Asymptote C

Percentage Accurate: 54.6% → 99.4%

Time: 7.6s

Alternatives: 7

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{x + 1}{1 - x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 5e-11)
   (/ (+ -3.0 (/ -1.0 x)) x)
   (fma (/ x (fma x x -1.0)) (+ x -1.0) (/ (+ x 1.0) (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-11) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = fma((x / fma(x, x, -1.0)), (x + -1.0), ((x + 1.0) / (1.0 - x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 5e-11)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = fma(Float64(x / fma(x, x, -1.0)), Float64(x + -1.0), Float64(Float64(x + 1.0) / Float64(1.0 - x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-11], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000018e-11

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{3 + \frac{1}{x}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(3 + \frac{1}{x}\right)}{x}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(3 + \frac{1}{x}\right)}{x}} \]

      3. neg-mul-1 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)}}{x} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}}{x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-3} + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}{x} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}}{x} \]

      7. distribute-neg-frac N/A

         \[\leadsto \frac{-3 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{x}}}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{-3 + \frac{\color{blue}{-1}}{x}}{x} \]

      9. lower-/.f64 100.0

         \[\leadsto \frac{-3 + \color{blue}{\frac{-1}{x}}}{x} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-3 + \frac{-1}{x}}{x}} \]

   if 5.00000000000000018e-11 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. lift-/.f64 N/A

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

      19. distribute-neg-frac2 N/A

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

      20. lower-/.f64 N/A

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

      21. lift--.f64 N/A

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

      22. sub-neg N/A

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

      23. distribute-neg-in N/A

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

      24. metadata-eval N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{x + 1}{\color{blue}{1 - x}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{x + 1}{1 - x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 5e-11) (/ (+ -3.0 (/ -1.0 x)) x) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 5e-11) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 5d-11) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 5e-11) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 5e-11:
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 5e-11)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 5e-11)
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-11], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000018e-11

   1. Initial program 7.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{3 + \frac{1}{x}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(3 + \frac{1}{x}\right)}{x}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(3 + \frac{1}{x}\right)}{x}} \]

      3. neg-mul-1 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(3 + \frac{1}{x}\right)\right)}}{x} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}}{x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-3} + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}{x} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{-3 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}}{x} \]

      7. distribute-neg-frac N/A

         \[\leadsto \frac{-3 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{x}}}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{-3 + \frac{\color{blue}{-1}}{x}}{x} \]

      9. lower-/.f64 99.2

         \[\leadsto \frac{-3 + \color{blue}{\frac{-1}{x}}}{x} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{-3 + \frac{-1}{x}}{x}} \]

   if 5.00000000000000018e-11 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.6%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))