3frac (problem 3.3.3)

Percentage Accurate: 69.6% → 99.5%

Time: 10.1s

Alternatives: 12

Speedup: 1.0×

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: 69.6% 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: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fma 2.0 (pow x -2.0) 2.0) (pow x -3.0)))

C

double code(double x) {
	return fma(2.0, pow(x, -2.0), 2.0) * pow(x, -3.0);
}

Julia

function code(x)
	return Float64(fma(2.0, (x ^ -2.0), 2.0) * (x ^ -3.0))
end

Wolfram

code[x_] := N[(N[(2.0 * N[Power[x, -2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around inf 98.6%

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

   1. associate-*r/ 98.6%

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

   2. metadata-eval 98.6%

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

7. Simplified 98.6%

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

   1. div-inv 98.6%

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

   2. +-commutative 98.6%

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

   3. div-inv 98.6%

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

   4. fma-define 98.6%

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

   5. pow-flip 98.6%

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

   6. metadata-eval 98.6%

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

   7. pow-flip 99.2%

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

   8. metadata-eval 99.2%

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

9. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-3}} \]
10. Add Preprocessing

Alternative 2: 73.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ x -1.0))))
   (if (<= (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))) 2e-29)
     (pow x -3.0)
     (/ (+ t_0 (* (+ x 1.0) (- 2.0 x))) (* (+ x 1.0) t_0)))))

C

double code(double x) {
	double t_0 = x * (x + -1.0);
	double tmp;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 2e-29) {
		tmp = pow(x, -3.0);
	} else {
		tmp = (t_0 + ((x + 1.0) * (2.0 - x))) / ((x + 1.0) * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x + (-1.0d0))
    if ((((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))) <= 2d-29) then
        tmp = x ** (-3.0d0)
    else
        tmp = (t_0 + ((x + 1.0d0) * (2.0d0 - x))) / ((x + 1.0d0) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x + -1.0);
	double tmp;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 2e-29) {
		tmp = Math.pow(x, -3.0);
	} else {
		tmp = (t_0 + ((x + 1.0) * (2.0 - x))) / ((x + 1.0) * t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x + -1.0)
	tmp = 0
	if (((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 2e-29:
		tmp = math.pow(x, -3.0)
	else:
		tmp = (t_0 + ((x + 1.0) * (2.0 - x))) / ((x + 1.0) * t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x + -1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0))) <= 2e-29)
		tmp = x ^ -3.0;
	else
		tmp = Float64(Float64(t_0 + Float64(Float64(x + 1.0) * Float64(2.0 - x))) / Float64(Float64(x + 1.0) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x + -1.0);
	tmp = 0.0;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 2e-29)
		tmp = x ^ -3.0;
	else
		tmp = (t_0 + ((x + 1.0) * (2.0 - x))) / ((x + 1.0) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 2e-29], N[Power[x, -3.0], $MachinePrecision], N[(N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] * N[(2.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 1.99999999999999989e-29

   1. Initial program 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in x around -inf 67.9%

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

      1. associate-*r/ 67.9%

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

      2. distribute-lft-in 67.9%

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

      3. metadata-eval 67.9%

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

      4. mul-1-neg 67.9%

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

      5. mul-1-neg 67.9%

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

      6. distribute-neg-frac2 67.9%

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

      7. distribute-frac-neg2 67.9%

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

      8. remove-double-neg 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in x around 0 71.8%

      \[\leadsto \color{blue}{\frac{1}{{x}^{3}}} \]
   9. Step-by-step derivation

      1. metadata-eval 71.8%

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

      2. cube-div 71.8%

         \[\leadsto \color{blue}{{\left(\frac{1}{x}\right)}^{3}} \]

   10. Simplified 71.8%

       \[\leadsto \color{blue}{{\left(\frac{1}{x}\right)}^{3}} \]
   11. Step-by-step derivation

       1. inv-pow 71.8%

          \[\leadsto {\color{blue}{\left({x}^{-1}\right)}}^{3} \]

       2. pow-pow 71.8%

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

       3. metadata-eval 71.8%

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

       4. *-un-lft-identity 71.8%

          \[\leadsto \color{blue}{1 \cdot {x}^{-3}} \]

   12. Applied egg-rr 71.8%

       \[\leadsto \color{blue}{1 \cdot {x}^{-3}} \]
   13. Step-by-step derivation

       1. *-lft-identity 71.8%

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

   14. Simplified 71.8%

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

   if 1.99999999999999989e-29 < (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 51.3%

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

   3. Simplified 51.4%

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

      1. +-commutative 51.4%

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

      2. frac-add 51.9%

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

      3. div-inv 52.0%

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

      4. fma-define 53.7%

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

      5. *-rgt-identity 53.7%

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

      6. +-commutative 53.7%

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

      7. distribute-rgt-in 53.7%

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

      8. metadata-eval 53.7%

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

      9. metadata-eval 53.7%

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

      10. fma-define 53.7%

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

      11. metadata-eval 53.7%

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

      12. distribute-rgt-in 53.7%

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

      13. neg-mul-1 53.7%

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

      14. sub-neg 53.7%

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

      15. pow2 53.7%

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

   6. Applied egg-rr 53.7%

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

   8. Simplified 51.1%

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

   9. Taylor expanded in x around 0 52.2%

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

       1. mul-1-neg 52.2%

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

       2. unsub-neg 52.2%

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

   11. Simplified 52.2%

       \[\leadsto \frac{1}{x + 1} + \frac{\color{blue}{\frac{2 - x}{x}}}{x + -1} \]
   12. Step-by-step derivation

       1. associate-/l/ 51.9%

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

       2. frac-add 83.1%

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

       3. *-un-lft-identity 83.1%

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

   13. Applied egg-rr 83.1%

       \[\leadsto \color{blue}{\frac{\left(x + -1\right) \cdot x + \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;{x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x + -1\right) + \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (pow x -3.0)))

C

double code(double x) {
	return 2.0 * pow(x, -3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * (x ** (-3.0d0))
end function

Java

public static double code(double x) {
	return 2.0 * Math.pow(x, -3.0);
}

Python

def code(x):
	return 2.0 * math.pow(x, -3.0)

Julia

function code(x)
	return Float64(2.0 * (x ^ -3.0))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * (x ^ -3.0);
end

Wolfram

code[x_] := N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around inf 98.0%

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

   1. div-inv 98.0%

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

   2. pow-flip 98.6%

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

   3. metadata-eval 98.6%

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

7. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{2 \cdot {x}^{-3}} \]
8. Add Preprocessing

Alternative 4: 69.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Simplified 67.7%

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

   1. +-commutative 67.7%

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

   2. frac-add 18.3%

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

   3. div-inv 17.6%

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

   4. fma-define 7.7%

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

   5. *-rgt-identity 7.7%

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

   6. +-commutative 7.7%

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

   7. distribute-rgt-in 7.7%

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

   8. metadata-eval 7.7%

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

   9. metadata-eval 7.7%

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

   10. fma-define 7.7%

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

   11. metadata-eval 7.7%

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

   12. distribute-rgt-in 7.7%

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

   13. neg-mul-1 7.7%

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

   14. sub-neg 7.7%

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

   15. pow2 7.7%

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

6. Applied egg-rr 7.7%

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

8. Simplified 67.4%

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

9. Taylor expanded in x around inf 67.8%

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

    1. sub-neg 67.8%

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

    2. associate-*r/ 67.8%

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

    3. metadata-eval 67.8%

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

    4. metadata-eval 67.8%

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

11. Simplified 67.8%

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

    1. frac-add 67.8%

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

    2. *-un-lft-identity 67.8%

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

    3. associate--l+ 67.8%

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

    4. metadata-eval 67.8%

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

13. Applied egg-rr 67.8%

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

Alternative 5: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} + \frac{\frac{2 - x}{x}}{x + -1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Simplified 67.7%

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

   1. +-commutative 67.7%

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

   2. frac-add 18.3%

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

   3. div-inv 17.6%

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

   4. fma-define 7.7%

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

   5. *-rgt-identity 7.7%

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

   6. +-commutative 7.7%

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

   7. distribute-rgt-in 7.7%

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

   8. metadata-eval 7.7%

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

   9. metadata-eval 7.7%

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

   10. fma-define 7.7%

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

   11. metadata-eval 7.7%

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

   12. distribute-rgt-in 7.7%

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

   13. neg-mul-1 7.7%

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

   14. sub-neg 7.7%

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

   15. pow2 7.7%

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

6. Applied egg-rr 7.7%

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

8. Simplified 67.4%

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

9. Taylor expanded in x around 0 67.8%

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

    1. mul-1-neg 67.8%

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

    2. unsub-neg 67.8%

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

11. Simplified 67.8%

    \[\leadsto \frac{1}{x + 1} + \frac{\color{blue}{\frac{2 - x}{x}}}{x + -1} \]
12. Add Preprocessing

Alternative 6: 69.6% accurate, 1.0× speedup

Math

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

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(1.0 / Float64(x + 1.0)) + Float64(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[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

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

3. Simplified 67.7%

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

5. Final simplification 67.7%

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

Alternative 7: 69.6% 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]

Derivation

1. Initial program 67.7%

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

3. Final simplification 67.7%

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

Alternative 8: 68.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} + \frac{-1 + \frac{1}{x}}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around inf 66.3%

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

6. Final simplification 66.3%

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

Alternative 9: 68.4% 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 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around inf 66.0%

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

6. Final simplification 66.0%

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

Alternative 10: 6.3% 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 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around inf 66.0%

   \[\leadsto \frac{1}{1 + x} + \color{blue}{\frac{-1}{x}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 66.0%

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

   2. +-commutative 66.0%

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

   3. add-sqr-sqrt 20.4%

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

   4. sqrt-unprod 12.7%

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

   5. frac-times 12.0%

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

   6. metadata-eval 12.0%

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

   7. metadata-eval 12.0%

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

   8. frac-times 12.7%

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

   9. sqrt-prod 3.0%

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

   10. add-sqr-sqrt 6.4%

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

7. Applied egg-rr 6.4%

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

   1. *-lft-identity 6.4%

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

   2. +-commutative 6.4%

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

9. Simplified 6.4%

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

10. Final simplification 6.4%

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

Alternative 11: 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 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around inf 66.0%

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

6. Taylor expanded in x around 0 5.0%

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

Alternative 12: 5.1% 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 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in x around 0 5.0%

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

Developer target: 99.1% 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 2024100 
(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))))