3frac (problem 3.3.3)

Percentage Accurate: 69.0% → 99.5%

Time: 9.5s

Alternatives: 12

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: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

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

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 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in x around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

7. Simplified 99.2%

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

   1. div-inv 99.1%

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

   2. +-commutative 99.1%

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

   3. div-inv 99.1%

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

   4. fma-define 99.1%

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

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

   7. pow-flip 99.4%

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

   8. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

    1. metadata-eval 99.4%

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

    2. pow-flip 99.4%

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

    3. pow2 99.4%

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

    4. fma-define 99.4%

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

    5. div-inv 99.4%

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

    6. pow2 99.4%

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

11. Applied egg-rr 99.4%

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

    1. unpow2 99.4%

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

13. Applied egg-rr 99.4%

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

14. Final simplification 99.4%

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

Alternative 2: 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 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in x around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

7. Simplified 99.2%

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

   1. div-inv 99.1%

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

   2. +-commutative 99.1%

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

   3. div-inv 99.1%

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

   4. fma-define 99.1%

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

   5. pow-flip 99.1%

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

   6. metadata-eval 99.1%

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

   7. pow-flip 99.4%

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

   8. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around inf 98.8%

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

Alternative 3: 69.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((1.0 / (1.0 - (-1.0 / x))) / 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 - ((-1.0d0) / x))) / x) + (((-2.0d0) / x) + (1.0d0 / (x + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in x around inf 68.0%

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

   1. inv-pow 68.0%

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

   2. unpow-prod-down 68.1%

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

   3. inv-pow 68.0%

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

7. Applied egg-rr 68.0%

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

   1. associate-*l/ 68.1%

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

   2. *-lft-identity 68.1%

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

   3. unpow-1 68.1%

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

9. Simplified 68.1%

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

10. Final simplification 68.1%

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

Alternative 4: 69.0% accurate, 0.8× speedup

Math

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

FPCore

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

Python

def code(x):
	return ((-2.0 / x) + (1.0 / (x + -1.0))) + (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 / Float64(x * Float64(1.0 - Float64(-1.0 / x)))))
end

MATLAB

function tmp = code(x)
	tmp = ((-2.0 / x) + (1.0 / (x + -1.0))) + (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 / N[(x * N[(1.0 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in x around inf 68.0%

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

6. Final simplification 68.0%

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

Alternative 5: 69.0% 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 67.9%

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

3. Simplified 67.9%

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

5. Final simplification 67.9%

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

Alternative 6: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Final simplification 67.9%

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

Alternative 7: 67.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in x around inf 66.5%

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

   1. *-un-lft-identity 66.5%

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

7. Applied egg-rr 66.5%

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

   1. *-lft-identity 66.5%

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

   2. associate-+r+ 66.5%

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

   3. metadata-eval 66.5%

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

   4. associate-*r/ 66.5%

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

   5. distribute-rgt1-in 66.5%

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

   6. metadata-eval 66.5%

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

   7. neg-mul-1 66.5%

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

   8. distribute-neg-frac 66.5%

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

   9. metadata-eval 66.5%

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

9. Simplified 66.5%

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

10. Final simplification 66.5%

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

Alternative 8: 67.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in x around inf 66.5%

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

6. Taylor expanded in x around 0 4.9%

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

   1. div-inv 4.9%

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

   2. mul-1-neg 4.9%

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

   3. add-sqr-sqrt 2.3%

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

   4. sqrt-prod 48.6%

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

   5. frac-times 50.8%

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

   6. metadata-eval 50.8%

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

   7. metadata-eval 50.8%

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

   8. frac-times 48.6%

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

   9. sqrt-unprod 3.4%

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

   10. add-sqr-sqrt 6.3%

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

8. Applied egg-rr 6.3%

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

   1. add-sqr-sqrt 3.4%

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

   2. sqrt-unprod 48.6%

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

   3. frac-times 50.8%

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

   4. metadata-eval 50.8%

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

   5. metadata-eval 50.8%

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

   6. frac-times 48.6%

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

   7. sqrt-prod 2.3%

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

   8. expm1-log1p-u 2.3%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right)\right)} \]

   9. add-sqr-sqrt 4.9%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{x}}\right)\right) \]

   10. expm1-undefine 66.2%

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

10. Applied egg-rr 66.8%

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

    1. associate--l+ 66.8%

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

12. Simplified 66.8%

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

13. Final simplification 66.8%

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

Alternative 9: 67.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in x around inf 66.5%

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

6. Taylor expanded in x around inf 66.3%

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

7. Final simplification 66.3%

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

Alternative 10: 6.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in x around inf 66.5%

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

6. Taylor expanded in x around 0 4.9%

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

   1. div-inv 4.9%

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

   2. mul-1-neg 4.9%

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

   3. add-sqr-sqrt 2.3%

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

   4. sqrt-prod 48.6%

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

   5. frac-times 50.8%

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

   6. metadata-eval 50.8%

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

   7. metadata-eval 50.8%

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

   8. frac-times 48.6%

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

   9. sqrt-unprod 3.4%

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

   10. add-sqr-sqrt 6.3%

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

8. Applied egg-rr 6.3%

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

9. Taylor expanded in x around 0 6.3%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
10. Add Preprocessing

Alternative 11: 5.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in x around inf 66.5%

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

6. Taylor expanded in x around 0 4.9%

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

Alternative 12: 5.0% 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 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in x around 0 4.9%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
6. 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 2024145 
(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))))