Asymptote A

Percentage Accurate: 76.6% → 99.9%

Time: 6.2s

Alternatives: 7

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: 76.6% 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 79.4%

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

   1. frac-sub 80.0%

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

   2. associate-/r* 80.0%

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

   3. *-un-lft-identity 80.0%

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

   4. *-rgt-identity 80.0%

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

   5. associate--l- 80.1%

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

   6. +-commutative 80.1%

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

   7. +-commutative 80.1%

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

   8. sub-neg 80.1%

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

   9. metadata-eval 80.1%

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

3. Applied egg-rr 80.1%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ -2.0 x) x)
   (if (<= x 1.55) (- (- 1.0 x) (/ 1.0 (+ x -1.0))) (/ -2.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-2.0 / x) / x;
	} else if (x <= 1.55) {
		tmp = (1.0 - x) - (1.0 / (x + -1.0));
	} 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) / x) / x
    else if (x <= 1.55d0) then
        tmp = (1.0d0 - x) - (1.0d0 / (x + (-1.0d0)))
    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 / x) / x;
	} else if (x <= 1.55) {
		tmp = (1.0 - x) - (1.0 / (x + -1.0));
	} else {
		tmp = -2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (-2.0 / x) / x
	elif x <= 1.55:
		tmp = (1.0 - x) - (1.0 / (x + -1.0))
	else:
		tmp = -2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(-2.0 / x) / x);
	elseif (x <= 1.55)
		tmp = Float64(Float64(1.0 - x) - Float64(1.0 / Float64(x + -1.0)));
	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 / x) / x;
	elseif (x <= 1.55)
		tmp = (1.0 - x) - (1.0 / (x + -1.0));
	else
		tmp = -2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.55], N[(N[(1.0 - x), $MachinePrecision] - N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 59.6%

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

      1. frac-sub 60.6%

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

      2. div-inv 60.6%

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

      3. *-un-lft-identity 60.6%

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

      4. *-rgt-identity 60.6%

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

      5. associate--l- 60.6%

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

      6. +-commutative 60.6%

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

      7. difference-of-sqr-1 60.6%

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

      8. fma-neg 60.6%

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

      9. metadata-eval 60.6%

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

   3. Applied egg-rr 60.6%

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

      1. associate-*r/ 60.6%

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

      2. *-rgt-identity 60.6%

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

      3. associate-+r+ 60.6%

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

      4. metadata-eval 60.6%

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

   5. Simplified 60.6%

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

   6. Taylor expanded in x around inf 97.3%

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

      1. unpow2 97.3%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   if -1 < x < 1.55000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.6%

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

      1. neg-mul-1 98.6%

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

      2. unsub-neg 98.6%

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

   4. Simplified 98.6%

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

   if 1.55000000000000004 < x

   1. Initial program 54.5%

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

   2. Taylor expanded in x around inf 98.1%

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

      1. unpow2 98.1%

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

   4. Simplified 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \end{array} \]

Alternative 3: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + x} + \left(x - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (/ (/ -2.0 x) x)
   (if (<= x 1.0) (+ (/ 1.0 (+ 1.0 x)) (- x -1.0)) (/ -2.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = (-2.0 / x) / x;
	} else if (x <= 1.0) {
		tmp = (1.0 / (1.0 + x)) + (x - -1.0);
	} else {
		tmp = -2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = ((-2.0d0) / x) / x
    else if (x <= 1.0d0) then
        tmp = (1.0d0 / (1.0d0 + x)) + (x - (-1.0d0))
    else
        tmp = (-2.0d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = (-2.0 / x) / x;
	} else if (x <= 1.0) {
		tmp = (1.0 / (1.0 + x)) + (x - -1.0);
	} else {
		tmp = -2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = (-2.0 / x) / x
	elif x <= 1.0:
		tmp = (1.0 / (1.0 + x)) + (x - -1.0)
	else:
		tmp = -2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(-2.0 / x) / x);
	elseif (x <= 1.0)
		tmp = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(x - -1.0));
	else
		tmp = Float64(-2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = (-2.0 / x) / x;
	elseif (x <= 1.0)
		tmp = (1.0 / (1.0 + x)) + (x - -1.0);
	else
		tmp = -2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 59.6%

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

      1. frac-sub 60.6%

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

      2. div-inv 60.6%

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

      3. *-un-lft-identity 60.6%

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

      4. *-rgt-identity 60.6%

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

      5. associate--l- 60.6%

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

      6. +-commutative 60.6%

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

      7. difference-of-sqr-1 60.6%

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

      8. fma-neg 60.6%

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

      9. metadata-eval 60.6%

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

   3. Applied egg-rr 60.6%

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

      1. associate-*r/ 60.6%

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

      2. *-rgt-identity 60.6%

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

      3. associate-+r+ 60.6%

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

      4. metadata-eval 60.6%

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

   5. Simplified 60.6%

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

   6. Taylor expanded in x around inf 97.3%

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

      1. unpow2 97.3%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   if -1.55000000000000004 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.6%

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

      1. sub-neg 98.6%

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

      2. metadata-eval 98.6%

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

      3. +-commutative 98.6%

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

      4. neg-mul-1 98.6%

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

      5. unsub-neg 98.6%

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

   4. Simplified 98.6%

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

   if 1 < x

   1. Initial program 54.5%

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

   2. Taylor expanded in x around inf 98.1%

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

      1. unpow2 98.1%

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

   4. Simplified 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + x} + \left(x - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 2.0;
	}
	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
    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;
	}
	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
	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 = 2.0;
	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;
	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], 2.0]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 57.3%

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

   2. Taylor expanded in x around inf 97.7%

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

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

4. Final simplification 98.1%

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

Alternative 5: 98.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-2.0 / x) / x;
	} else 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) / x) / x
    else 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 / x) / x;
	} else 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 / x) / x
	elif 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 = Float64(Float64(-2.0 / x) / x);
	elseif (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 / x) / x;
	elseif (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], N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.0], 2.0, N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 59.6%

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

      1. frac-sub 60.6%

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

      2. div-inv 60.6%

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

      3. *-un-lft-identity 60.6%

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

      4. *-rgt-identity 60.6%

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

      5. associate--l- 60.6%

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

      6. +-commutative 60.6%

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

      7. difference-of-sqr-1 60.6%

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

      8. fma-neg 60.6%

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

      9. metadata-eval 60.6%

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

   3. Applied egg-rr 60.6%

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

      1. associate-*r/ 60.6%

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

      2. *-rgt-identity 60.6%

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

      3. associate-+r+ 60.6%

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

      4. metadata-eval 60.6%

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

   5. Simplified 60.6%

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

   6. Taylor expanded in x around inf 97.3%

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

      1. unpow2 97.3%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

   if 1 < x

   1. Initial program 54.5%

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

   2. Taylor expanded in x around inf 98.1%

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

      1. unpow2 98.1%

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

   4. Simplified 98.1%

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

4. Final simplification 98.8%

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

Alternative 6: 10.7% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 79.4%

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

2. Taylor expanded in x around 0 51.8%

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

3. Taylor expanded in x around inf 11.1%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 11.1%

   \[\leadsto 1 \]

Alternative 7: 50.6% 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 79.4%

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

2. Taylor expanded in x around 0 52.5%

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

3. Final simplification 52.5%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023189 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))