3frac (problem 3.3.3)

Percentage Accurate: 69.1% → 99.8%

Time: 9.9s

Alternatives: 7

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: 99.8% 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 69.5%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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 21.3%

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

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 98.8%

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

   1. lift-/.f64 N/A

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

   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. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

9. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.6× speedup

Math

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

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

Wolfram

code[x_] := N[(N[(2.0 / x), $MachinePrecision] / N[(x * x + -1.0), $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 20.3%

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

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 99.1%

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

   1. lift-/.f64 N/A

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

   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. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

9. Applied rewrites 99.8%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. distribute-lft1-in N/A

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

    5. *-commutative N/A

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

    6. associate-/r* N/A

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

    7. associate-/r* N/A

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

    8. lift-+.f64 N/A

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

    9. metadata-eval N/A

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

11. Applied rewrites 99.8%

    \[\leadsto \color{blue}{\frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, -1\right)}} \]
12. 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 2024219 
(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))))