3frac (problem 3.3.3)

Percentage Accurate: 84.9% → 98.3%

Time: 14.8s

Alternatives: 10

Speedup: 1.0×

Specification

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: 84.9% 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: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ t_1 := x \cdot \left(x + -1\right)\\ \mathbf{if}\;t_0 \leq -50000000000000:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \mathsf{fma}\left(2, x + -1, -x\right) \cdot \left(-1 - x\right)}{\left(1 + x\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))))
        (t_1 (* x (+ x -1.0))))
   (if (<= t_0 -50000000000000.0)
     (/ -2.0 x)
     (if (<= t_0 5e-29)
       (+ (/ 2.0 (pow x 5.0)) (/ 2.0 (pow x 3.0)))
       (/
        (+ t_1 (* (fma 2.0 (+ x -1.0) (- x)) (- -1.0 x)))
        (* (+ 1.0 x) t_1))))))

C

double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_1 = x * (x + -1.0);
	double tmp;
	if (t_0 <= -50000000000000.0) {
		tmp = -2.0 / x;
	} else if (t_0 <= 5e-29) {
		tmp = (2.0 / pow(x, 5.0)) + (2.0 / pow(x, 3.0));
	} else {
		tmp = (t_1 + (fma(2.0, (x + -1.0), -x) * (-1.0 - x))) / ((1.0 + x) * t_1);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	t_1 = Float64(x * Float64(x + -1.0))
	tmp = 0.0
	if (t_0 <= -50000000000000.0)
		tmp = Float64(-2.0 / x);
	elseif (t_0 <= 5e-29)
		tmp = Float64(Float64(2.0 / (x ^ 5.0)) + Float64(2.0 / (x ^ 3.0)));
	else
		tmp = Float64(Float64(t_1 + Float64(fma(2.0, Float64(x + -1.0), Float64(-x)) * Float64(-1.0 - x))) / Float64(Float64(1.0 + x) * t_1));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000000000.0], N[(-2.0 / x), $MachinePrecision], If[LessEqual[t$95$0, 5e-29], N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(N[(2.0 * N[(x + -1.0), $MachinePrecision] + (-x)), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -5e13

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   if -5e13 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 4.99999999999999986e-29

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in x around inf 98.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 98.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + 2 \cdot \frac{1}{{x}^{3}} \]

      2. metadata-eval 98.4%

         \[\leadsto \frac{\color{blue}{2}}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]

      3. associate-*r/ 98.4%

         \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} \]

      4. metadata-eval 98.4%

         \[\leadsto \frac{2}{{x}^{5}} + \frac{\color{blue}{2}}{{x}^{3}} \]

   6. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}} \]

   if 4.99999999999999986e-29 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

      1. flip-+ 99.2%

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

      2. sub-neg 99.2%

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

      3. metadata-eval 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. +-commutative 99.2%

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

      6. associate-/r/ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. distribute-neg-in 99.2%

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

      10. metadata-eval 99.2%

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

      11. sub-neg 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. associate-*l/ 99.2%

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

      2. *-lft-identity 99.2%

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

   7. Simplified 99.2%

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

      1. clear-num 99.2%

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

      2. frac-sub 99.3%

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

      3. frac-sub 99.3%

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

      4. *-un-lft-identity 99.3%

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

      5. metadata-eval 99.3%

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

      6. flip-+ 100.0%

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

      7. fma-neg 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. metadata-eval 100.0%

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

      10. flip-+ 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -50000000000000:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x + -1\right) + \mathsf{fma}\left(2, x + -1, -x\right) \cdot \left(-1 - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ t_1 := x \cdot x - x\\ t_2 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-21}:\\ \;\;\;\;\frac{t_1 + \left(1 + x\right) \cdot \left(x - \mathsf{fma}\left(x, 2, -2\right)\right)}{\left(1 + x\right) \cdot t_1}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \mathsf{fma}\left(2, x + -1, -x\right) \cdot \left(-1 - x\right)}{\left(1 + x\right) \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ x -1.0)))
        (t_1 (- (* x x) x))
        (t_2 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_2 -5e-21)
     (/ (+ t_1 (* (+ 1.0 x) (- x (fma x 2.0 -2.0)))) (* (+ 1.0 x) t_1))
     (if (<= t_2 5e-29)
       (/ 2.0 (pow x 3.0))
       (/
        (+ t_0 (* (fma 2.0 (+ x -1.0) (- x)) (- -1.0 x)))
        (* (+ 1.0 x) t_0))))))

C

double code(double x) {
	double t_0 = x * (x + -1.0);
	double t_1 = (x * x) - x;
	double t_2 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_2 <= -5e-21) {
		tmp = (t_1 + ((1.0 + x) * (x - fma(x, 2.0, -2.0)))) / ((1.0 + x) * t_1);
	} else if (t_2 <= 5e-29) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = (t_0 + (fma(2.0, (x + -1.0), -x) * (-1.0 - x))) / ((1.0 + x) * t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x + -1.0))
	t_1 = Float64(Float64(x * x) - x)
	t_2 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_2 <= -5e-21)
		tmp = Float64(Float64(t_1 + Float64(Float64(1.0 + x) * Float64(x - fma(x, 2.0, -2.0)))) / Float64(Float64(1.0 + x) * t_1));
	elseif (t_2 <= 5e-29)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(Float64(t_0 + Float64(fma(2.0, Float64(x + -1.0), Float64(-x)) * Float64(-1.0 - x))) / Float64(Float64(1.0 + x) * t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-21], N[(N[(t$95$1 + N[(N[(1.0 + x), $MachinePrecision] * N[(x - N[(x * 2.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-29], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(N[(2.0 * N[(x + -1.0), $MachinePrecision] + (-x)), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -4.99999999999999973e-21

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

      1. frac-sub 98.9%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. sub-neg 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. fma-def 100.0%

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

      12. metadata-eval 100.0%

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

      13. distribute-rgt-in 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub-neg 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-udef 100.0%

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

      8. distribute-lft-neg-in 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. fma-udef 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. remove-double-neg 100.0%

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

      15. metadata-eval 100.0%

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

      16. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -4.99999999999999973e-21 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 4.99999999999999986e-29

   1. Initial program 66.4%

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

   3. Simplified 66.4%

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

   4. Taylor expanded in x around inf 98.4%

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

   if 4.99999999999999986e-29 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

      1. flip-+ 99.2%

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

      2. sub-neg 99.2%

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

      3. metadata-eval 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. +-commutative 99.2%

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

      6. associate-/r/ 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. distribute-neg-in 99.2%

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

      10. metadata-eval 99.2%

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

      11. sub-neg 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. associate-*l/ 99.2%

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

      2. *-lft-identity 99.2%

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

   7. Simplified 99.2%

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

      1. clear-num 99.2%

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

      2. frac-sub 99.3%

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

      3. frac-sub 99.3%

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

      4. *-un-lft-identity 99.3%

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

      5. metadata-eval 99.3%

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

      6. flip-+ 100.0%

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

      7. fma-neg 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. metadata-eval 100.0%

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

      10. flip-+ 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(1 + x\right) \cdot \left(x - \mathsf{fma}\left(x, 2, -2\right)\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x + -1\right) + \mathsf{fma}\left(2, x + -1, -x\right) \cdot \left(-1 - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - x\\ t_1 := \frac{1}{1 + x}\\ t_2 := \left(t_1 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-21}:\\ \;\;\;\;\frac{t_0 + \left(1 + x\right) \cdot \left(x - \mathsf{fma}\left(x, 2, -2\right)\right)}{\left(1 + x\right) \cdot t_0}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{x + -2 \cdot \left(x + -1\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* x x) x))
        (t_1 (/ 1.0 (+ 1.0 x)))
        (t_2 (+ (- t_1 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_2 -5e-21)
     (/ (+ t_0 (* (+ 1.0 x) (- x (fma x 2.0 -2.0)))) (* (+ 1.0 x) t_0))
     (if (<= t_2 2e-15)
       (/ 2.0 (pow x 3.0))
       (+ t_1 (/ (+ x (* -2.0 (+ x -1.0))) t_0))))))

C

double code(double x) {
	double t_0 = (x * x) - x;
	double t_1 = 1.0 / (1.0 + x);
	double t_2 = (t_1 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_2 <= -5e-21) {
		tmp = (t_0 + ((1.0 + x) * (x - fma(x, 2.0, -2.0)))) / ((1.0 + x) * t_0);
	} else if (t_2 <= 2e-15) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = t_1 + ((x + (-2.0 * (x + -1.0))) / t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) - x)
	t_1 = Float64(1.0 / Float64(1.0 + x))
	t_2 = Float64(Float64(t_1 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_2 <= -5e-21)
		tmp = Float64(Float64(t_0 + Float64(Float64(1.0 + x) * Float64(x - fma(x, 2.0, -2.0)))) / Float64(Float64(1.0 + x) * t_0));
	elseif (t_2 <= 2e-15)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(t_1 + Float64(Float64(x + Float64(-2.0 * Float64(x + -1.0))) / t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-21], N[(N[(t$95$0 + N[(N[(1.0 + x), $MachinePrecision] * N[(x - N[(x * 2.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-15], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(x + N[(-2.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -4.99999999999999973e-21

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

      1. frac-sub 98.9%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. sub-neg 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. fma-def 100.0%

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

      12. metadata-eval 100.0%

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

      13. distribute-rgt-in 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub-neg 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-udef 100.0%

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

      8. distribute-lft-neg-in 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. fma-udef 100.0%

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

      11. *-commutative 100.0%

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

      12. neg-mul-1 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. remove-double-neg 100.0%

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

      15. metadata-eval 100.0%

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

      16. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -4.99999999999999973e-21 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 2.0000000000000002e-15

   1. Initial program 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in x around inf 98.2%

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

   if 2.0000000000000002e-15 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-sub 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-neg-in 100.0%

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

      12. metadata-eval 100.0%

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

      13. sub-neg 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. cancel-sign-sub 100.0%

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

      2. *-commutative 100.0%

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

      3. neg-mul-1 100.0%

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

      4. unsub-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. sqr-neg 100.0%

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

      9. unpow2 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. sub-neg 100.0%

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

      12. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(1 + x\right) \cdot \left(x - \mathsf{fma}\left(x, 2, -2\right)\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{x + -2 \cdot \left(x + -1\right)}{x \cdot x - x}\\ \end{array} \]

Alternative 4: 98.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -50000000000000:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{x + -2 \cdot \left(x + -1\right)}{x \cdot x - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 x))) (t_1 (+ (- t_0 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -50000000000000.0)
     (/ -2.0 x)
     (if (<= t_1 2e-15)
       (/ 2.0 (pow x 3.0))
       (+ t_0 (/ (+ x (* -2.0 (+ x -1.0))) (- (* x x) x)))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -50000000000000.0) {
		tmp = -2.0 / x;
	} else if (t_1 <= 2e-15) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + x)
    t_1 = (t_0 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_1 <= (-50000000000000.0d0)) then
        tmp = (-2.0d0) / x
    else if (t_1 <= 2d-15) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = t_0 + ((x + ((-2.0d0) * (x + (-1.0d0)))) / ((x * x) - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -50000000000000.0) {
		tmp = -2.0 / x;
	} else if (t_1 <= 2e-15) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + x)
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_1 <= -50000000000000.0:
		tmp = -2.0 / x
	elif t_1 <= 2e-15:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + x))
	t_1 = Float64(Float64(t_0 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -50000000000000.0)
		tmp = Float64(-2.0 / x);
	elseif (t_1 <= 2e-15)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(t_0 + Float64(Float64(x + Float64(-2.0 * Float64(x + -1.0))) / Float64(Float64(x * x) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (1.0 + x);
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= -50000000000000.0)
		tmp = -2.0 / x;
	elseif (t_1 <= 2e-15)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = t_0 + ((x + (-2.0 * (x + -1.0))) / ((x * x) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000000000.0], N[(-2.0 / x), $MachinePrecision], If[LessEqual[t$95$1, 2e-15], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(x + N[(-2.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -5e13

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   if -5e13 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 2.0000000000000002e-15

   1. Initial program 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in x around inf 98.1%

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

   if 2.0000000000000002e-15 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-sub 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-neg-in 100.0%

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

      12. metadata-eval 100.0%

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

      13. sub-neg 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. cancel-sign-sub 100.0%

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

      2. *-commutative 100.0%

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

      3. neg-mul-1 100.0%

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

      4. unsub-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. sqr-neg 100.0%

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

      9. unpow2 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. sub-neg 100.0%

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

      12. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -50000000000000:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{x + -2 \cdot \left(x + -1\right)}{x \cdot x - x}\\ \end{array} \]

Alternative 5: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

2. Final simplification 83.4%

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

Alternative 6: 76.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.2 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.2e+77))) (/ -1.0 (* x x)) (- 1.0 (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.2e+77)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.2d+77))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = 1.0d0 - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.2e+77)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.2e+77):
		tmp = -1.0 / (x * x)
	else:
		tmp = 1.0 - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.2e+77))
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(1.0 - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.2e+77)))
		tmp = -1.0 / (x * x);
	else
		tmp = 1.0 - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.2e+77]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.1999999999999999e77 < x

   1. Initial program 73.3%

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

   3. Simplified 73.3%

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

      1. flip-+ 18.3%

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

      2. sub-neg 18.3%

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

      3. metadata-eval 18.3%

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

      4. distribute-neg-in 18.3%

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

      5. +-commutative 18.3%

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

      6. associate-/r/ 15.4%

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

      7. metadata-eval 15.4%

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

      8. +-commutative 15.4%

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

      9. distribute-neg-in 15.4%

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

      10. metadata-eval 15.4%

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

      11. sub-neg 15.4%

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

   5. Applied egg-rr 15.4%

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

      1. associate-*l/ 17.7%

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

      2. *-lft-identity 17.7%

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

   7. Simplified 17.7%

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

   8. Taylor expanded in x around inf 17.6%

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

      1. mul-1-neg 17.6%

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

      2. unpow2 17.6%

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

      3. distribute-rgt-neg-out 17.6%

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

   10. Simplified 17.6%

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

       1. flip-- 68.9%

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

   12. Applied egg-rr 68.9%

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

       1. associate-+r- 68.9%

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

   14. Simplified 68.9%

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

   15. Taylor expanded in x around 0 59.8%

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

       1. unpow2 59.8%

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

   17. Simplified 59.8%

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

   if -1 < x < 1.1999999999999999e77

   1. Initial program 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in x around 0 90.3%

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

   5. Taylor expanded in x around 0 90.1%

      \[\leadsto 1 - \color{blue}{\frac{2}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.2 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{x}\\ \end{array} \]

Alternative 7: 76.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0 / (x * x);
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 / (x * x);
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0 / (x * x);
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0 / (x * x)
	elif x <= 1.0:
		tmp = -2.0 / x
	else:
		tmp = 1.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-1.0 / Float64(x * x));
	elseif (x <= 1.0)
		tmp = Float64(-2.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0 / (x * x);
	elseif (x <= 1.0)
		tmp = -2.0 / x;
	else
		tmp = 1.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 62.9%

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

   3. Simplified 62.9%

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

      1. flip-+ 14.6%

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

      2. sub-neg 14.6%

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

      3. metadata-eval 14.6%

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

      4. distribute-neg-in 14.6%

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

      5. +-commutative 14.6%

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

      6. associate-/r/ 11.0%

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

      7. metadata-eval 11.0%

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

      8. +-commutative 11.0%

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

      9. distribute-neg-in 11.0%

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

      10. metadata-eval 11.0%

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

      11. sub-neg 11.0%

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

   5. Applied egg-rr 11.0%

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

      1. associate-*l/ 12.2%

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

      2. *-lft-identity 12.2%

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

   7. Simplified 12.2%

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

   8. Taylor expanded in x around inf 12.0%

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

      1. mul-1-neg 12.0%

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

      2. unpow2 12.0%

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

      3. distribute-rgt-neg-out 12.0%

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

   10. Simplified 12.0%

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

       1. flip-- 58.6%

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

   12. Applied egg-rr 58.6%

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

       1. associate-+r- 58.6%

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

   14. Simplified 58.6%

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

   15. Taylor expanded in x around 0 55.1%

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

       1. unpow2 55.1%

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

   17. Simplified 55.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1 < x

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

      1. flip-+ 21.1%

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

      2. sub-neg 21.1%

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

      3. metadata-eval 21.1%

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

      4. distribute-neg-in 21.1%

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

      5. +-commutative 21.1%

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

      6. associate-/r/ 19.7%

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

      7. metadata-eval 19.7%

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

      8. +-commutative 19.7%

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

      9. distribute-neg-in 19.7%

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

      10. metadata-eval 19.7%

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

      11. sub-neg 19.7%

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

   5. Applied egg-rr 19.7%

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

      1. associate-*l/ 22.5%

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

      2. *-lft-identity 22.5%

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

   7. Simplified 22.5%

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

   8. Taylor expanded in x around inf 68.1%

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

   9. Taylor expanded in x around inf 52.4%

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

       1. unpow2 52.4%

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

   11. Simplified 52.4%

       \[\leadsto \color{blue}{\frac{1}{x \cdot x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x}\\ \end{array} \]

Alternative 8: 83.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Simplified 83.4%

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

4. Taylor expanded in x around 0 52.5%

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

5. Taylor expanded in x around 0 82.7%

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

6. Final simplification 82.7%

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

Alternative 9: 52.7% 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 83.4%

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

3. Simplified 83.4%

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

4. Taylor expanded in x around 0 53.2%

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

5. Final simplification 53.2%

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

Alternative 10: 3.3% 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 83.4%

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

3. Simplified 83.4%

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

4. Taylor expanded in x around 0 52.5%

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

5. Taylor expanded in x around inf 11.2%

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

6. Taylor expanded in x around inf 3.4%

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

7. Final simplification 3.4%

   \[\leadsto 1 \]

Developer target: 99.6% 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 2023279 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

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