Numeric.Integration.TanhSinh:nonNegative from integration-0.2.1

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{1 - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (- 1.0 x)))

C

double code(double x) {
	return x / (1.0 - x);
}

Fortran

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

Java

public static double code(double x) {
	return x / (1.0 - x);
}

Python

def code(x):
	return x / (1.0 - x)

Julia

function code(x)
	return Float64(x / Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 - x);
end

Wolfram

code[x_] := N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (- 1.0 x)))

C

double code(double x) {
	return x / (1.0 - x);
}

Fortran

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

Java

public static double code(double x) {
	return x / (1.0 - x);
}

Python

def code(x):
	return x / (1.0 - x)

Julia

function code(x)
	return Float64(x / Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 - x);
end

Wolfram

code[x_] := N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (- 1.0 x)))

C

double code(double x) {
	return x / (1.0 - x);
}

Fortran

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

Java

public static double code(double x) {
	return x / (1.0 - x);
}

Python

def code(x):
	return x / (1.0 - x)

Julia

function code(x)
	return Float64(x / Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 - x);
end

Wolfram

code[x_] := N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x}{1 - x} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 - x} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x, x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (- 1.0 x)) -0.5) -1.0 (fma x (fma x x x) x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = fma(x, fma(x, x, x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = fma(x, fma(x, x, x), x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (-.f64 #s(literal 1 binary64) x)) < -0.5

   1. Initial program 100.0%

      \[\frac{x}{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.6%

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

   if -0.5 < (/.f64 x (-.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{x}{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x, x\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 - x} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (- 1.0 x)) -0.5) -1.0 (fma x x x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = fma(x, x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = fma(x, x, x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (-.f64 #s(literal 1 binary64) x)) < -0.5

   1. Initial program 100.0%

      \[\frac{x}{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.6%

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

   if -0.5 < (/.f64 x (-.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{x}{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 - x} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (/ x (- 1.0 x)) -0.5) -1.0 x))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x / (1.0d0 - x)) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (1.0 - x)) <= -0.5:
		tmp = -1.0
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x / (1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (-.f64 #s(literal 1 binary64) x)) < -0.5

   1. Initial program 100.0%

      \[\frac{x}{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.6%

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

   if -0.5 < (/.f64 x (-.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{x}{1 - x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} \]

      2. lower-*.f64 5.7

         \[\leadsto \color{blue}{x \cdot x} \]

   8. Applied rewrites 5.7%

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

      1. rem-exp-log N/A

         \[\leadsto \color{blue}{e^{\log x}} \cdot x \]

      2. rem-exp-log N/A

         \[\leadsto e^{\log x} \cdot \color{blue}{e^{\log x}} \]

      3. prod-exp N/A

         \[\leadsto \color{blue}{e^{\log x + \log x}} \]

      4. flip-+ N/A

         \[\leadsto e^{\color{blue}{\frac{\log x \cdot \log x - \log x \cdot \log x}{\log x - \log x}}} \]

      5. +-inverses N/A

         \[\leadsto e^{\frac{\color{blue}{0}}{\log x - \log x}} \]

      6. +-inverses N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. flip-+ N/A

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

      10. log-prod N/A

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

      11. sqr-pow N/A

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

      12. metadata-eval N/A

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

      13. unpow1 N/A

          \[\leadsto e^{\log \color{blue}{x}} \]

      14. rem-exp-log 98.4

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

   10. Applied rewrites 98.4%

       \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 52.0% accurate, 15.0× 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 100.0%

   \[\frac{x}{1 - x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Applied rewrites 50.8%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "Numeric.Integration.TanhSinh:nonNegative from integration-0.2.1"
  :precision binary64
  (/ x (- 1.0 x)))