Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto \frac{x}{x + 1} + \frac{1}{x - 1} \]
4. Add Preprocessing

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{1}{x - 1} \leq -1:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -2, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ 1.0 (- x 1.0))) -1.0)
   (fma (* x x) -2.0 -1.0)
   (+ (/ 2.0 (* x x)) 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + (1.0 / (x - 1.0))) <= -1.0) {
		tmp = fma((x * x), -2.0, -1.0);
	} else {
		tmp = (2.0 / (x * x)) + 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(1.0 / Float64(x - 1.0))) <= -1.0)
		tmp = fma(Float64(x * x), -2.0, -1.0);
	else
		tmp = Float64(Float64(2.0 / Float64(x * x)) + 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(x * x), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision], N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-2 \cdot {x}^{2} - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

         \[\leadsto {x}^{2} \cdot -2 + \color{blue}{-1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, -2, -1\right)} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, -2, -1\right) \]

      6. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, -2, -1\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -2, -1\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{2}} + 1} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{2}} + 1} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + 1 \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{x}^{2}} + 1 \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{x}^{2}}} + 1 \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot x}} + 1 \]

      7. lower-*.f64 99.6

         \[\leadsto \frac{2}{\color{blue}{x \cdot x}} + 1 \]

   5. Applied rewrites 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{1}{x - 1} \leq -1:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -2, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.3%

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

Alternative 4: 51.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.3%

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

6. Taylor expanded in x around 0

   \[\leadsto -1 + \frac{x}{\color{blue}{1}} \]
7. Step-by-step derivation

8. Applied rewrites 48.0%

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

9. Final simplification 48.0%

   \[\leadsto \frac{x}{1} + -1 \]
10. Add Preprocessing

Alternative 5: 50.7% accurate, 32.0× 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%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1 + 2 \cdot \frac{1}{{x}^{2}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{2}} + 1} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{2}} + 1} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + 1 \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{2}}{{x}^{2}} + 1 \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{{x}^{2}}} + 1 \]

   6. unpow2 N/A

      \[\leadsto \frac{2}{\color{blue}{x \cdot x}} + 1 \]

   7. lower-*.f64 53.3

      \[\leadsto \frac{2}{\color{blue}{x \cdot x}} + 1 \]

5. Applied rewrites 53.3%

   \[\leadsto \color{blue}{\frac{2}{x \cdot x} + 1} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 47.6%

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

Reproduce

herbie shell --seed 2024312 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))