3frac (problem 3.3.3)

Percentage Accurate: 70.1% → 99.3%

Time: 9.3s

Alternatives: 4

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: 70.1% 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 67.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
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 99.0

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.5%

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

Alternative 2: 98.8% accurate, 1.6× speedup

Math

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

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 / x) / Float64(x * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

3. Taylor expanded in x around inf

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

   1. lower-/.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. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 98.0

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

5. Applied rewrites 98.0%

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

7. Applied rewrites 98.8%

   \[\leadsto \frac{\frac{2}{x}}{\color{blue}{x \cdot x}} \]
8. 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 2024226 
(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))))