3frac (problem 3.3.3)

Percentage Accurate: 69.8% → 99.8%

Time: 12.0s

Alternatives: 6

Speedup: 2.0×

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: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

   1. +-commutative N/A

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

   2. frac-sub N/A

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

   3. frac-add N/A

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

   4. /-lowering-/.f64 N/A

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

4. Applied egg-rr 18.7%

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

5. Taylor expanded in x around 0

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

7. Simplified 98.9%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. +-lowering-+.f64 99.8

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

9. Applied egg-rr 99.8%

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

Alternative 2: 98.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

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

   2. remove-double-neg N/A

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

   3. sub-neg N/A

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

   4. --lowering--.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. /-lowering-/.f64 N/A

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

   10. +-rgt-identity N/A

       \[\leadsto \frac{2 - \frac{-2}{\color{blue}{{x}^{2} + 0}}}{{x}^{3}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 - \frac{-2}{\color{blue}{x \cdot x} + 0}}{{x}^{3}} \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{2 - \frac{-2}{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}}}{{x}^{3}} \]

   13. cube-mult N/A

       \[\leadsto \frac{2 - \frac{-2}{\mathsf{fma}\left(x, x, 0\right)}}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

   14. unpow2 N/A

       \[\leadsto \frac{2 - \frac{-2}{\mathsf{fma}\left(x, x, 0\right)}}{x \cdot \color{blue}{{x}^{2}}} \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 - \frac{-2}{\mathsf{fma}\left(x, x, 0\right)}}{\color{blue}{x \cdot {x}^{2}}} \]

   16. +-rgt-identity N/A

       \[\leadsto \frac{2 - \frac{-2}{\mathsf{fma}\left(x, x, 0\right)}}{x \cdot \color{blue}{\left({x}^{2} + 0\right)}} \]

   17. unpow2 N/A

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

   18. accelerator-lowering-fma.f64 98.6

       \[\leadsto \frac{2 - \frac{-2}{\mathsf{fma}\left(x, x, 0\right)}}{x \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \]

5. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{2 - \frac{-2}{\mathsf{fma}\left(x, x, 0\right)}}{x \cdot \mathsf{fma}\left(x, x, 0\right)}} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2 - \frac{-2}{x \cdot x + 0}}{x}}{x \cdot x}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{2 - \frac{-2}{x \cdot x + 0}}{x}}{x}}{x}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{2 - \frac{-2}{x \cdot x + 0}}{x}}{x}}{x}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{2 - \frac{-2}{x \cdot x + 0}}{x}}{x}}}{x} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{2 - \frac{-2}{x \cdot x + 0}}{x}}}{x}}{x} \]

   7. sub-neg N/A

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

   8. +-lowering-+.f64 N/A

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

   9. +-rgt-identity N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f64 N/A

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

   13. +-rgt-identity N/A

       \[\leadsto \frac{\frac{\frac{2 + \frac{2}{\color{blue}{x \cdot x + 0}}}{x}}{x}}{x} \]

   14. accelerator-lowering-fma.f64 99.6

       \[\leadsto \frac{\frac{\frac{2 + \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}}}{x}}{x}}{x} \]

7. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2 + \frac{2}{\mathsf{fma}\left(x, x, 0\right)}}{x}}{x}}{x}} \]

8. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

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

   2. +-rgt-identity N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{{x}^{2} + 0}}}{x} \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{x \cdot x} + 0}}{x} \]

   4. accelerator-lowering-fma.f64 99.1

      \[\leadsto \frac{\frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}}}{x} \]

10. Simplified 99.1%

    \[\leadsto \frac{\color{blue}{\frac{2}{\mathsf{fma}\left(x, x, 0\right)}}}{x} \]
11. Step-by-step derivation

    1. +-rgt-identity N/A

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

    2. *-lowering-*.f64 99.1

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

12. Applied egg-rr 99.1%

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

Alternative 3: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

   1. +-commutative N/A

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

   2. frac-sub N/A

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

   3. frac-add N/A

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

   4. /-lowering-/.f64 N/A

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

4. Applied egg-rr 18.7%

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

5. Taylor expanded in x around 0

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

7. Simplified 98.9%

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

8. Taylor expanded in x around 0

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

10. Simplified 98.9%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x, -1\right), 0\right)}} \]
11. Step-by-step derivation

    1. +-rgt-identity N/A

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

    2. *-commutative N/A

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

    3. metadata-eval N/A

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

    4. metadata-eval N/A

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

    5. sub-neg N/A

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

    6. *-lowering-*.f64 N/A

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

    7. sub-neg N/A

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

    8. metadata-eval N/A

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

    9. metadata-eval N/A

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

    10. accelerator-lowering-fma.f64 98.9

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

12. Applied egg-rr 98.9%

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

13. Final simplification 98.9%

    \[\leadsto \frac{2}{x \cdot \mathsf{fma}\left(x, x, -1\right)} \]
14. Add Preprocessing

Alternative 4: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot \mathsf{fma}\left(x, x, 0\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* x (fma x x 0.0))))

C

double code(double x) {
	return 2.0 / (x * fma(x, x, 0.0));
}

Julia

function code(x)
	return Float64(2.0 / Float64(x * fma(x, x, 0.0)))
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(x * x + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-rgt-identity N/A

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

   6. unpow2 N/A

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

   7. accelerator-lowering-fma.f64 98.1

      \[\leadsto \frac{2}{x \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{\frac{2}{x \cdot \mathsf{fma}\left(x, x, 0\right)}} \]
6. Add Preprocessing

Alternative 5: 52.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

   1. frac-sub N/A

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

   2. associate-/r* N/A

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

   3. frac-add N/A

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

   4. /-lowering-/.f64 N/A

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

4. Applied egg-rr 70.8%

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

5. Taylor expanded in x around 0

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

7. Simplified 55.0%

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

8. Taylor expanded in x around inf

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

10. Simplified 55.0%

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

Alternative 6: 5.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 5.2

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

5. Simplified 5.2%

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

Developer Target 1: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024197 
(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))))