Numeric.Integration.TanhSinh:nonNegative from integration-0.2.1

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 7

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: 99.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -0.5) {
		tmp = -1.0 - pow(x, -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)) <= -0.5)
		tmp = Float64(-1.0 - (x ^ -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], -0.5], N[(-1.0 - N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision], 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)) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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-rgt-in N/A

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

      3. *-lft-identity N/A

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

      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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -0.5) {
		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)) <= -0.5)
		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], -0.5], -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)) < -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 98.8%

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

   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-rgt-in N/A

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

      3. *-lft-identity N/A

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

      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 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 4: 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 98.8%

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

   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 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 5: 97.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = x * 1.0;
	}
	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 * 1.0d0
    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 * 1.0;
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = Float64(x * 1.0);
	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 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(x * 1.0), $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 98.8%

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

   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 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 98.1%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 - x} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (/ x (- 1.0 x)) -5e-149) -1.0 (* x x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -5e-149) {
		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)) <= (-5d-149)) 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)) <= -5e-149) {
		tmp = -1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (1.0 - x)) <= -5e-149:
		tmp = -1.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 - x)) <= -5e-149)
		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)) <= -5e-149)
		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], -5e-149], -1.0, N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (-.f64 #s(literal 1 binary64) x)) < -4.99999999999999968e-149

   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 82.7%

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

   if -4.99999999999999968e-149 < (/.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

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

   8. Applied rewrites 7.3%

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

Alternative 7: 51.2% 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 49.6%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (x)
  :name "Numeric.Integration.TanhSinh:nonNegative from integration-0.2.1"
  :precision binary64
  (/ x (- 1.0 x)))