Asymptote A

Percentage Accurate: 76.9% → 99.9%

Time: 7.2s

Alternatives: 5

Speedup: 1.2×

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: 76.9% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. clear-num N/A

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

   2. frac-sub N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/r* N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\frac{x + 1}{1}}\right), \color{blue}{\left(x - 1\right)}\right) \]

4. Applied egg-rr 81.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{-2}, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(x, -1\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.9%

   \[\leadsto \frac{\frac{\color{blue}{-2}}{1 + x}}{x + -1} \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = 2.0d0 + (2.0d0 * (x_m * x_m))
    else
        tmp = ((-2.0d0) / x_m) / x_m
    end if
    code = tmp
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = 2.0 + (2.0 * (x_m * x_m))
	else:
		tmp = (-2.0 / x_m) / x_m
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = 2.0 + (2.0 * (x_m * x_m));
	else
		tmp = (-2.0 / x_m) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(2, \color{blue}{\left(2 \cdot {x}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 67.4%

         \[\leadsto \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{2 + 2 \cdot \left(x \cdot x\right)} \]

   if 1 < x

   1. Initial program 56.6%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 98.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{x}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, x\right), x\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{-2}{x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = 2.0d0
    else
        tmp = ((-2.0d0) / x_m) / x_m
    end if
    code = tmp
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = 2.0
	else:
		tmp = (-2.0 / x_m) / x_m
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = 2.0;
	else
		tmp = (-2.0 / x_m) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.2%

      \[\leadsto \color{blue}{2} \]

   if 1 < x

   1. Initial program 56.6%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 98.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{x}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, x\right), x\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{-2}{x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = 2.0d0
    else
        tmp = (-2.0d0) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = 2.0
	else:
		tmp = -2.0 / (x_m * x_m)
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = 2.0;
	else
		tmp = -2.0 / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.2%

      \[\leadsto \color{blue}{2} \]

   if 1 < x

   1. Initial program 56.6%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 50.6% accurate, 11.0× 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 77.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 49.6%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))