qlog (example 3.10)

Percentage Accurate: 18.3% → 99.8%

Time: 27.5s

Alternatives: 6

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: 18.3% 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: 99.8% 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 18.3%

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

   1. sub-neg 18.3%

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

   2. log1p-def 15.6%

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

   3. log1p-def 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)} \]
6. Add Preprocessing

Alternative 2: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(-0.5 \cdot \left(x \cdot x\right) + -0.4166666666666667 \cdot {x}^{3}\right) - x\right) + -1 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (+ (* -0.5 (* x x)) (* -0.4166666666666667 (pow x 3.0))) x) -1.0))

C

double code(double x) {
	return (((-0.5 * (x * x)) + (-0.4166666666666667 * pow(x, 3.0))) - x) + -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((-0.5d0) * (x * x)) + ((-0.4166666666666667d0) * (x ** 3.0d0))) - x) + (-1.0d0)
end function

Java

public static double code(double x) {
	return (((-0.5 * (x * x)) + (-0.4166666666666667 * Math.pow(x, 3.0))) - x) + -1.0;
}

Python

def code(x):
	return (((-0.5 * (x * x)) + (-0.4166666666666667 * math.pow(x, 3.0))) - x) + -1.0

Julia

function code(x)
	return Float64(Float64(Float64(Float64(-0.5 * Float64(x * x)) + Float64(-0.4166666666666667 * (x ^ 3.0))) - x) + -1.0)
end

MATLAB

function tmp = code(x)
	tmp = (((-0.5 * (x * x)) + (-0.4166666666666667 * (x ^ 3.0))) - x) + -1.0;
end

Wolfram

code[x_] := N[(N[(N[(N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.4166666666666667 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 18.3%

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

3. Taylor expanded in x around 0 97.7%

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

   1. unpow2 96.9%

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

5. Applied egg-rr 97.7%

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

6. Final simplification 97.7%

   \[\leadsto \left(\left(-0.5 \cdot \left(x \cdot x\right) + -0.4166666666666667 \cdot {x}^{3}\right) - x\right) + -1 \]
7. Add Preprocessing

Alternative 3: 97.2% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

3. Taylor expanded in x around 0 96.9%

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

   1. expm1-log1p-u 96.9%

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

   2. expm1-udef 96.9%

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

5. Applied egg-rr 96.9%

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

   1. associate--l+ 96.9%

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

7. Simplified 96.9%

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

   1. unpow2 96.9%

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

9. Applied egg-rr 96.9%

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

10. Final simplification 96.9%

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

Alternative 4: 97.2% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

3. Taylor expanded in x around 0 96.9%

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

   1. unpow2 96.9%

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

5. Applied egg-rr 96.9%

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

6. Final simplification 96.9%

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

Alternative 5: 95.7% 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 18.3%

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

3. Taylor expanded in x around 0 95.2%

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

   1. sub-neg 95.2%

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

   2. metadata-eval 95.2%

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

   3. +-commutative 95.2%

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

   4. mul-1-neg 95.2%

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

   5. unsub-neg 95.2%

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

5. Simplified 95.2%

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

6. Final simplification 95.2%

   \[\leadsto -1 - x \]
7. Add Preprocessing

Alternative 6: 90.7% 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 18.3%

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

3. Taylor expanded in x around 0 90.7%

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

4. Final simplification 90.7%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 99.8% 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 2024024 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (<= (fabs x) 1.0)

  :herbie-target
  (/ (log1p (- x)) (log1p x))

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