Asymptote A

Percentage Accurate: 77.1% → 99.9%

Time: 12.8s

Alternatives: 6

Speedup: 1.6×

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, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{1 + x}}{1 - x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (2.0 / (1.0 + x)) / (1.0 - x);
}

Python

def code(x):
	return (2.0 / (1.0 + x)) / (1.0 - x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

4. Applied egg-rr 78.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.9%

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

Alternative 2: 75.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 84.6%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 71.2%

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

   5. Simplified 71.2%

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

   if 1 < x

   1. Initial program 53.8%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 99.0%

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

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]
   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 99.8%

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

   7. Applied egg-rr 99.8%

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

Alternative 3: 75.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 71.2%

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

   if 1 < x

   1. Initial program 53.8%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 99.0%

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

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]
   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 99.8%

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

   7. Applied egg-rr 99.8%

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

Alternative 4: 74.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 71.2%

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

   if 1 < x

   1. Initial program 53.8%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   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 99.0%

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

   5. Simplified 99.0%

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

Alternative 5: 99.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{1 - x \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 2.0 / (1.0 - (x * x));
}

Python

def code(x):
	return 2.0 / (1.0 - (x * x))

Julia

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

MATLAB

function tmp = code(x)
	tmp = 2.0 / (1.0 - (x * x));
end

Wolfram

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

Derivation

1. Initial program 77.4%

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

4. Applied egg-rr 78.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.9%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

      \[\leadsto \frac{\frac{2}{1 + x}}{1 \cdot 1 - x \cdot x} \cdot \color{blue}{\left(1 + x\right)} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\frac{2}{1 + x} \cdot \left(1 + x\right)}{\color{blue}{1 \cdot 1 - x \cdot x}} \]

9. Applied egg-rr 99.7%

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

Alternative 6: 50.5% accurate, 11.0× 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 77.4%

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

3. Taylor expanded in x around 0

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

5. Simplified 55.1%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))