Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 7

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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ 1.0 (- x 1.0))) -1.0)
   (/ (fma x x 1.0) (fma x x -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, 1.0) / fma(x, x, -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 = Float64(fma(x, x, 1.0) / fma(x, x, -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 + 1.0), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-add N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. *-lft-identity N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + {x}^{2}}}{\mathsf{fma}\left(x, x, -1\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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%

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

Alternative 3: 99.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ 1.0 (- x 1.0))) -1.0)
   (* (fma x x 1.0) (fma (- x) x -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, 1.0) * fma(-x, x, -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 = Float64(fma(x, x, 1.0) * fma(Float64(-x), x, -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 + 1.0), $MachinePrecision] * N[((-x) * x + -1.0), $MachinePrecision]), $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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. frac-add N/A

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

      7. div-inv N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-neg-out N/A

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

      10. distribute-neg-in N/A

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

      11. *-lft-identity N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(-1 \cdot {x}^{2} - 1\right)} \cdot \frac{1}{\mathsf{fma}\left(-x, x, 1\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(-x, x, -1\right) \cdot \color{blue}{\left(1 + {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.4

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

   10. Applied rewrites 99.4%

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

   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 99.5%

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

Alternative 4: 99.3% 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. metadata-eval N/A

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

      3. *-commutative N/A

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

      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 99.4

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

   5. Applied rewrites 99.4%

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

   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 99.5%

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

Alternative 5: 99.1% accurate, 0.7× speedup

Math

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

FPCore

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

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. metadata-eval N/A

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

      3. *-commutative N/A

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

      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 99.4

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

   5. Applied rewrites 99.4%

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

   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

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.4%

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

Alternative 6: 98.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

MATLAB

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

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

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

   5. Applied rewrites 98.5%

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

   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

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.0%

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

Alternative 7: 49.7% 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 0

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

5. Applied rewrites 48.9%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))