3frac (problem 3.3.3)

Percentage Accurate: 85.0% → 99.6%

Time: 8.7s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -560:\\ \;\;\;\;\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\\ \mathbf{elif}\;x \leq 12000:\\ \;\;\;\;\frac{-2}{x} + \left(\frac{1}{x + -1} + \frac{-1}{-1 - x}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -560.0)
   (+ (/ 2.0 (pow x 3.0)) (/ 2.0 (pow x 5.0)))
   (if (<= x 12000.0)
     (+ (/ -2.0 x) (+ (/ 1.0 (+ x -1.0)) (/ -1.0 (- -1.0 x))))
     (* 2.0 (pow x -3.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -560.0) {
		tmp = (2.0 / pow(x, 3.0)) + (2.0 / pow(x, 5.0));
	} else if (x <= 12000.0) {
		tmp = (-2.0 / x) + ((1.0 / (x + -1.0)) + (-1.0 / (-1.0 - x)));
	} else {
		tmp = 2.0 * pow(x, -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-560.0d0)) then
        tmp = (2.0d0 / (x ** 3.0d0)) + (2.0d0 / (x ** 5.0d0))
    else if (x <= 12000.0d0) then
        tmp = ((-2.0d0) / x) + ((1.0d0 / (x + (-1.0d0))) + ((-1.0d0) / ((-1.0d0) - x)))
    else
        tmp = 2.0d0 * (x ** (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -560.0) {
		tmp = (2.0 / Math.pow(x, 3.0)) + (2.0 / Math.pow(x, 5.0));
	} else if (x <= 12000.0) {
		tmp = (-2.0 / x) + ((1.0 / (x + -1.0)) + (-1.0 / (-1.0 - x)));
	} else {
		tmp = 2.0 * Math.pow(x, -3.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -560.0:
		tmp = (2.0 / math.pow(x, 3.0)) + (2.0 / math.pow(x, 5.0))
	elif x <= 12000.0:
		tmp = (-2.0 / x) + ((1.0 / (x + -1.0)) + (-1.0 / (-1.0 - x)))
	else:
		tmp = 2.0 * math.pow(x, -3.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -560.0)
		tmp = Float64(Float64(2.0 / (x ^ 3.0)) + Float64(2.0 / (x ^ 5.0)));
	elseif (x <= 12000.0)
		tmp = Float64(Float64(-2.0 / x) + Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(-1.0 / Float64(-1.0 - x))));
	else
		tmp = Float64(2.0 * (x ^ -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -560.0)
		tmp = (2.0 / (x ^ 3.0)) + (2.0 / (x ^ 5.0));
	elseif (x <= 12000.0)
		tmp = (-2.0 / x) + ((1.0 / (x + -1.0)) + (-1.0 / (-1.0 - x)));
	else
		tmp = 2.0 * (x ^ -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -560.0], N[(N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 12000.0], N[(N[(-2.0 / x), $MachinePrecision] + N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -560

   1. Initial program 69.3%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} + 2 \cdot \frac{1}{{x}^{5}} \]

      2. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{2}}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}} \]

      3. associate-*r/ 99.8%

         \[\leadsto \frac{2}{{x}^{3}} + \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{2}{{x}^{3}} + \frac{\color{blue}{2}}{{x}^{5}} \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}} \]

   if -560 < x < 12000

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

   if 12000 < x

   1. Initial program 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in x around inf 99.0%

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

      1. div-inv 99.0%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}}} \]

      2. *-commutative 99.0%

         \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot 2} \]

      3. pow-flip 100.0%

         \[\leadsto \color{blue}{{x}^{\left(-3\right)}} \cdot 2 \]

      4. metadata-eval 100.0%

         \[\leadsto {x}^{\color{blue}{-3}} \cdot 2 \]

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -560:\\ \;\;\;\;\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\\ \mathbf{elif}\;x \leq 12000:\\ \;\;\;\;\frac{-2}{x} + \left(\frac{1}{x + -1} + \frac{-1}{-1 - x}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + -1}\\ t_1 := t_0 + \left(\frac{1}{x + 1} - \frac{2}{x}\right)\\ \mathbf{if}\;t_1 \leq -50:\\ \;\;\;\;\frac{-2}{x} + \left(t_0 + \frac{-1}{-1 - x}\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x -1.0))) (t_1 (+ t_0 (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)))))
   (if (<= t_1 -50.0)
     (+ (/ -2.0 x) (+ t_0 (/ -1.0 (- -1.0 x))))
     (if (<= t_1 0.0) (* 2.0 (pow x -3.0)) t_1))))

C

double code(double x) {
	double t_0 = 1.0 / (x + -1.0);
	double t_1 = t_0 + ((1.0 / (x + 1.0)) - (2.0 / x));
	double tmp;
	if (t_1 <= -50.0) {
		tmp = (-2.0 / x) + (t_0 + (-1.0 / (-1.0 - x)));
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * pow(x, -3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (x + (-1.0d0))
    t_1 = t_0 + ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x))
    if (t_1 <= (-50.0d0)) then
        tmp = ((-2.0d0) / x) + (t_0 + ((-1.0d0) / ((-1.0d0) - x)))
    else if (t_1 <= 0.0d0) then
        tmp = 2.0d0 * (x ** (-3.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (x + -1.0);
	double t_1 = t_0 + ((1.0 / (x + 1.0)) - (2.0 / x));
	double tmp;
	if (t_1 <= -50.0) {
		tmp = (-2.0 / x) + (t_0 + (-1.0 / (-1.0 - x)));
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * Math.pow(x, -3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (x + -1.0)
	t_1 = t_0 + ((1.0 / (x + 1.0)) - (2.0 / x))
	tmp = 0
	if t_1 <= -50.0:
		tmp = (-2.0 / x) + (t_0 + (-1.0 / (-1.0 - x)))
	elif t_1 <= 0.0:
		tmp = 2.0 * math.pow(x, -3.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(x + -1.0))
	t_1 = Float64(t_0 + Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)))
	tmp = 0.0
	if (t_1 <= -50.0)
		tmp = Float64(Float64(-2.0 / x) + Float64(t_0 + Float64(-1.0 / Float64(-1.0 - x))));
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 * (x ^ -3.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (x + -1.0);
	t_1 = t_0 + ((1.0 / (x + 1.0)) - (2.0 / x));
	tmp = 0.0;
	if (t_1 <= -50.0)
		tmp = (-2.0 / x) + (t_0 + (-1.0 / (-1.0 - x)));
	elseif (t_1 <= 0.0)
		tmp = 2.0 * (x ^ -3.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50.0], N[(N[(-2.0 / x), $MachinePrecision] + N[(t$95$0 + N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -50

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   if -50 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 68.5%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in x around inf 99.1%

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

      1. div-inv 99.1%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}}} \]

      2. *-commutative 99.1%

         \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot 2} \]

      3. pow-flip 99.6%

         \[\leadsto \color{blue}{{x}^{\left(-3\right)}} \cdot 2 \]

      4. metadata-eval 99.6%

         \[\leadsto {x}^{\color{blue}{-3}} \cdot 2 \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{x}^{-3} \cdot 2} \]

   if 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.5%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x + -1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \leq -50:\\ \;\;\;\;\frac{-2}{x} + \left(\frac{1}{x + -1} + \frac{-1}{-1 - x}\right)\\ \mathbf{elif}\;\frac{1}{x + -1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + -1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)\\ \end{array} \]

Alternative 3: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + -1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

2. Final simplification 84.1%

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

Alternative 4: 83.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 1.0) (- (* x -2.0) (/ 2.0 x)) 0.0)))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 1.0d0) then
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 1.0:
		tmp = (x * -2.0) - (2.0 / x)
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 1.0], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 68.5%

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

   3. Simplified 68.5%

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

   5. Applied egg-rr 6.1%

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

      1. fma-udef 6.1%

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

      2. *-rgt-identity 6.1%

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

      3. count-2 6.1%

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

      4. count-2 6.1%

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

      5. associate-*r/ 6.1%

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

      6. metadata-eval 6.1%

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

      7. +-commutative 6.1%

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

   7. Simplified 6.1%

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

      1. +-commutative 6.1%

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

      2. associate-*r/ 6.1%

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

      3. frac-2neg 6.1%

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

      4. metadata-eval 6.1%

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

      5. frac-add 54.7%

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

      6. +-commutative 54.7%

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

      7. metadata-eval 54.7%

         \[\leadsto \frac{\left(x + 1\right) \cdot 2 + \color{blue}{4} \cdot \left(-x\right)}{\left(x + 1\right) \cdot \left(-x\right)} \]

   9. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) \cdot 2 + 4 \cdot \left(-x\right)}{\left(x + 1\right) \cdot \left(-x\right)}} \]

   10. Taylor expanded in x around 0 55.0%

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

   12. Applied egg-rr 67.2%

       \[\leadsto \color{blue}{0} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 83.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

3. Simplified 84.1%

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

5. Applied egg-rr 62.6%

   \[\leadsto \frac{-2}{x} + \color{blue}{\left(\left(\frac{1}{1 + x} + \frac{1}{1 + x}\right) + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)} \]
6. Step-by-step derivation

   1. associate-+l+ 62.3%

      \[\leadsto \frac{-2}{x} + \color{blue}{\left(\frac{1}{1 + x} + \left(\frac{1}{1 + x} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)} \]

   2. expm1-log1p 62.3%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + x}\right)\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

   3. expm1-def 49.8%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - 1\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

   4. associate-+l- 49.8%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)}\right) \]

   5. fma-udef 49.8%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) \cdot {\left(1 + x\right)}^{-0.5} + \frac{1}{1 + x}\right)}\right)\right)\right) \]

   6. distribute-lft-neg-out 49.8%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\color{blue}{\left(-{\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}\right)} + \frac{1}{1 + x}\right)\right)\right)\right) \]

   7. pow-sqr 51.2%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{{\left(1 + x\right)}^{\left(2 \cdot -0.5\right)}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   8. metadata-eval 51.2%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-{\left(1 + x\right)}^{\color{blue}{-1}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   9. unpow-1 51.2%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{\frac{1}{1 + x}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   10. +-commutative 51.2%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} + \left(-\frac{1}{1 + x}\right)\right)}\right)\right)\right) \]

   11. sub-neg 51.2%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} - \frac{1}{1 + x}\right)}\right)\right)\right) \]

   12. +-inverses 51.2%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{0}\right)\right)\right) \]

   13. metadata-eval 51.2%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \color{blue}{1}\right)\right) \]

7. Simplified 82.6%

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

8. Final simplification 82.6%

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

Alternative 6: 83.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 1.0) (/ -2.0 x) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 1.0d0) then
        tmp = (-2.0d0) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 1.0:
		tmp = -2.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 1.0], N[(-2.0 / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 68.5%

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

   3. Simplified 68.5%

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

   5. Applied egg-rr 6.1%

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

      1. fma-udef 6.1%

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

      2. *-rgt-identity 6.1%

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

      3. count-2 6.1%

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

      4. count-2 6.1%

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

      5. associate-*r/ 6.1%

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

      6. metadata-eval 6.1%

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

      7. +-commutative 6.1%

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

   7. Simplified 6.1%

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

      1. +-commutative 6.1%

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

      2. associate-*r/ 6.1%

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

      3. frac-2neg 6.1%

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

      4. metadata-eval 6.1%

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

      5. frac-add 54.7%

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

      6. +-commutative 54.7%

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

      7. metadata-eval 54.7%

         \[\leadsto \frac{\left(x + 1\right) \cdot 2 + \color{blue}{4} \cdot \left(-x\right)}{\left(x + 1\right) \cdot \left(-x\right)} \]

   9. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) \cdot 2 + 4 \cdot \left(-x\right)}{\left(x + 1\right) \cdot \left(-x\right)}} \]

   10. Taylor expanded in x around 0 55.0%

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

   12. Applied egg-rr 67.2%

       \[\leadsto \color{blue}{0} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 3.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -8 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -8.0)

C

double code(double x) {
	return -8.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -8.0d0
end function

Java

public static double code(double x) {
	return -8.0;
}

Python

def code(x):
	return -8.0

Julia

function code(x)
	return -8.0
end

MATLAB

function tmp = code(x)
	tmp = -8.0;
end

Wolfram

code[x_] := -8.0

Derivation

1. Initial program 84.1%

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

3. Simplified 84.1%

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

5. Applied egg-rr 51.7%

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

   1. fma-udef 51.7%

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

   2. *-rgt-identity 51.7%

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

   3. count-2 51.7%

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

   4. count-2 51.7%

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

   5. associate-*r/ 51.7%

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

   6. metadata-eval 51.7%

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

   7. +-commutative 51.7%

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

7. Simplified 51.7%

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

   1. +-commutative 51.7%

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

   2. associate-*r/ 51.7%

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

   3. frac-2neg 51.7%

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

   4. metadata-eval 51.7%

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

   5. frac-add 76.2%

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

   6. +-commutative 76.2%

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

   7. metadata-eval 76.2%

      \[\leadsto \frac{\left(x + 1\right) \cdot 2 + \color{blue}{4} \cdot \left(-x\right)}{\left(x + 1\right) \cdot \left(-x\right)} \]

9. Applied egg-rr 76.2%

   \[\leadsto \color{blue}{\frac{\left(x + 1\right) \cdot 2 + 4 \cdot \left(-x\right)}{\left(x + 1\right) \cdot \left(-x\right)}} \]

10. Taylor expanded in x around 0 76.3%

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

12. Applied egg-rr 3.5%

    \[\leadsto \color{blue}{-8} \]

13. Final simplification 3.5%

    \[\leadsto -8 \]

Alternative 8: 35.9% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 84.1%

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

3. Simplified 84.1%

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

5. Applied egg-rr 51.7%

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

   1. fma-udef 51.7%

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

   2. *-rgt-identity 51.7%

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

   3. count-2 51.7%

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

   4. count-2 51.7%

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

   5. associate-*r/ 51.7%

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

   6. metadata-eval 51.7%

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

   7. +-commutative 51.7%

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

7. Simplified 51.7%

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

   1. +-commutative 51.7%

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

   2. associate-*r/ 51.7%

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

   3. frac-2neg 51.7%

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

   4. metadata-eval 51.7%

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

   5. frac-add 76.2%

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

   6. +-commutative 76.2%

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

   7. metadata-eval 76.2%

      \[\leadsto \frac{\left(x + 1\right) \cdot 2 + \color{blue}{4} \cdot \left(-x\right)}{\left(x + 1\right) \cdot \left(-x\right)} \]

9. Applied egg-rr 76.2%

   \[\leadsto \color{blue}{\frac{\left(x + 1\right) \cdot 2 + 4 \cdot \left(-x\right)}{\left(x + 1\right) \cdot \left(-x\right)}} \]

10. Taylor expanded in x around 0 76.3%

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

12. Applied egg-rr 35.0%

    \[\leadsto \color{blue}{0} \]

13. Final simplification 35.0%

    \[\leadsto 0 \]

Developer target: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot \left(x \cdot x - 1\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))