%

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

   1. sub-neg 5.3%

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

   2. log1p-def 6.0%

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

   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, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

   1. sub-neg 5.3%

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

   2. log1p-def 6.0%

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

   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. Taylor expanded in x around 0 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

   1. associate-+r+ 99.5%

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

   2. +-commutative 99.5%

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

   3. unpow2 99.5%

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

   4. *-commutative 99.5%

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

   5. associate-*r* 99.5%

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

   6. fma-udef 99.5%

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

   7. mul-1-neg 99.5%

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

   8. unsub-neg 99.5%

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

   9. fma-udef 99.5%

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

   10. *-commutative 99.5%

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

   11. unpow3 99.5%

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

   12. associate-*r* 99.5%

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

   13. *-commutative 99.5%

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

   14. associate-*r* 99.5%

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

   15. distribute-lft-out 99.5%

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

   16. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 99.2% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

   1. sub-neg 5.3%

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

   2. log1p-def 6.0%

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

   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. Taylor expanded in x around 0 99.5%

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

5. Taylor expanded in x around 0 99.2%

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

   1. unpow2 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*r* 99.2%

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

   4. mul-1-neg 99.2%

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

   5. sub-neg 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 4: 98.9% 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 5.3%

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

   1. sub-neg 5.3%

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

   2. log1p-def 6.0%

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

   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. Taylor expanded in x around 0 98.6%

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

   1. sub-neg 98.6%

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

   2. metadata-eval 98.6%

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

   3. +-commutative 98.6%

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

   4. mul-1-neg 98.6%

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

   5. unsub-neg 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

   \[\leadsto -1 - x \]

Alternative 5: 97.8% 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 5.3%

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

   1. sub-neg 5.3%

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

   2. log1p-def 6.0%

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

   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. Taylor expanded in x around 0 97.3%

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

5. Final simplification 97.3%

   \[\leadsto -1 \]

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 2023192 
(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))))