3frac (problem 3.3.3)

Percentage Accurate: 69.6% → 99.7%

Time: 10.2s

Alternatives: 10

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: 69.6% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(2 + 2 \cdot \frac{\frac{1}{x}}{x}\right) \cdot \left(1 + {x}^{-4}\right)\right) \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (+ 2.0 (* 2.0 (/ (/ 1.0 x) x))) (+ 1.0 (pow x -4.0))) (pow x -3.0)))

C

double code(double x) {
	return ((2.0 + (2.0 * ((1.0 / x) / x))) * (1.0 + pow(x, -4.0))) * pow(x, -3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.0d0 + (2.0d0 * ((1.0d0 / x) / x))) * (1.0d0 + (x ** (-4.0d0)))) * (x ** (-3.0d0))
end function

Java

public static double code(double x) {
	return ((2.0 + (2.0 * ((1.0 / x) / x))) * (1.0 + Math.pow(x, -4.0))) * Math.pow(x, -3.0);
}

Python

def code(x):
	return ((2.0 + (2.0 * ((1.0 / x) / x))) * (1.0 + math.pow(x, -4.0))) * math.pow(x, -3.0)

Julia

function code(x)
	return Float64(Float64(Float64(2.0 + Float64(2.0 * Float64(Float64(1.0 / x) / x))) * Float64(1.0 + (x ^ -4.0))) * (x ^ -3.0))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 + (2.0 * ((1.0 / x) / x))) * (1.0 + (x ^ -4.0))) * (x ^ -3.0);
end

Wolfram

code[x_] := N[(N[(N[(2.0 + N[(2.0 * N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in x around -inf 99.0%

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

   1. mul-1-neg 99.0%

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

   2. distribute-neg-frac 99.0%

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

7. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. div-inv 99.0%

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

   3. fma-define 99.0%

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

   4. +-commutative 99.0%

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

   5. div-inv 99.0%

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

   6. fma-define 99.0%

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

   7. pow-flip 99.0%

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

   8. metadata-eval 99.0%

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

   9. pow-flip 99.0%

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

   10. metadata-eval 99.0%

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

   11. +-commutative 99.0%

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

   12. div-inv 99.0%

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

   13. fma-define 99.0%

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

   14. pow-flip 99.0%

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

   15. metadata-eval 99.0%

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

   16. pow-flip 99.5%

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

   17. metadata-eval 99.5%

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

9. Applied egg-rr 99.5%

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

    1. fma-undefine 99.5%

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

    2. +-commutative 99.5%

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

    3. *-rgt-identity 99.5%

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

    4. distribute-lft-out 99.5%

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

11. Simplified 99.5%

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

    1. fma-undefine 99.5%

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

13. Applied egg-rr 99.5%

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

    1. metadata-eval 99.5%

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

    2. pow-div 99.5%

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

    3. inv-pow 99.5%

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

    4. pow1 99.5%

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

15. Applied egg-rr 99.5%

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

16. Final simplification 99.5%

    \[\leadsto \left(\left(2 + 2 \cdot \frac{\frac{1}{x}}{x}\right) \cdot \left(1 + {x}^{-4}\right)\right) \cdot {x}^{-3} \]
17. Add Preprocessing

Alternative 2: 98.9% 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 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in x around inf 98.7%

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

   1. associate-*r/ 98.7%

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

   2. metadata-eval 98.7%

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

7. Simplified 98.7%

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

   1. unpow2 98.7%

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

9. Applied egg-rr 98.7%

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

Alternative 3: 99.0% 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 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in x around -inf 99.0%

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

   1. mul-1-neg 99.0%

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

   2. distribute-neg-frac 99.0%

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

7. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. div-inv 99.0%

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

   3. fma-define 99.0%

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

   4. +-commutative 99.0%

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

   5. div-inv 99.0%

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

   6. fma-define 99.0%

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

   7. pow-flip 99.0%

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

   8. metadata-eval 99.0%

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

   9. pow-flip 99.0%

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

   10. metadata-eval 99.0%

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

   11. +-commutative 99.0%

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

   12. div-inv 99.0%

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

   13. fma-define 99.0%

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

   14. pow-flip 99.0%

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

   15. metadata-eval 99.0%

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

   16. pow-flip 99.5%

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

   17. metadata-eval 99.5%

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

9. Applied egg-rr 99.5%

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

    1. fma-undefine 99.5%

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

    2. +-commutative 99.5%

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

    3. *-rgt-identity 99.5%

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

    4. distribute-lft-out 99.5%

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

11. Simplified 99.5%

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

12. Taylor expanded in x around inf 98.5%

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

Alternative 4: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

3. Simplified 73.0%

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

   1. frac-2neg 73.0%

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

   2. metadata-eval 73.0%

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

   3. frac-add 19.1%

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

   4. *-commutative 19.1%

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

   5. neg-mul-1 19.1%

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

   6. fma-define 19.2%

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

   7. +-commutative 19.2%

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

   8. distribute-neg-in 19.2%

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

   9. metadata-eval 19.2%

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

   10. sub-neg 19.2%

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

   11. +-commutative 19.2%

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

   12. distribute-neg-in 19.2%

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

   13. metadata-eval 19.2%

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

   14. sub-neg 19.2%

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

6. Applied egg-rr 19.2%

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

   1. associate-/r* 73.0%

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

   2. fmm-undef 72.6%

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

   3. *-commutative 72.6%

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

   4. div-sub 72.6%

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

   5. +-rgt-identity 72.6%

      \[\leadsto \frac{1}{1 + x} + \frac{\frac{\color{blue}{\left(1 - x\right) \cdot -2 + 0}}{x} - \frac{x}{x}}{1 - x} \]

   6. +-rgt-identity 72.6%

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

   7. *-commutative 72.6%

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

   8. *-lft-identity 72.6%

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

   9. associate-*l/ 72.6%

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

   10. lft-mult-inverse 72.6%

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

8. Simplified 72.6%

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

9. Taylor expanded in x around 0 73.0%

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

    1. div-sub 73.0%

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

    2. *-inverses 73.0%

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

    3. metadata-eval 73.0%

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

    4. associate-*r/ 73.0%

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

    5. cancel-sign-sub-inv 73.0%

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

    6. metadata-eval 73.0%

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

    7. associate-*r/ 73.0%

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

    8. metadata-eval 73.0%

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

11. Simplified 73.0%

    \[\leadsto \frac{1}{1 + x} + \frac{\color{blue}{1 + \frac{-2}{x}}}{1 - x} \]
12. Add Preprocessing

Alternative 5: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

3. Simplified 73.0%

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

Alternative 6: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

3. Final simplification 73.0%

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

Alternative 7: 68.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in x around inf 71.4%

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

6. Taylor expanded in x around inf 71.0%

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

7. Final simplification 71.0%

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

Alternative 8: 68.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in x around inf 71.2%

   \[\leadsto \frac{1}{1 + x} + \color{blue}{\frac{-1}{x}} \]
6. Add Preprocessing

Alternative 9: 5.1% accurate, 5.0× 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 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in x around 0 5.2%

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

Alternative 10: 3.4% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in x around inf 71.4%

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

6. Taylor expanded in x around 0 3.3%

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

7. Taylor expanded in x around inf 3.3%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Developer Target 1: 99.3% 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 2024150 
(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))))