qlog (example 3.10)

Percentage Accurate: 4.2% → 100.0%

Time: 11.1s

Alternatives: 7

Speedup: 207.0×

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: 4.2% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.1%

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

3. Taylor expanded in x around inf 4.2%

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

   1. expm1-log1p-u 3.8%

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

   2. expm1-undefine 3.6%

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

   3. sub-neg 3.6%

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

   4. distribute-lft-in 4.1%

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

   5. div-inv 4.1%

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

   6. *-inverses 4.1%

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

   7. metadata-eval 4.1%

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

5. Applied egg-rr 4.1%

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

   1. sub-neg 4.1%

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

   2. metadata-eval 4.1%

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

   3. +-commutative 4.1%

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

   4. log1p-undefine 3.9%

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

   5. rem-exp-log 3.9%

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

   6. *-commutative 3.9%

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

   7. neg-mul-1 3.9%

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

   8. associate-+r+ 4.3%

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

   9. metadata-eval 4.3%

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

   10. sub-neg 4.3%

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

7. Simplified 4.3%

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

   1. add-log-exp 4.3%

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

   2. associate-+r- 4.1%

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

   3. metadata-eval 4.1%

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

   4. sub-neg 4.1%

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

   5. log1p-define 4.6%

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

   6. log1p-define 100.0%

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

9. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 4.1%

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

   1. sub-neg 4.1%

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

   2. log1p-define 4.6%

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

   3. log1p-define 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. Add Preprocessing
5. Add Preprocessing

Alternative 3: 99.6% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * ((x * ((x * ((x * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0)) + (-1.0d0))) / (x * (1.0d0 + (x * ((x * ((x * (-0.25d0)) + 0.3333333333333333d0)) - 0.5d0))))
end function

Java

public static double code(double x) {
	return (x * ((x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0)) / (x * (1.0 + (x * ((x * ((x * -0.25) + 0.3333333333333333)) - 0.5))));
}

Python

def code(x):
	return (x * ((x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0)) / (x * (1.0 + (x * ((x * ((x * -0.25) + 0.3333333333333333)) - 0.5))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * ((x * ((x * ((x * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0)) / (x * (1.0 + (x * ((x * ((x * -0.25) + 0.3333333333333333)) - 0.5))));
end

Wolfram

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

Derivation

1. Initial program 4.1%

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

3. Taylor expanded in x around 0 5.6%

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

4. Taylor expanded in x around 0 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot -0.25 - 0.3333333333333333\right) - 0.5\right) + -1\right)}{x \cdot \left(1 + x \cdot \left(x \cdot \left(x \cdot -0.25 + 0.3333333333333333\right) - 0.5\right)\right)} \]
6. Add Preprocessing

Alternative 4: 99.5% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 4.1%

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

3. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.6%

   \[\leadsto x \cdot \left(x \cdot \left(x \cdot -0.4166666666666667 - 0.5\right) + -1\right) + -1 \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 4.1%

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

3. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.5%

   \[\leadsto x \cdot \left(x \cdot -0.5 + -1\right) + -1 \]
5. Add Preprocessing

Alternative 6: 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 4.1%

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

3. Taylor expanded in x around 0 99.3%

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

   1. sub-neg 99.3%

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

   2. metadata-eval 99.3%

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

   3. +-commutative 99.3%

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

   4. mul-1-neg 99.3%

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

   5. unsub-neg 99.3%

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

5. Simplified 99.3%

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

Alternative 7: 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 4.1%

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

3. Taylor expanded in x around 0 98.2%

   \[\leadsto \color{blue}{-1} \]
4. 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 2024169 
(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))))