Asymptote A

Percentage Accurate: 77.3% → 99.9%

Time: 5.2s

Alternatives: 5

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: 77.3% 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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. frac-sub 76.9%

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

   2. associate-/r* 76.9%

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

   3. *-un-lft-identity 76.9%

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

   4. *-rgt-identity 76.9%

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

   5. associate--l- 76.9%

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

   6. +-commutative 76.9%

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

   7. +-commutative 76.9%

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

   8. sub-neg 76.9%

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

   9. metadata-eval 76.9%

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

3. Applied egg-rr 76.9%

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

4. Taylor expanded in x around 0 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.2% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ -2.0 (* x x)) (+ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -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)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-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) || !(x <= 1.0)) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 2.0 + (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -2.0 / (x * x)
	else:
		tmp = 2.0 + (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-2.0 / Float64(x * x));
	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) || ~((x <= 1.0)))
		tmp = -2.0 / (x * x);
	else
		tmp = 2.0 + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 52.0%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. unpow2 99.4%

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

   4. Simplified 99.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.8%

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

      1. neg-mul-1 98.8%

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

      2. unsub-neg 98.8%

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

   4. Simplified 98.8%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. unpow2 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{2 + x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

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

Alternative 3: 98.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ (/ -2.0 x) x) (+ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-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)) .or. (.not. (x <= 1.0d0))) then
        tmp = ((-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) || !(x <= 1.0)) {
		tmp = (-2.0 / x) / x;
	} else {
		tmp = 2.0 + (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (-2.0 / x) / x
	else:
		tmp = 2.0 + (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(-2.0 / x) / x);
	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) || ~((x <= 1.0)))
		tmp = (-2.0 / x) / x;
	else
		tmp = 2.0 + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 52.0%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. unpow2 99.4%

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

   4. Simplified 99.4%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   8. Simplified 99.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.8%

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

      1. neg-mul-1 98.8%

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

      2. unsub-neg 98.8%

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

   4. Simplified 98.8%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. unpow2 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{2 + x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

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

Alternative 4: 99.4% 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 75.0%

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

   1. frac-sub 76.9%

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

   2. associate-/r* 76.9%

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

   3. *-un-lft-identity 76.9%

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

   4. *-rgt-identity 76.9%

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

   5. associate--l- 76.9%

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

   6. +-commutative 76.9%

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

   7. +-commutative 76.9%

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

   8. sub-neg 76.9%

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

   9. metadata-eval 76.9%

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

3. Applied egg-rr 76.9%

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

4. Taylor expanded in x around 0 99.8%

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

   1. frac-2neg 99.8%

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

   2. div-inv 99.8%

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

   3. distribute-neg-frac 99.8%

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

   4. metadata-eval 99.8%

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

   5. +-commutative 99.8%

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

   6. distribute-neg-in 99.8%

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

   7. metadata-eval 99.8%

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

   8. sub-neg 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. metadata-eval 99.8%

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

   4. distribute-neg-frac 99.8%

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

   5. distribute-neg-frac 99.8%

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

   6. associate-/l/ 99.7%

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

   7. distribute-neg-frac 99.7%

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

   8. metadata-eval 99.7%

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

   9. *-commutative 99.7%

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

   10. +-commutative 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in x around 0 99.7%

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

    1. mul-1-neg 99.7%

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

    2. sub-neg 99.7%

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

    3. unpow2 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 5: 51.0% 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 75.0%

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

2. Taylor expanded in x around 0 48.8%

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

3. Final simplification 48.8%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))