Asymptote C

Percentage Accurate: 54.3% → 99.6%

Time: 13.2s

Alternatives: 15

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-3}{{x}^{3}} + \left(\frac{\frac{-1}{x}}{x} + \left(\frac{-1}{{x}^{4}} - \frac{3}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} + \frac{-1 - x}{x}}{\frac{1}{\frac{\frac{\mathsf{fma}\left(x, x, x\right)}{x + 1}}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))) 5e-9)
   (+
    (/ -3.0 (pow x 3.0))
    (+ (/ (/ -1.0 x) x) (- (/ -1.0 (pow x 4.0)) (/ 3.0 x))))
   (/
    (+ (/ (+ x -1.0) (+ x 1.0)) (/ (- -1.0 x) x))
    (/ 1.0 (/ (/ (fma x x x) (+ x 1.0)) (+ x -1.0))))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = (-3.0 / pow(x, 3.0)) + (((-1.0 / x) / x) + ((-1.0 / pow(x, 4.0)) - (3.0 / x)));
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / (1.0 / ((fma(x, x, x) / (x + 1.0)) / (x + -1.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 5e-9)
		tmp = Float64(Float64(-3.0 / (x ^ 3.0)) + Float64(Float64(Float64(-1.0 / x) / x) + Float64(Float64(-1.0 / (x ^ 4.0)) - Float64(3.0 / x))));
	else
		tmp = Float64(Float64(Float64(Float64(x + -1.0) / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / x)) / Float64(1.0 / Float64(Float64(fma(x, x, x) / Float64(x + 1.0)) / Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision] + N[(N[(-1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[(3.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(N[(x * x + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-9

   1. Initial program 8.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 99.2%

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. distribute-neg-in 99.2%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{{x}^{3}}\right) + \left(-\left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)\right)} \]

      2. unsub-neg 99.2%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{{x}^{3}}\right) - \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)} \]

      3. associate-*r/ 99.2%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{{x}^{3}}}\right) - \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      4. metadata-eval 99.2%

         \[\leadsto \left(-\frac{\color{blue}{3}}{{x}^{3}}\right) - \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      5. distribute-neg-frac 99.2%

         \[\leadsto \color{blue}{\frac{-3}{{x}^{3}}} - \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      6. metadata-eval 99.2%

         \[\leadsto \frac{\color{blue}{-3}}{{x}^{3}} - \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      7. +-commutative 99.2%

         \[\leadsto \frac{-3}{{x}^{3}} - \color{blue}{\left(\left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right) + 3 \cdot \frac{1}{x}\right)} \]

      8. associate-+l+ 99.2%

         \[\leadsto \frac{-3}{{x}^{3}} - \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\frac{1}{{x}^{4}} + 3 \cdot \frac{1}{x}\right)\right)} \]

      9. unpow2 99.2%

         \[\leadsto \frac{-3}{{x}^{3}} - \left(\frac{1}{\color{blue}{x \cdot x}} + \left(\frac{1}{{x}^{4}} + 3 \cdot \frac{1}{x}\right)\right) \]

      10. associate-/r* 99.2%

          \[\leadsto \frac{-3}{{x}^{3}} - \left(\color{blue}{\frac{\frac{1}{x}}{x}} + \left(\frac{1}{{x}^{4}} + 3 \cdot \frac{1}{x}\right)\right) \]

      11. associate-*r/ 99.7%

          \[\leadsto \frac{-3}{{x}^{3}} - \left(\frac{\frac{1}{x}}{x} + \left(\frac{1}{{x}^{4}} + \color{blue}{\frac{3 \cdot 1}{x}}\right)\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \frac{-3}{{x}^{3}} - \left(\frac{\frac{1}{x}}{x} + \left(\frac{1}{{x}^{4}} + \frac{\color{blue}{3}}{x}\right)\right) \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{-3}{{x}^{3}} - \left(\frac{\frac{1}{x}}{x} + \left(\frac{1}{{x}^{4}} + \frac{3}{x}\right)\right)} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. clear-num 100.0%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. clear-num 100.0%

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

      3. frac-times 99.9%

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

      4. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-3}{{x}^{3}} + \left(\frac{\frac{-1}{x}}{x} + \left(\frac{-1}{{x}^{4}} - \frac{3}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} + \frac{-1 - x}{x}}{\frac{1}{\frac{\frac{\mathsf{fma}\left(x, x, x\right)}{x + 1}}{x + -1}}}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} + \frac{-1 - x}{x}}{\frac{1}{\frac{\frac{\mathsf{fma}\left(x, x, x\right)}{x + 1}}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))) 5e-9)
   (/
    1.0
    (-
     (+
      0.1111111111111111
      (fma -0.3333333333333333 x (/ 0.2962962962962963 x)))
     (/ 0.09876543209876543 (* x x))))
   (/
    (+ (/ (+ x -1.0) (+ x 1.0)) (/ (- -1.0 x) x))
    (/ 1.0 (/ (/ (fma x x x) (+ x 1.0)) (+ x -1.0))))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = 1.0 / ((0.1111111111111111 + fma(-0.3333333333333333, x, (0.2962962962962963 / x))) - (0.09876543209876543 / (x * x)));
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / (1.0 / ((fma(x, x, x) / (x + 1.0)) / (x + -1.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 5e-9)
		tmp = Float64(1.0 / Float64(Float64(0.1111111111111111 + fma(-0.3333333333333333, x, Float64(0.2962962962962963 / x))) - Float64(0.09876543209876543 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(x + -1.0) / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / x)) / Float64(1.0 / Float64(Float64(fma(x, x, x) / Float64(x + 1.0)) / Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(1.0 / N[(N[(0.1111111111111111 + N[(-0.3333333333333333 * x + N[(0.2962962962962963 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.09876543209876543 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(N[(x * x + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-9

   1. Initial program 8.3%

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

      1. flip-- 8.3%

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

      2. clear-num 8.3%

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

      3. sub-neg 8.3%

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

      4. metadata-eval 8.3%

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

      5. pow2 8.3%

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

      6. pow1 8.3%

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

      7. pow-plus 8.3%

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

      8. clear-num 8.3%

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

      9. inv-pow 8.3%

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

      10. metadata-eval 8.3%

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

      11. pow-pow 8.5%

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

   3. Applied egg-rr 8.5%

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

   4. Taylor expanded in x around inf 99.2%

      \[\leadsto \frac{1}{\color{blue}{\left(0.1111111111111111 + \left(-0.3333333333333333 \cdot x + 0.2962962962962963 \cdot \frac{1}{x}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. fma-def 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.2962962962962963 \cdot \frac{1}{x}\right)}\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      2. associate-*r/ 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \color{blue}{\frac{0.2962962962962963 \cdot 1}{x}}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      3. metadata-eval 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{\color{blue}{0.2962962962962963}}{x}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \color{blue}{\frac{0.09876543209876543 \cdot 1}{{x}^{2}}}} \]

      5. metadata-eval 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{\color{blue}{0.09876543209876543}}{{x}^{2}}} \]

      6. unpow2 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{\color{blue}{x \cdot x}}} \]

   6. Simplified 99.2%

      \[\leadsto \frac{1}{\color{blue}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{x \cdot x}}} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. clear-num 100.0%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. clear-num 100.0%

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

      3. frac-times 99.9%

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

      4. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} + \frac{-1 - x}{x}}{\frac{1}{\frac{\frac{\mathsf{fma}\left(x, x, x\right)}{x + 1}}{x + -1}}}\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} + \frac{-1 - x}{x}}{\frac{x + -1}{\frac{x + x \cdot x}{x + 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))) 5e-9)
   (/
    1.0
    (-
     (+
      0.1111111111111111
      (fma -0.3333333333333333 x (/ 0.2962962962962963 x)))
     (/ 0.09876543209876543 (* x x))))
   (/
    (+ (/ (+ x -1.0) (+ x 1.0)) (/ (- -1.0 x) x))
    (/ (+ x -1.0) (/ (+ x (* x x)) (+ x 1.0))))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = 1.0 / ((0.1111111111111111 + fma(-0.3333333333333333, x, (0.2962962962962963 / x))) - (0.09876543209876543 / (x * x)));
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / ((x + -1.0) / ((x + (x * x)) / (x + 1.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 5e-9)
		tmp = Float64(1.0 / Float64(Float64(0.1111111111111111 + fma(-0.3333333333333333, x, Float64(0.2962962962962963 / x))) - Float64(0.09876543209876543 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(x + -1.0) / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / x)) / Float64(Float64(x + -1.0) / Float64(Float64(x + Float64(x * x)) / Float64(x + 1.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(1.0 / N[(N[(0.1111111111111111 + N[(-0.3333333333333333 * x + N[(0.2962962962962963 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.09876543209876543 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] / N[(N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-9

   1. Initial program 8.3%

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

      1. flip-- 8.3%

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

      2. clear-num 8.3%

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

      3. sub-neg 8.3%

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

      4. metadata-eval 8.3%

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

      5. pow2 8.3%

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

      6. pow1 8.3%

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

      7. pow-plus 8.3%

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

      8. clear-num 8.3%

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

      9. inv-pow 8.3%

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

      10. metadata-eval 8.3%

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

      11. pow-pow 8.5%

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

   3. Applied egg-rr 8.5%

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

   4. Taylor expanded in x around inf 99.2%

      \[\leadsto \frac{1}{\color{blue}{\left(0.1111111111111111 + \left(-0.3333333333333333 \cdot x + 0.2962962962962963 \cdot \frac{1}{x}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. fma-def 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.2962962962962963 \cdot \frac{1}{x}\right)}\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      2. associate-*r/ 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \color{blue}{\frac{0.2962962962962963 \cdot 1}{x}}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      3. metadata-eval 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{\color{blue}{0.2962962962962963}}{x}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \color{blue}{\frac{0.09876543209876543 \cdot 1}{{x}^{2}}}} \]

      5. metadata-eval 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{\color{blue}{0.09876543209876543}}{{x}^{2}}} \]

      6. unpow2 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{\color{blue}{x \cdot x}}} \]

   6. Simplified 99.2%

      \[\leadsto \frac{1}{\color{blue}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{x \cdot x}}} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. clear-num 100.0%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. frac-times 100.0%

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

      3. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. sub-neg 100.0%

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

      2. *-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. sub-neg 100.0%

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

       2. +-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} + \frac{-1 - x}{x}}{\frac{x + -1}{\frac{x + x \cdot x}{x + 1}}}\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(0.1111111111111111 + \left(\frac{0.2962962962962963}{x} + x \cdot -0.3333333333333333\right)\right) - \frac{0.09876543209876543}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} + \frac{-1 - x}{x}}{\frac{x + -1}{\frac{x + x \cdot x}{x + 1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))) 5e-9)
   (/
    1.0
    (-
     (+
      0.1111111111111111
      (+ (/ 0.2962962962962963 x) (* x -0.3333333333333333)))
     (/ 0.09876543209876543 (* x x))))
   (/
    (+ (/ (+ x -1.0) (+ x 1.0)) (/ (- -1.0 x) x))
    (/ (+ x -1.0) (/ (+ x (* x x)) (+ x 1.0))))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)));
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / ((x + -1.0) / ((x + (x * x)) / (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) - ((x + 1.0d0) / (x + (-1.0d0)))) <= 5d-9) then
        tmp = 1.0d0 / ((0.1111111111111111d0 + ((0.2962962962962963d0 / x) + (x * (-0.3333333333333333d0)))) - (0.09876543209876543d0 / (x * x)))
    else
        tmp = (((x + (-1.0d0)) / (x + 1.0d0)) + (((-1.0d0) - x) / x)) / ((x + (-1.0d0)) / ((x + (x * x)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)));
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / ((x + -1.0) / ((x + (x * x)) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9:
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)))
	else:
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / ((x + -1.0) / ((x + (x * x)) / (x + 1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 5e-9)
		tmp = Float64(1.0 / Float64(Float64(0.1111111111111111 + Float64(Float64(0.2962962962962963 / x) + Float64(x * -0.3333333333333333))) - Float64(0.09876543209876543 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(x + -1.0) / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / x)) / Float64(Float64(x + -1.0) / Float64(Float64(x + Float64(x * x)) / Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9)
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)));
	else
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / ((x + -1.0) / ((x + (x * x)) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(1.0 / N[(N[(0.1111111111111111 + N[(N[(0.2962962962962963 / x), $MachinePrecision] + N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.09876543209876543 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] / N[(N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-9

   1. Initial program 8.3%

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

      1. flip-- 8.3%

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

      2. clear-num 8.3%

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

      3. sub-neg 8.3%

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

      4. metadata-eval 8.3%

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

      5. pow2 8.3%

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

      6. pow1 8.3%

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

      7. pow-plus 8.3%

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

      8. clear-num 8.3%

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

      9. inv-pow 8.3%

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

      10. metadata-eval 8.3%

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

      11. pow-pow 8.5%

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

   3. Applied egg-rr 8.5%

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

   4. Taylor expanded in x around inf 99.2%

      \[\leadsto \frac{1}{\color{blue}{\left(0.1111111111111111 + \left(-0.3333333333333333 \cdot x + 0.2962962962962963 \cdot \frac{1}{x}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. fma-def 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.2962962962962963 \cdot \frac{1}{x}\right)}\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      2. associate-*r/ 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \color{blue}{\frac{0.2962962962962963 \cdot 1}{x}}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      3. metadata-eval 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{\color{blue}{0.2962962962962963}}{x}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \color{blue}{\frac{0.09876543209876543 \cdot 1}{{x}^{2}}}} \]

      5. metadata-eval 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{\color{blue}{0.09876543209876543}}{{x}^{2}}} \]

      6. unpow2 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{\color{blue}{x \cdot x}}} \]

   6. Simplified 99.2%

      \[\leadsto \frac{1}{\color{blue}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{x \cdot x}}} \]
   7. Step-by-step derivation

      1. fma-udef 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\left(-0.3333333333333333 \cdot x + \frac{0.2962962962962963}{x}\right)}\right) - \frac{0.09876543209876543}{x \cdot x}} \]

   8. Applied egg-rr 99.2%

      \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\left(-0.3333333333333333 \cdot x + \frac{0.2962962962962963}{x}\right)}\right) - \frac{0.09876543209876543}{x \cdot x}} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. clear-num 100.0%

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

      3. frac-sub 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. frac-times 100.0%

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

      3. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. sub-neg 100.0%

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

      2. *-rgt-identity 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. sub-neg 100.0%

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

       2. +-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(0.1111111111111111 + \left(\frac{0.2962962962962963}{x} + x \cdot -0.3333333333333333\right)\right) - \frac{0.09876543209876543}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} + \frac{-1 - x}{x}}{\frac{x + -1}{\frac{x + x \cdot x}{x + 1}}}\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(0.1111111111111111 + \left(\frac{0.2962962962962963}{x} + x \cdot -0.3333333333333333\right)\right) - \frac{0.09876543209876543}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} + \frac{-1 - x}{x}}{1 + \frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))) 5e-9)
   (/
    1.0
    (-
     (+
      0.1111111111111111
      (+ (/ 0.2962962962962963 x) (* x -0.3333333333333333)))
     (/ 0.09876543209876543 (* x x))))
   (/ (+ (/ (+ x -1.0) (+ x 1.0)) (/ (- -1.0 x) x)) (+ 1.0 (/ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)));
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / (1.0 + (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) - ((x + 1.0d0) / (x + (-1.0d0)))) <= 5d-9) then
        tmp = 1.0d0 / ((0.1111111111111111d0 + ((0.2962962962962963d0 / x) + (x * (-0.3333333333333333d0)))) - (0.09876543209876543d0 / (x * x)))
    else
        tmp = (((x + (-1.0d0)) / (x + 1.0d0)) + (((-1.0d0) - x) / x)) / (1.0d0 + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)));
	} else {
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / (1.0 + (-1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9:
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)))
	else:
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / (1.0 + (-1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 5e-9)
		tmp = Float64(1.0 / Float64(Float64(0.1111111111111111 + Float64(Float64(0.2962962962962963 / x) + Float64(x * -0.3333333333333333))) - Float64(0.09876543209876543 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(x + -1.0) / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / x)) / Float64(1.0 + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9)
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)));
	else
		tmp = (((x + -1.0) / (x + 1.0)) + ((-1.0 - x) / x)) / (1.0 + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(1.0 / N[(N[(0.1111111111111111 + N[(N[(0.2962962962962963 / x), $MachinePrecision] + N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.09876543209876543 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-9

   1. Initial program 8.3%

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

      1. flip-- 8.3%

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

      2. clear-num 8.3%

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

      3. sub-neg 8.3%

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

      4. metadata-eval 8.3%

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

      5. pow2 8.3%

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

      6. pow1 8.3%

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

      7. pow-plus 8.3%

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

      8. clear-num 8.3%

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

      9. inv-pow 8.3%

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

      10. metadata-eval 8.3%

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

      11. pow-pow 8.5%

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

   3. Applied egg-rr 8.5%

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

   4. Taylor expanded in x around inf 99.2%

      \[\leadsto \frac{1}{\color{blue}{\left(0.1111111111111111 + \left(-0.3333333333333333 \cdot x + 0.2962962962962963 \cdot \frac{1}{x}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. fma-def 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.2962962962962963 \cdot \frac{1}{x}\right)}\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      2. associate-*r/ 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \color{blue}{\frac{0.2962962962962963 \cdot 1}{x}}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      3. metadata-eval 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{\color{blue}{0.2962962962962963}}{x}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \color{blue}{\frac{0.09876543209876543 \cdot 1}{{x}^{2}}}} \]

      5. metadata-eval 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{\color{blue}{0.09876543209876543}}{{x}^{2}}} \]

      6. unpow2 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{\color{blue}{x \cdot x}}} \]

   6. Simplified 99.2%

      \[\leadsto \frac{1}{\color{blue}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{x \cdot x}}} \]
   7. Step-by-step derivation

      1. fma-udef 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\left(-0.3333333333333333 \cdot x + \frac{0.2962962962962963}{x}\right)}\right) - \frac{0.09876543209876543}{x \cdot x}} \]

   8. Applied egg-rr 99.2%

      \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\left(-0.3333333333333333 \cdot x + \frac{0.2962962962962963}{x}\right)}\right) - \frac{0.09876543209876543}{x \cdot x}} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. clear-num 99.9%

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

      4. *-rgt-identity 99.9%

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

      5. clear-num 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. frac-sub 100.0%

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

      9. *-un-lft-identity 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. *-rgt-identity 100.0%

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

      12. *-commutative 100.0%

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

      13. *-rgt-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(0.1111111111111111 + \left(\frac{0.2962962962962963}{x} + x \cdot -0.3333333333333333\right)\right) - \frac{0.09876543209876543}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + -1}{x + 1} + \frac{-1 - x}{x}}{1 + \frac{-1}{x}}\\ \end{array} \]

Alternative 6: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(0.1111111111111111 + \left(\frac{0.2962962962962963}{x} + x \cdot -0.3333333333333333\right)\right) - \frac{0.09876543209876543}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(x + 1\right) \cdot \frac{-1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (- t_0 (/ (+ x 1.0) (+ x -1.0))) 5e-9)
     (/
      1.0
      (-
       (+
        0.1111111111111111
        (+ (/ 0.2962962962962963 x) (* x -0.3333333333333333)))
       (/ 0.09876543209876543 (* x x))))
     (+ t_0 (* (+ x 1.0) (/ -1.0 (+ x -1.0)))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)));
	} else {
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)));
	}
	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 ((t_0 - ((x + 1.0d0) / (x + (-1.0d0)))) <= 5d-9) then
        tmp = 1.0d0 / ((0.1111111111111111d0 + ((0.2962962962962963d0 / x) + (x * (-0.3333333333333333d0)))) - (0.09876543209876543d0 / (x * x)))
    else
        tmp = t_0 + ((x + 1.0d0) * ((-1.0d0) / (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)));
	} else {
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 - ((x + 1.0) / (x + -1.0))) <= 5e-9:
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)))
	else:
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 5e-9)
		tmp = Float64(1.0 / Float64(Float64(0.1111111111111111 + Float64(Float64(0.2962962962962963 / x) + Float64(x * -0.3333333333333333))) - Float64(0.09876543209876543 / Float64(x * x))));
	else
		tmp = Float64(t_0 + Float64(Float64(x + 1.0) * Float64(-1.0 / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 5e-9)
		tmp = 1.0 / ((0.1111111111111111 + ((0.2962962962962963 / x) + (x * -0.3333333333333333))) - (0.09876543209876543 / (x * x)));
	else
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(1.0 / N[(N[(0.1111111111111111 + N[(N[(0.2962962962962963 / x), $MachinePrecision] + N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.09876543209876543 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] * N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-9

   1. Initial program 8.3%

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

      1. flip-- 8.3%

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

      2. clear-num 8.3%

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

      3. sub-neg 8.3%

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

      4. metadata-eval 8.3%

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

      5. pow2 8.3%

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

      6. pow1 8.3%

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

      7. pow-plus 8.3%

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

      8. clear-num 8.3%

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

      9. inv-pow 8.3%

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

      10. metadata-eval 8.3%

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

      11. pow-pow 8.5%

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

   3. Applied egg-rr 8.5%

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

   4. Taylor expanded in x around inf 99.2%

      \[\leadsto \frac{1}{\color{blue}{\left(0.1111111111111111 + \left(-0.3333333333333333 \cdot x + 0.2962962962962963 \cdot \frac{1}{x}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. fma-def 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x, 0.2962962962962963 \cdot \frac{1}{x}\right)}\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      2. associate-*r/ 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \color{blue}{\frac{0.2962962962962963 \cdot 1}{x}}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      3. metadata-eval 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{\color{blue}{0.2962962962962963}}{x}\right)\right) - 0.09876543209876543 \cdot \frac{1}{{x}^{2}}} \]

      4. associate-*r/ 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \color{blue}{\frac{0.09876543209876543 \cdot 1}{{x}^{2}}}} \]

      5. metadata-eval 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{\color{blue}{0.09876543209876543}}{{x}^{2}}} \]

      6. unpow2 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{\color{blue}{x \cdot x}}} \]

   6. Simplified 99.2%

      \[\leadsto \frac{1}{\color{blue}{\left(0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333, x, \frac{0.2962962962962963}{x}\right)\right) - \frac{0.09876543209876543}{x \cdot x}}} \]
   7. Step-by-step derivation

      1. fma-udef 99.2%

         \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\left(-0.3333333333333333 \cdot x + \frac{0.2962962962962963}{x}\right)}\right) - \frac{0.09876543209876543}{x \cdot x}} \]

   8. Applied egg-rr 99.2%

      \[\leadsto \frac{1}{\left(0.1111111111111111 + \color{blue}{\left(-0.3333333333333333 \cdot x + \frac{0.2962962962962963}{x}\right)}\right) - \frac{0.09876543209876543}{x \cdot x}} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(0.1111111111111111 + \left(\frac{0.2962962962962963}{x} + x \cdot -0.3333333333333333\right)\right) - \frac{0.09876543209876543}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{-1}{x + -1}\\ \end{array} \]

Alternative 7: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))) 5e-9)
   (+ (/ -3.0 x) (/ (/ -1.0 x) x))
   (+ (* x (/ 1.0 (+ x 1.0))) (/ (- -1.0 x) (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = (x * (1.0 / (x + 1.0))) + ((-1.0 - x) / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) - ((x + 1.0d0) / (x + (-1.0d0)))) <= 5d-9) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = (x * (1.0d0 / (x + 1.0d0))) + (((-1.0d0) - x) / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = (x * (1.0 / (x + 1.0))) + ((-1.0 - x) / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = (x * (1.0 / (x + 1.0))) + ((-1.0 - x) / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 5e-9)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = Float64(Float64(x * Float64(1.0 / Float64(x + 1.0))) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) <= 5e-9)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = (x * (1.0 / (x + 1.0))) + ((-1.0 - x) / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-9

   1. Initial program 8.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 98.7%

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

      1. distribute-neg-in 98.7%

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

      2. unsub-neg 98.7%

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

      3. associate-*r/ 99.2%

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

      4. metadata-eval 99.2%

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

      5. distribute-neg-frac 99.2%

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

      6. metadata-eval 99.2%

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

      7. unpow2 99.2%

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

      8. associate-/r* 99.2%

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

   4. Simplified 99.2%

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

   if 5.0000000000000001e-9 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 8: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(x + 1\right) \cdot \frac{-1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (- t_0 (/ (+ x 1.0) (+ x -1.0))) 5e-9)
     (+ (/ -3.0 x) (/ (/ -1.0 x) x))
     (+ t_0 (* (+ x 1.0) (/ -1.0 (+ x -1.0)))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)));
	}
	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 ((t_0 - ((x + 1.0d0) / (x + (-1.0d0)))) <= 5d-9) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = t_0 + ((x + 1.0d0) * ((-1.0d0) / (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 5e-9) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 - ((x + 1.0) / (x + -1.0))) <= 5e-9:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 - Float64(Float64(x + 1.0) / Float64(x + -1.0))) <= 5e-9)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = Float64(t_0 + Float64(Float64(x + 1.0) * Float64(-1.0 / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 - ((x + 1.0) / (x + -1.0))) <= 5e-9)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] * N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-9

   1. Initial program 8.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 98.7%

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

      1. distribute-neg-in 98.7%

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

      2. unsub-neg 98.7%

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

      3. associate-*r/ 99.2%

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

      4. metadata-eval 99.2%

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

      5. distribute-neg-frac 99.2%

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

      6. metadata-eval 99.2%

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

      7. unpow2 99.2%

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

      8. associate-/r* 99.2%

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

   4. Simplified 99.2%

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

   if 5.0000000000000001e-9 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{-1}{x + -1}\\ \end{array} \]

Alternative 9: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0)))))
   (if (<= t_0 5e-9) (+ (/ -3.0 x) (/ (/ -1.0 x) x)) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_0 <= 5e-9) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = 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)) - ((x + 1.0d0) / (x + (-1.0d0)))
    if (t_0 <= 5d-9) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	double tmp;
	if (t_0 <= 5e-9) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))
	tmp = 0
	if t_0 <= 5e-9:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 5e-9)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) - ((x + 1.0) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 5e-9)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-9], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-9

   1. Initial program 8.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 98.7%

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

      1. distribute-neg-in 98.7%

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

      2. unsub-neg 98.7%

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

      3. associate-*r/ 99.2%

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

      4. metadata-eval 99.2%

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

      5. distribute-neg-frac 99.2%

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

      6. metadata-eval 99.2%

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

      7. unpow2 99.2%

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

      8. associate-/r* 99.2%

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

   4. Simplified 99.2%

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

   if 5.0000000000000001e-9 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x + -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \end{array} \]

Alternative 10: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (+ (/ -3.0 x) (/ (/ -1.0 x) x))
   (+ 1.0 (* x (+ x 3.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else
        tmp = 1.0d0 + (x * (x + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	else:
		tmp = 1.0 + (x * (x + 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	else
		tmp = 1.0 + (x * (x + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 97.6%

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

      1. distribute-neg-in 97.6%

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

      2. unsub-neg 97.6%

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

      3. associate-*r/ 98.0%

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

      4. metadata-eval 98.0%

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

      5. distribute-neg-frac 98.0%

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

      6. metadata-eval 98.0%

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

      7. unpow2 98.0%

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

      8. associate-/r* 98.0%

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

   4. Simplified 98.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1 + \left(3 \cdot x + {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

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

      2. distribute-rgt-out 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \]

Alternative 11: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ -3.0 x) (+ 1.0 (* x (+ x 3.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-3.0d0) / x
    else
        tmp = 1.0d0 + (x * (x + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -3.0 / x
	else:
		tmp = 1.0 + (x * (x + 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -3.0 / x;
	else
		tmp = 1.0 + (x * (x + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-3.0 / x), $MachinePrecision], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 97.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1 + \left(3 \cdot x + {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

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

      2. distribute-rgt-out 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \]

Alternative 12: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{0.1111111111111111 + x \cdot -0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ 1.0 (+ 0.1111111111111111 (* x -0.3333333333333333)))
   (+ 1.0 (* x (+ x 3.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333));
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 1.0d0 / (0.1111111111111111d0 + (x * (-0.3333333333333333d0)))
    else
        tmp = 1.0d0 + (x * (x + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333));
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333))
	else:
		tmp = 1.0 + (x * (x + 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(1.0 / Float64(0.1111111111111111 + Float64(x * -0.3333333333333333)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = 1.0 / (0.1111111111111111 + (x * -0.3333333333333333));
	else
		tmp = 1.0 + (x * (x + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(1.0 / N[(0.1111111111111111 + N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.7%

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

      1. flip-- 9.7%

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

      2. clear-num 9.8%

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

      3. sub-neg 9.8%

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

      4. metadata-eval 9.8%

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

      5. pow2 9.8%

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

      6. pow1 9.8%

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

      7. pow-plus 9.8%

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

      8. clear-num 9.8%

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

      9. inv-pow 9.8%

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

      10. metadata-eval 9.8%

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

      11. pow-pow 9.9%

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

   3. Applied egg-rr 9.9%

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

   4. Taylor expanded in x around inf 97.5%

      \[\leadsto \frac{1}{\color{blue}{0.1111111111111111 + -0.3333333333333333 \cdot x}} \]
   5. Step-by-step derivation

      1. +-commutative 97.5%

         \[\leadsto \frac{1}{\color{blue}{-0.3333333333333333 \cdot x + 0.1111111111111111}} \]

      2. *-commutative 97.5%

         \[\leadsto \frac{1}{\color{blue}{x \cdot -0.3333333333333333} + 0.1111111111111111} \]

   6. Simplified 97.5%

      \[\leadsto \frac{1}{\color{blue}{x \cdot -0.3333333333333333 + 0.1111111111111111}} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1 + \left(3 \cdot x + {x}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

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

      2. distribute-rgt-out 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{0.1111111111111111 + x \cdot -0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \]

Alternative 13: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ -3.0 x) (+ 1.0 (* x 3.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-3.0d0) / x
    else
        tmp = 1.0d0 + (x * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -3.0 / x
	else:
		tmp = 1.0 + (x * 3.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(1.0 + Float64(x * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -3.0 / x;
	else
		tmp = 1.0 + (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-3.0 / x), $MachinePrecision], N[(1.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 97.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around 0 99.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array} \]

Alternative 14: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (/ -3.0 x) (if (<= x 1.0) 1.0 (/ -3.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = (-3.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -3.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 1.0:
		tmp = 1.0
	else:
		tmp = -3.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-3.0 / x);
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = Float64(-3.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -3.0 / x;
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = -3.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 1.0], 1.0, N[(-3.0 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 9.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around inf 97.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Taylor expanded in x around 0 99.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 15: 50.9% accurate, 13.0× 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 55.2%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

2. Taylor expanded in x around 0 51.9%

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

3. Final simplification 51.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023285 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))