Numeric.Integration.TanhSinh:nonNegative from integration-0.2.1

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 6

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: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   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 99.2%

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

   if -1 < (/.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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 3: 97.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -1.0) {
		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)) <= -1.0)
		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], -1.0], -1.0, N[(x * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 99.2%

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

   if -1 < (/.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 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 4: 97.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   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 99.2%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(1 - x\right) - \left(-1 - x\right) \cdot \left(-x\right)}{\left(-1 - x\right) \cdot \left(1 - x\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \left(1 - x\right) - \left(-1 - x\right) \cdot \left(-x\right)}{\left(-1 - x\right) \cdot \left(1 - x\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{0 \cdot \left(1 - x\right) - \left(-1 - x\right) \cdot \left(-x\right)}}{\left(-1 - x\right) \cdot \left(1 - x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{0 \cdot \left(1 - x\right)} - \left(-1 - x\right) \cdot \left(-x\right)}{\left(-1 - x\right) \cdot \left(1 - x\right)} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0} - \left(-1 - x\right) \cdot \left(-x\right)}{\left(-1 - x\right) \cdot \left(1 - x\right)} \]

      5. neg-sub0 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. remove-double-neg N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 96.6%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 - x} \leq -1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 - x} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (/ x (- 1.0 x)) -2e-151) -1.0 (* x x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -2e-151) {
		tmp = -1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x / (1.0d0 - x)) <= (-2d-151)) then
        tmp = -1.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -2e-151) {
		tmp = -1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (1.0 - x)) <= -2e-151:
		tmp = -1.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 - x)) <= -2e-151)
		tmp = -1.0;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x / (1.0 - x)) <= -2e-151)
		tmp = -1.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -2e-151], -1.0, N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (-.f64 #s(literal 1 binary64) x)) < -1.9999999999999999e-151

   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 85.9%

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

   if -1.9999999999999999e-151 < (/.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 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 7.1%

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

Alternative 6: 50.4% 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 49.8%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (x)
  :name "Numeric.Integration.TanhSinh:nonNegative from integration-0.2.1"
  :precision binary64
  (/ x (- 1.0 x)))