3frac (problem 3.3.3)

Percentage Accurate: 68.6% → 99.7%

Time: 8.8s

Alternatives: 11

Speedup: 1.0×

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: 68.6% 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: 99.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma 2.0 (pow x -2.0) (fma 2.0 (pow x -4.0) 2.0)) (pow x -3.0)))

C

double code(double x) {
	return fma(2.0, pow(x, -2.0), fma(2.0, pow(x, -4.0), 2.0)) * pow(x, -3.0);
}

Julia

function code(x)
	return Float64(fma(2.0, (x ^ -2.0), fma(2.0, (x ^ -4.0), 2.0)) * (x ^ -3.0))
end

Wolfram

code[x_] := N[(N[(2.0 * N[Power[x, -2.0], $MachinePrecision] + N[(2.0 * N[Power[x, -4.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 99.1%

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

   1. associate-+r+ 99.1%

      \[\leadsto \frac{\color{blue}{\left(2 + 2 \cdot \frac{1}{{x}^{2}}\right) + \frac{2}{{x}^{4}}}}{{x}^{3}} \]

   2. +-commutative 99.1%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{x}^{2}} + 2\right)} + \frac{2}{{x}^{4}}}{{x}^{3}} \]

   3. associate-+l+ 99.1%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}}{{x}^{3}} \]

   4. associate-*r/ 99.1%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

   5. metadata-eval 99.1%

      \[\leadsto \frac{\frac{\color{blue}{2}}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

7. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 99.1%

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}}} \]

   2. div-inv 99.1%

      \[\leadsto \left(\color{blue}{2 \cdot \frac{1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}} \]

   3. fma-define 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2 + \frac{2}{{x}^{4}}\right)} \cdot \frac{1}{{x}^{3}} \]

   4. pow-flip 99.1%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2 + \frac{2}{{x}^{4}}\right) \cdot \frac{1}{{x}^{3}} \]

   5. metadata-eval 99.1%

      \[\leadsto \mathsf{fma}\left(2, {x}^{\color{blue}{-2}}, 2 + \frac{2}{{x}^{4}}\right) \cdot \frac{1}{{x}^{3}} \]

   6. +-commutative 99.1%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\frac{2}{{x}^{4}} + 2}\right) \cdot \frac{1}{{x}^{3}} \]

   7. div-inv 99.1%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{2 \cdot \frac{1}{{x}^{4}}} + 2\right) \cdot \frac{1}{{x}^{3}} \]

   8. fma-define 99.1%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{4}}, 2\right)}\right) \cdot \frac{1}{{x}^{3}} \]

   9. pow-flip 99.1%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-4\right)}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

   10. metadata-eval 99.1%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{\color{blue}{-4}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

   11. pow-flip 99.3%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot \color{blue}{{x}^{\left(-3\right)}} \]

   12. metadata-eval 99.3%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{\color{blue}{-3}} \]

9. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{-3}} \]
10. Add Preprocessing

Alternative 2: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{x \cdot x} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ 2.0 (* x x)) (+ 2.0 (/ 2.0 (pow x 4.0)))) (pow x 3.0)))

C

double code(double x) {
	return ((2.0 / (x * x)) + (2.0 + (2.0 / pow(x, 4.0)))) / pow(x, 3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.0d0 / (x * x)) + (2.0d0 + (2.0d0 / (x ** 4.0d0)))) / (x ** 3.0d0)
end function

Java

public static double code(double x) {
	return ((2.0 / (x * x)) + (2.0 + (2.0 / Math.pow(x, 4.0)))) / Math.pow(x, 3.0);
}

Python

def code(x):
	return ((2.0 / (x * x)) + (2.0 + (2.0 / math.pow(x, 4.0)))) / math.pow(x, 3.0)

Julia

function code(x)
	return Float64(Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 + Float64(2.0 / (x ^ 4.0)))) / (x ^ 3.0))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 / (x * x)) + (2.0 + (2.0 / (x ^ 4.0)))) / (x ^ 3.0);
end

Wolfram

code[x_] := N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(2.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 99.1%

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

   1. associate-+r+ 99.1%

      \[\leadsto \frac{\color{blue}{\left(2 + 2 \cdot \frac{1}{{x}^{2}}\right) + \frac{2}{{x}^{4}}}}{{x}^{3}} \]

   2. +-commutative 99.1%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{x}^{2}} + 2\right)} + \frac{2}{{x}^{4}}}{{x}^{3}} \]

   3. associate-+l+ 99.1%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}}{{x}^{3}} \]

   4. associate-*r/ 99.1%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

   5. metadata-eval 99.1%

      \[\leadsto \frac{\frac{\color{blue}{2}}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

7. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
8. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{x \cdot x}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

9. Applied egg-rr 99.1%

   \[\leadsto \frac{\frac{2}{\color{blue}{x \cdot x}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]
10. Add Preprocessing

Alternative 3: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 98.9%

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

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

7. Simplified 98.9%

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

   1. unpow2 99.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{x \cdot x}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

9. Applied egg-rr 98.9%

   \[\leadsto \frac{2 + \frac{2}{\color{blue}{x \cdot x}}}{{x}^{3}} \]
10. Add Preprocessing

Alternative 4: 99.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 99.1%

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

   1. associate-+r+ 99.1%

      \[\leadsto \frac{\color{blue}{\left(2 + 2 \cdot \frac{1}{{x}^{2}}\right) + \frac{2}{{x}^{4}}}}{{x}^{3}} \]

   2. +-commutative 99.1%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{x}^{2}} + 2\right)} + \frac{2}{{x}^{4}}}{{x}^{3}} \]

   3. associate-+l+ 99.1%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}}{{x}^{3}} \]

   4. associate-*r/ 99.1%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

   5. metadata-eval 99.1%

      \[\leadsto \frac{\frac{\color{blue}{2}}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

7. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 99.1%

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}}} \]

   2. div-inv 99.1%

      \[\leadsto \left(\color{blue}{2 \cdot \frac{1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)\right) \cdot \frac{1}{{x}^{3}} \]

   3. fma-define 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2 + \frac{2}{{x}^{4}}\right)} \cdot \frac{1}{{x}^{3}} \]

   4. pow-flip 99.1%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2 + \frac{2}{{x}^{4}}\right) \cdot \frac{1}{{x}^{3}} \]

   5. metadata-eval 99.1%

      \[\leadsto \mathsf{fma}\left(2, {x}^{\color{blue}{-2}}, 2 + \frac{2}{{x}^{4}}\right) \cdot \frac{1}{{x}^{3}} \]

   6. +-commutative 99.1%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\frac{2}{{x}^{4}} + 2}\right) \cdot \frac{1}{{x}^{3}} \]

   7. div-inv 99.1%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{2 \cdot \frac{1}{{x}^{4}}} + 2\right) \cdot \frac{1}{{x}^{3}} \]

   8. fma-define 99.1%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{4}}, 2\right)}\right) \cdot \frac{1}{{x}^{3}} \]

   9. pow-flip 99.1%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-4\right)}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

   10. metadata-eval 99.1%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{\color{blue}{-4}}, 2\right)\right) \cdot \frac{1}{{x}^{3}} \]

   11. pow-flip 99.3%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot \color{blue}{{x}^{\left(-3\right)}} \]

   12. metadata-eval 99.3%

       \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{\color{blue}{-3}} \]

9. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{-3}} \]

10. Taylor expanded in x around inf 98.6%

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

Alternative 5: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

3. Simplified 68.0%

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

5. Final simplification 68.0%

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

Alternative 6: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

3. Simplified 68.0%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative 68.0%

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

   2. *-un-lft-identity 68.0%

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

   3. fma-define 68.0%

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

   4. frac-2neg 68.0%

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

   5. metadata-eval 68.0%

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

   6. distribute-frac-neg2 68.0%

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

   7. fmm-undef 68.0%

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

   8. *-un-lft-identity 68.0%

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

6. Applied egg-rr 68.0%

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

   1. associate--l+ 68.0%

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

   2. sub-neg 68.0%

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

   3. distribute-neg-frac 68.0%

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

   4. metadata-eval 68.0%

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

   5. +-commutative 68.0%

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

8. Simplified 68.0%

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

Alternative 7: 68.6% 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 68.0%

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

3. Final simplification 68.0%

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

Alternative 8: 67.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 66.2%

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

6. Final simplification 66.2%

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

Alternative 9: 67.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

3. Simplified 68.0%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-exp-log 66.8%

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

   2. +-commutative 66.8%

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

   3. associate-+l+ 66.8%

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

6. Applied egg-rr 66.8%

   \[\leadsto \color{blue}{e^{\log \left(\frac{-2}{x} + \left(\frac{1}{x + -1} + \frac{1}{1 + x}\right)\right)}} \]
7. Step-by-step derivation

   1. rem-exp-log 68.0%

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

   2. +-commutative 68.0%

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

   3. +-commutative 68.0%

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

8. Applied egg-rr 68.0%

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

9. Taylor expanded in x around inf 66.2%

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

10. Final simplification 66.2%

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

Alternative 10: 67.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 66.4%

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

6. Taylor expanded in x around inf 66.2%

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

7. Final simplification 66.2%

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

Alternative 11: 67.2% 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 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 66.2%

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

6. Taylor expanded in x around inf 65.9%

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

7. Taylor expanded in x around 0 65.9%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.2% 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 2024177 
(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))))