Asymptote A

Percentage Accurate: 77.3% → 99.4%

Time: 8.6s

Alternatives: 5

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{\mathsf{fma}\left(x, x, -1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 (fma x x -1.0)))

C

double code(double x) {
	return -2.0 / fma(x, x, -1.0);
}

Julia

function code(x)
	return Float64(-2.0 / fma(x, x, -1.0))
end

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. lift-+.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift--.f64 N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. frac-add N/A

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

   16. *-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. lift--.f64 N/A

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

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

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

   20. metadata-eval N/A

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

   21. lower-/.f64 N/A

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

4. Applied egg-rr 80.6%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.3%

   \[\leadsto \frac{\color{blue}{-2}}{\mathsf{fma}\left(x, x, -1\right)} \]
8. Add Preprocessing

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{x + 1} + \frac{-1}{x + -1} \leq 0:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x \cdot x, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (+ x 1.0)) (/ -1.0 (+ x -1.0))) 0.0)
   (/ -2.0 (* x x))
   (fma 2.0 (* x x) 2.0)))

C

double code(double x) {
	double tmp;
	if (((1.0 / (x + 1.0)) + (-1.0 / (x + -1.0))) <= 0.0) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = fma(2.0, (x * x), 2.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 0.0

   1. Initial program 53.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.5

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

   5. Simplified 97.5%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x + 1} + \frac{-1}{x + -1} \leq 0:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x \cdot x, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 89.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 74.3

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

   5. Simplified 74.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.2

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

   8. Simplified 74.2%

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

   if 1 < x

   1. Initial program 50.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 3.1

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

   5. Simplified 3.1%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 47.5

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

   8. Simplified 47.5%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 47.5

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

   11. Simplified 47.5%

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

4. Final simplification 67.4%

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

Alternative 4: 75.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) + \left(x - -1\right) \end{array} \]

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(Float64(1.0 - x) + Float64(x - -1.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 56.2

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

5. Simplified 56.2%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 77.5

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

8. Simplified 77.5%

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

9. Final simplification 77.5%

   \[\leadsto \left(1 - x\right) + \left(x - -1\right) \]
10. Add Preprocessing

Alternative 5: 50.6% accurate, 32.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 79.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 56.2%

   \[\leadsto \color{blue}{2} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))