3frac (problem 3.3.3)

Percentage Accurate: 68.7% → 99.8%

Time: 6.6s

Alternatives: 6

Speedup: 1.4×

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: 68.7% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

3. Simplified 64.9%

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

   1. frac-add 18.7%

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

   2. frac-add 21.9%

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

   3. *-un-lft-identity 21.9%

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

   4. fma-define 20.9%

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

   5. *-rgt-identity 20.9%

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

   6. +-commutative 20.9%

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

   7. distribute-rgt-in 20.9%

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

   8. metadata-eval 20.9%

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

   9. metadata-eval 20.9%

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

   10. fma-define 20.9%

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

   11. metadata-eval 20.9%

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

   12. distribute-rgt-in 20.9%

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

   13. neg-mul-1 20.9%

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

   14. sub-neg 20.9%

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

   15. pow2 20.9%

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

6. Applied egg-rr 20.9%

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

   1. +-commutative 20.9%

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

   2. fma-undefine 20.9%

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

   3. *-commutative 20.9%

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

   4. fma-define 20.9%

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

   5. +-commutative 20.9%

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

   6. sub-neg 20.9%

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

   7. unpow2 20.9%

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

   8. neg-mul-1 20.9%

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

   9. distribute-rgt-in 20.9%

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

8. Simplified 20.9%

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

9. Taylor expanded in x around 0 99.1%

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

    1. associate-/r* 99.8%

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

    2. div-inv 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

3. Simplified 64.9%

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

   1. frac-add 18.7%

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

   2. frac-add 21.9%

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

   3. *-un-lft-identity 21.9%

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

   4. fma-define 20.9%

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

   5. *-rgt-identity 20.9%

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

   6. +-commutative 20.9%

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

   7. distribute-rgt-in 20.9%

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

   8. metadata-eval 20.9%

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

   9. metadata-eval 20.9%

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

   10. fma-define 20.9%

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

   11. metadata-eval 20.9%

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

   12. distribute-rgt-in 20.9%

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

   13. neg-mul-1 20.9%

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

   14. sub-neg 20.9%

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

   15. pow2 20.9%

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

6. Applied egg-rr 20.9%

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

   1. +-commutative 20.9%

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

   2. fma-undefine 20.9%

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

   3. *-commutative 20.9%

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

   4. fma-define 20.9%

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

   5. +-commutative 20.9%

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

   6. sub-neg 20.9%

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

   7. unpow2 20.9%

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

   8. neg-mul-1 20.9%

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

   9. distribute-rgt-in 20.9%

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

8. Simplified 20.9%

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

9. Taylor expanded in x around 0 99.1%

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

10. Final simplification 99.1%

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

Alternative 3: 67.4% 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 64.9%

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

3. Simplified 64.9%

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

5. Taylor expanded in x around inf 63.1%

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

6. Final simplification 63.1%

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

Alternative 4: 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 64.9%

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

3. Simplified 64.9%

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

5. Taylor expanded in x around 0 4.8%

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

6. Final simplification 4.8%

   \[\leadsto \frac{-2}{x} \]
7. Add Preprocessing

Alternative 5: 5.1% 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 64.9%

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

3. Simplified 64.9%

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

5. Taylor expanded in x around inf 63.1%

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

6. Taylor expanded in x around 0 4.9%

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

7. Final simplification 4.9%

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

Alternative 6: 3.4% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 64.9%

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

3. Simplified 64.9%

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

5. Taylor expanded in x around inf 63.1%

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

6. Taylor expanded in x around 0 3.5%

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

7. Taylor expanded in x around inf 3.5%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 3.5%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer target: 99.0% 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 2024044 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))