3frac (problem 3.3.3)

Percentage Accurate: 69.1% → 99.3%

Time: 10.2s

Alternatives: 5

Speedup: 2.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% 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: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.6%

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

Alternative 2: 99.2% accurate, 1.8× speedup

Math

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

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(2.0 / Float64(fma(x, x, x) * Float64(x + -1.0)))
end

Wolfram

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

Derivation

1. Initial program 69.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-sub N/A

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

   8. frac-add N/A

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

   9. lower-/.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 rewrites 20.1%

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

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. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. *-lft-identity N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-lft-identity N/A

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

   12. +-commutative N/A

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

   13. distribute-rgt-in N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f64 N/A

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. lift-+.f64 99.2

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

9. Applied rewrites 99.2%

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

Developer Target 1: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024222 
(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))))