qlog (example 3.10)

Percentage Accurate: 3.8% → 100.0%

Time: 7.0s

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: 3.8% 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 2.5%

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

   1. sub-neg 2.5%

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

   2. log1p-def 4.1%

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

   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: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.5%

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

   1. sub-neg 2.5%

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

   2. log1p-def 4.1%

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

   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. Taylor expanded in x around 0 99.7%

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

   1. associate--l+ 99.7%

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

   2. +-commutative 99.7%

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

   3. mul-1-neg 99.7%

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

   4. unsub-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. unpow2 99.7%

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

   7. associate-*l* 99.7%

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

   8. fma-neg 99.7%

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

   9. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, -1\right) - x \]

Alternative 3: 99.4% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 2.5%

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

   1. sub-neg 2.5%

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

   2. log1p-def 4.1%

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

   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. Taylor expanded in x around 0 99.7%

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

   1. associate--l+ 99.7%

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

   2. +-commutative 99.7%

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

   3. mul-1-neg 99.7%

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

   4. unsub-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. unpow2 99.7%

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

   7. associate-*l* 99.7%

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

   8. fma-neg 99.7%

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

   9. metadata-eval 99.7%

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

6. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, -1\right) - x} \]
7. Step-by-step derivation

   1. fma-udef 99.7%

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

   2. associate--l+ 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 99.1% 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 2.5%

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

   1. sub-neg 2.5%

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

   2. log1p-def 4.1%

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

   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. Taylor expanded in x around 0 99.4%

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

   1. sub-neg 99.4%

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

   2. metadata-eval 99.4%

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

   3. +-commutative 99.4%

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

   4. mul-1-neg 99.4%

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

   5. unsub-neg 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

   \[\leadsto -1 - x \]

Alternative 5: 98.1% 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 2.5%

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

   1. sub-neg 2.5%

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

   2. log1p-def 4.1%

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

   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. Taylor expanded in x around 0 98.5%

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

5. Final simplification 98.5%

   \[\leadsto -1 \]

Developer target: 99.6% 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 2023293 
(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))))