3frac (problem 3.3.3)

Percentage Accurate: 68.5% → 99.8%

Time: 9.4s

Alternatives: 11

Speedup: 2.1×

Specification

\[\left|x\right| > 1\]

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: 68.5% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} - {x}^{-3} \cdot -2 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (- (/ 2.0 (* x x)) -2.0) (pow x 5.0)) (* (pow x -3.0) -2.0)))

C

double code(double x) {
	return (((2.0 / (x * x)) - -2.0) / pow(x, 5.0)) - (pow(x, -3.0) * -2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((2.0d0 / (x * x)) - (-2.0d0)) / (x ** 5.0d0)) - ((x ** (-3.0d0)) * (-2.0d0))
end function

Java

public static double code(double x) {
	return (((2.0 / (x * x)) - -2.0) / Math.pow(x, 5.0)) - (Math.pow(x, -3.0) * -2.0);
}

Python

def code(x):
	return (((2.0 / (x * x)) - -2.0) / math.pow(x, 5.0)) - (math.pow(x, -3.0) * -2.0)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(2.0 / Float64(x * x)) - -2.0) / (x ^ 5.0)) - Float64((x ^ -3.0) * -2.0))
end

MATLAB

function tmp = code(x)
	tmp = (((2.0 / (x * x)) - -2.0) / (x ^ 5.0)) - ((x ^ -3.0) * -2.0);
end

Wolfram

code[x_] := N[(N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, -3.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2}{{x}^{3}}\right)} \]

   2. div-sub N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \frac{2}{{x}^{3}}\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} + \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)}\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)\right)} \]

   5. unsub-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right)}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

5. Applied rewrites 98.2%

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

7. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} - {x}^{-3} \cdot \color{blue}{-2} \]
8. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} + \frac{\frac{\frac{2}{x}}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ (- (/ 2.0 (* x x)) -2.0) (pow x 5.0)) (/ (/ (/ 2.0 x) x) x)))

C

double code(double x) {
	return (((2.0 / (x * x)) - -2.0) / pow(x, 5.0)) + (((2.0 / x) / x) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((2.0d0 / (x * x)) - (-2.0d0)) / (x ** 5.0d0)) + (((2.0d0 / x) / x) / x)
end function

Java

public static double code(double x) {
	return (((2.0 / (x * x)) - -2.0) / Math.pow(x, 5.0)) + (((2.0 / x) / x) / x);
}

Python

def code(x):
	return (((2.0 / (x * x)) - -2.0) / math.pow(x, 5.0)) + (((2.0 / x) / x) / x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(2.0 / Float64(x * x)) - -2.0) / (x ^ 5.0)) + Float64(Float64(Float64(2.0 / x) / x) / x))
end

MATLAB

function tmp = code(x)
	tmp = (((2.0 / (x * x)) - -2.0) / (x ^ 5.0)) + (((2.0 / x) / x) / x);
end

Wolfram

code[x_] := N[(N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2}{{x}^{3}}\right)} \]

   2. div-sub N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \frac{2}{{x}^{3}}\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} + \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)}\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)\right)} \]

   5. unsub-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right)}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

5. Applied rewrites 98.2%

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

7. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} - \frac{\frac{\frac{2}{x}}{x}}{\color{blue}{-x}} \]

8. Final simplification 99.4%

   \[\leadsto \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} + \frac{\frac{\frac{2}{x}}{x}}{x} \]
9. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} - \frac{\frac{-2}{x \cdot x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (- (/ 2.0 (* x x)) -2.0) (pow x 5.0)) (/ (/ -2.0 (* x x)) x)))

C

double code(double x) {
	return (((2.0 / (x * x)) - -2.0) / pow(x, 5.0)) - ((-2.0 / (x * x)) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((2.0d0 / (x * x)) - (-2.0d0)) / (x ** 5.0d0)) - (((-2.0d0) / (x * x)) / x)
end function

Java

public static double code(double x) {
	return (((2.0 / (x * x)) - -2.0) / Math.pow(x, 5.0)) - ((-2.0 / (x * x)) / x);
}

Python

def code(x):
	return (((2.0 / (x * x)) - -2.0) / math.pow(x, 5.0)) - ((-2.0 / (x * x)) / x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(2.0 / Float64(x * x)) - -2.0) / (x ^ 5.0)) - Float64(Float64(-2.0 / Float64(x * x)) / x))
end

MATLAB

function tmp = code(x)
	tmp = (((2.0 / (x * x)) - -2.0) / (x ^ 5.0)) - ((-2.0 / (x * x)) / x);
end

Wolfram

code[x_] := N[(N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2}{{x}^{3}}\right)} \]

   2. div-sub N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \frac{2}{{x}^{3}}\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} + \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)}\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)\right)} \]

   5. unsub-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right)}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

5. Applied rewrites 98.2%

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

7. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} - \frac{\frac{-2}{x \cdot x}}{\color{blue}{x}} \]
8. Add Preprocessing

Alternative 4: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} - \frac{\frac{-2}{x}}{x \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (- (/ 2.0 (* x x)) -2.0) (pow x 5.0)) (/ (/ -2.0 x) (* x x))))

C

double code(double x) {
	return (((2.0 / (x * x)) - -2.0) / pow(x, 5.0)) - ((-2.0 / x) / (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((2.0d0 / (x * x)) - (-2.0d0)) / (x ** 5.0d0)) - (((-2.0d0) / x) / (x * x))
end function

Java

public static double code(double x) {
	return (((2.0 / (x * x)) - -2.0) / Math.pow(x, 5.0)) - ((-2.0 / x) / (x * x));
}

Python

def code(x):
	return (((2.0 / (x * x)) - -2.0) / math.pow(x, 5.0)) - ((-2.0 / x) / (x * x))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(2.0 / Float64(x * x)) - -2.0) / (x ^ 5.0)) - Float64(Float64(-2.0 / x) / Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (((2.0 / (x * x)) - -2.0) / (x ^ 5.0)) - ((-2.0 / x) / (x * x));
end

Wolfram

code[x_] := N[(N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2}{{x}^{3}}\right)} \]

   2. div-sub N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \frac{2}{{x}^{3}}\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} + \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)}\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)\right)\right)} \]

   5. unsub-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}}{{x}^{3}}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}\right)\right)}\right)\right) - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{3}} - \left(\mathsf{neg}\left(\frac{2}{{x}^{3}}\right)\right)} \]

5. Applied rewrites 98.2%

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

7. Applied rewrites 99.3%

   \[\leadsto \frac{\frac{2}{x \cdot x} - -2}{{x}^{5}} - \frac{\frac{-2}{x}}{\color{blue}{x \cdot x}} \]
8. Add Preprocessing

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {x}^{-3} \cdot 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x -3.0) 2.0))

C

double code(double x) {
	return pow(x, -3.0) * 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-3.0d0)) * 2.0d0
end function

Java

public static double code(double x) {
	return Math.pow(x, -3.0) * 2.0;
}

Python

def code(x):
	return math.pow(x, -3.0) * 2.0

Julia

function code(x)
	return Float64((x ^ -3.0) * 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -3.0) * 2.0;
end

Wolfram

code[x_] := N[(N[Power[x, -3.0], $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 97.3

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

5. Applied rewrites 97.3%

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

7. Applied rewrites 98.7%

   \[\leadsto \color{blue}{{x}^{-3} \cdot 2} \]
8. Add Preprocessing

Alternative 6: 99.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 97.3

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

5. Applied rewrites 97.3%

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

7. Applied rewrites 98.5%

   \[\leadsto \frac{\frac{2}{x \cdot x}}{\color{blue}{x}} \]
8. Add Preprocessing

Alternative 7: 99.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 97.3

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

5. Applied rewrites 97.3%

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

7. Applied rewrites 98.5%

   \[\leadsto \frac{\frac{2}{x}}{\color{blue}{x \cdot x}} \]
8. Add Preprocessing

Alternative 8: 98.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 97.3

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

5. Applied rewrites 97.3%

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

7. Applied rewrites 97.2%

   \[\leadsto \frac{2}{\left(x \cdot x\right) \cdot \color{blue}{x}} \]
8. Add Preprocessing

Alternative 9: 52.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 65.6%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 52.6

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

7. Applied rewrites 52.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2}}{x \cdot \left(x - 1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 54.5%

    \[\leadsto \frac{\color{blue}{2}}{x \cdot \left(x - 1\right)} \]
11. Add Preprocessing

Alternative 10: 52.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 65.6%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 52.6

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

7. Applied rewrites 52.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2}}{x \cdot \left(x - 1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 54.5%

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

11. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 54.5

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

13. Applied rewrites 54.5%

    \[\leadsto \frac{2}{\color{blue}{x \cdot x}} \]
14. Add Preprocessing

Alternative 11: 5.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 5.2

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

5. Applied rewrites 5.2%

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

Developer Target 1: 99.2% accurate, 1.8× 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 2024313 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

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