Asymptote A

Percentage Accurate: 77.1% → 99.9%

Time: 8.5s

Alternatives: 6

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: 77.1% 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: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{-2}{x\_m + -1}}{x\_m + 1} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (/ -2.0 (+ x_m -1.0)) (+ x_m 1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return (-2.0 / (x_m + -1.0)) / (x_m + 1.0);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = ((-2.0d0) / (x_m + (-1.0d0))) / (x_m + 1.0d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (-2.0 / (x_m + -1.0)) / (x_m + 1.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (-2.0 / (x_m + -1.0)) / (x_m + 1.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(-2.0 / Float64(x_m + -1.0)) / Float64(x_m + 1.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (-2.0 / (x_m + -1.0)) / (x_m + 1.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(-2.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.9%

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

   1. clear-num N/A

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

   2. frac-sub N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. /-lowering-/.f64 N/A

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

4. Applied egg-rr 78.8%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 98.6% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (+ (/ 1.0 (+ x_m 1.0)) (/ -1.0 (+ x_m -1.0))) 0.0)
   (/ -2.0 (* x_m x_m))
   (fma 2.0 (* x_m x_m) 2.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[(1.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-2.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(x$95$m * x$95$m), $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 55.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 97.3

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

   5. Simplified 97.3%

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

   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. accelerator-lowering-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. *-lowering-*.f64 98.8

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

   5. Simplified 98.8%

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

4. Final simplification 98.1%

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

Alternative 3: 99.4% accurate, 1.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ -2.0 (fma x_m x_m -1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(-2.0 / N[(x$95$m * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.9%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-neg-frac N/A

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

   4. metadata-eval N/A

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

   5. clear-num N/A

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

   6. frac-add N/A

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

   7. *-commutative N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-rgt-identity N/A

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

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

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 N/A

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

4. Applied egg-rr 78.9%

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

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 99.0%

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

Alternative 4: 51.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{2}{x\_m + 1} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 2.0 (+ x_m 1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return 2.0 / (x_m + 1.0);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 2.0d0 / (x_m + 1.0d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0 / (x_m + 1.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0 / (x_m + 1.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(2.0 / Float64(x_m + 1.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 2.0 / (x_m + 1.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(2.0 / N[(x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.9%

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

   1. clear-num N/A

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

   2. frac-sub N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. /-lowering-/.f64 N/A

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

4. Applied egg-rr 78.8%

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

5. Taylor expanded in x around 0

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

7. Simplified 51.6%

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

8. Final simplification 51.6%

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

Alternative 5: 50.8% accurate, 32.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 2 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 2.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 2.0;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0

Julia

x_m = abs(x)
function code(x_m)
	return 2.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 2.0

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.9%

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

Alternative 6: 10.7% accurate, 32.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.0%

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

6. Taylor expanded in x around inf

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

8. Simplified 10.8%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))