Numeric.Integration.TanhSinh:nonNegative from integration-0.2.1

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

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, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (- 1.0 x)) -0.5) (- (/ -1.0 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 / 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(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[(N[(-1.0 / x), $MachinePrecision] - 1.0), $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. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   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

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

      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 99.4

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

   5. Applied rewrites 99.4%

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

Reproduce

herbie shell --seed 2024229 
(FPCore (x)
  :name "Numeric.Integration.TanhSinh:nonNegative from integration-0.2.1"
  :precision binary64
  (/ x (- 1.0 x)))