Asymptote C

Percentage Accurate: 53.9% → 99.7%

Time: 9.7s

Alternatives: 14

Speedup: 0.7×

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: 53.9% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{x + 1}\\ \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{x} \cdot \left(-1 + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(x + -1\right)\right) \cdot t\_0 - \mathsf{fma}\left(x, x, -1\right)}{t\_0 \cdot \mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) (+ x 1.0))))
   (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 5e-6)
     (* (/ (+ 3.0 (/ 1.0 x)) x) (+ -1.0 (/ -1.0 (* x x))))
     (/ (- (* (* x (+ x -1.0)) t_0) (fma x x -1.0)) (* t_0 (fma x x -1.0))))))

C

double code(double x) {
	double t_0 = (x + -1.0) / (x + 1.0);
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-6) {
		tmp = ((3.0 + (1.0 / x)) / x) * (-1.0 + (-1.0 / (x * x)));
	} else {
		tmp = (((x * (x + -1.0)) * t_0) - fma(x, x, -1.0)) / (t_0 * fma(x, x, -1.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x + -1.0) / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 5e-6)
		tmp = Float64(Float64(Float64(3.0 + Float64(1.0 / x)) / x) * Float64(-1.0 + Float64(-1.0 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(x * Float64(x + -1.0)) * t_0) - fma(x, x, -1.0)) / Float64(t_0 * fma(x, x, -1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(-1.0 + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000041e-6

   1. Initial program 9.0%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. unpow2 N/A

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

   5. Simplified 100.0%

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

   if 5.00000000000000041e-6 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. flip-+ N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

      1. clear-num N/A

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

      2. frac-sub N/A

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

      3. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{x} \cdot \left(-1 + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(x + -1\right)\right) \cdot \frac{x + -1}{x + 1} - \mathsf{fma}\left(x, x, -1\right)}{\frac{x + -1}{x + 1} \cdot \mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{x} \cdot \left(-1 + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -1\right) \cdot \frac{x \cdot \left(x + -1\right)}{x + 1} - \mathsf{fma}\left(x, x, -1\right)}{\left(x + -1\right) \cdot \left(x + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 5e-6)
   (* (/ (+ 3.0 (/ 1.0 x)) x) (+ -1.0 (/ -1.0 (* x x))))
   (/
    (- (* (+ x -1.0) (/ (* x (+ x -1.0)) (+ x 1.0))) (fma x x -1.0))
    (* (+ x -1.0) (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-6) {
		tmp = ((3.0 + (1.0 / x)) / x) * (-1.0 + (-1.0 / (x * x)));
	} else {
		tmp = (((x + -1.0) * ((x * (x + -1.0)) / (x + 1.0))) - fma(x, x, -1.0)) / ((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(-1.0 - x) / Float64(x + -1.0))) <= 5e-6)
		tmp = Float64(Float64(Float64(3.0 + Float64(1.0 / x)) / x) * Float64(-1.0 + Float64(-1.0 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(x + -1.0) * Float64(Float64(x * Float64(x + -1.0)) / Float64(x + 1.0))) - fma(x, x, -1.0)) / Float64(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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(-1.0 + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + -1.0), $MachinePrecision] * N[(N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000041e-6

   1. Initial program 9.0%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. unpow2 N/A

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

   5. Simplified 100.0%

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

   if 5.00000000000000041e-6 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. flip-+ N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

      1. frac-2neg N/A

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

      2. frac-sub N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. frac-sub N/A

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

      8. difference-of-sqr-1 N/A

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

      9. associate-/r* N/A

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

      10. frac-2neg N/A

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{x} \cdot \left(-1 + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -1\right) \cdot \frac{x \cdot \left(x + -1\right)}{x + 1} - \mathsf{fma}\left(x, x, -1\right)}{\left(x + -1\right) \cdot \left(x + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{x} \cdot \left(-1 + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x \cdot x, 1\right)}, \mathsf{fma}\left(x, x, 1\right) - x, \frac{x + 1}{1 - x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 5e-6)
   (* (/ (+ 3.0 (/ 1.0 x)) x) (+ -1.0 (/ -1.0 (* x x))))
   (fma
    (/ x (fma x (* x x) 1.0))
    (- (fma x x 1.0) x)
    (/ (+ x 1.0) (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-6) {
		tmp = ((3.0 + (1.0 / x)) / x) * (-1.0 + (-1.0 / (x * x)));
	} else {
		tmp = fma((x / fma(x, (x * x), 1.0)), (fma(x, x, 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(-1.0 - x) / Float64(x + -1.0))) <= 5e-6)
		tmp = Float64(Float64(Float64(3.0 + Float64(1.0 / x)) / x) * Float64(-1.0 + Float64(-1.0 / Float64(x * x))));
	else
		tmp = fma(Float64(x / fma(x, Float64(x * x), 1.0)), Float64(fma(x, x, 1.0) - x), Float64(Float64(x + 1.0) / Float64(1.0 - x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(-1.0 + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x + 1.0), $MachinePrecision] - x), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000041e-6

   1. Initial program 9.0%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. unpow2 N/A

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

   5. Simplified 100.0%

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

   if 5.00000000000000041e-6 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. flip3-+ N/A

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

      3. associate-/r/ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. cube-mult N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

      12. associate-+r- N/A

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

      13. --lowering--.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. distribute-neg-frac2 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. +-lowering-+.f64 N/A

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

      18. sub-neg N/A

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

      19. distribute-neg-in N/A

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

      20. metadata-eval N/A

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

   4. Applied egg-rr 99.9%

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. --lowering--.f64 99.9

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{x} \cdot \left(-1 + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x \cdot x, 1\right)}, \mathsf{fma}\left(x, x, 1\right) - x, \frac{x + 1}{1 - x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{x} \cdot \left(-1 + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x + -1, \left(x + 1\right) \cdot \left(-1 - x\right)\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 5e-6)
   (* (/ (+ 3.0 (/ 1.0 x)) x) (+ -1.0 (/ -1.0 (* x x))))
   (/ (fma x (+ x -1.0) (* (+ x 1.0) (- -1.0 x))) (fma x x -1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-6) {
		tmp = ((3.0 + (1.0 / x)) / x) * (-1.0 + (-1.0 / (x * x)));
	} else {
		tmp = fma(x, (x + -1.0), ((x + 1.0) * (-1.0 - x))) / fma(x, x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 5e-6)
		tmp = Float64(Float64(Float64(3.0 + Float64(1.0 / x)) / x) * Float64(-1.0 + Float64(-1.0 / Float64(x * x))));
	else
		tmp = Float64(fma(x, Float64(x + -1.0), Float64(Float64(x + 1.0) * Float64(-1.0 - x))) / fma(x, x, -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(-1.0 + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x + -1.0), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000041e-6

   1. Initial program 9.0%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. unpow2 N/A

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

   5. Simplified 100.0%

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

   if 5.00000000000000041e-6 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. frac-add N/A

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

      4. difference-of-sqr-1 N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. +-lowering-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. accelerator-lowering-fma.f64 N/A

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

      20. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{3 + \frac{1}{x}}{x} \cdot \left(-1 + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x + -1, \left(x + 1\right) \cdot \left(-1 - x\right)\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-3 + \frac{-3 - x}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x + -1, \left(x + 1\right) \cdot \left(-1 - x\right)\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 5e-6)
   (/ (+ -3.0 (/ (- -3.0 x) (* x x))) x)
   (/ (fma x (+ x -1.0) (* (+ x 1.0) (- -1.0 x))) (fma x x -1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-6) {
		tmp = (-3.0 + ((-3.0 - x) / (x * x))) / x;
	} else {
		tmp = fma(x, (x + -1.0), ((x + 1.0) * (-1.0 - x))) / fma(x, x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 5e-6)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-3.0 - x) / Float64(x * x))) / x);
	else
		tmp = Float64(fma(x, Float64(x + -1.0), Float64(Float64(x + 1.0) * Float64(-1.0 - x))) / fma(x, x, -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(-3.0 + N[(N[(-3.0 - x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * N[(x + -1.0), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000041e-6

   1. Initial program 9.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 99.8%

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

   if 5.00000000000000041e-6 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. frac-add N/A

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

      4. difference-of-sqr-1 N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. +-lowering-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. accelerator-lowering-fma.f64 N/A

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

      20. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-3 + \frac{-3 - x}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x + -1, \left(x + 1\right) \cdot \left(-1 - x\right)\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-3 + \frac{-3 - x}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x + 1}, x, \frac{x + 1}{1 - x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 5e-6)
   (/ (+ -3.0 (/ (- -3.0 x) (* x x))) x)
   (fma (/ 1.0 (+ x 1.0)) x (/ (+ x 1.0) (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 5e-6) {
		tmp = (-3.0 + ((-3.0 - x) / (x * x))) / x;
	} else {
		tmp = fma((1.0 / (x + 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(-1.0 - x) / Float64(x + -1.0))) <= 5e-6)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-3.0 - x) / Float64(x * x))) / x);
	else
		tmp = fma(Float64(1.0 / Float64(x + 1.0)), x, Float64(Float64(x + 1.0) / Float64(1.0 - x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(-3.0 + N[(N[(-3.0 - x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000041e-6

   1. Initial program 9.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 99.8%

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

   if 5.00000000000000041e-6 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. flip-+ N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. clear-num N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. flip-+ N/A

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

      12. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-3 + \frac{-3 - x}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x + 1}, x, \frac{x + 1}{1 - x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-3 + \frac{-3 - x}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 5e-6) (/ (+ -3.0 (/ (- -3.0 x) (* x x))) x) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = (-3.0 + ((-3.0 - x) / (x * 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)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 5d-6) then
        tmp = ((-3.0d0) + (((-3.0d0) - x) / (x * 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)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = (-3.0 + ((-3.0 - x) / (x * x))) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 5e-6:
		tmp = (-3.0 + ((-3.0 - x) / (x * x))) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-3.0 - x) / Float64(x * x))) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 5e-6)
		tmp = (-3.0 + ((-3.0 - x) / (x * 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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(N[(-3.0 + N[(N[(-3.0 - x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000041e-6

   1. Initial program 9.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 99.8%

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

   if 5.00000000000000041e-6 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-3 + \frac{-3 - x}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3 + \frac{-3 - x}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right) - x, x, \frac{x + 1}{1 - x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.005)
   (/ (+ -3.0 (/ (- -3.0 x) (* x x))) x)
   (fma (- (fma x x 1.0) x) x (/ (+ x 1.0) (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.005) {
		tmp = (-3.0 + ((-3.0 - x) / (x * x))) / x;
	} else {
		tmp = fma((fma(x, x, 1.0) - x), 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(-1.0 - x) / Float64(x + -1.0))) <= 0.005)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-3.0 - x) / Float64(x * x))) / x);
	else
		tmp = fma(Float64(fma(x, x, 1.0) - x), x, Float64(Float64(x + 1.0) / Float64(1.0 - x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(-3.0 + N[(N[(-3.0 - x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x * x + 1.0), $MachinePrecision] - x), $MachinePrecision] * x + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 0.0050000000000000001

   1. Initial program 9.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 99.5%

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

   if 0.0050000000000000001 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. flip-+ N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. clear-num N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. flip-+ N/A

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

      12. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

      2. unpow2 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 98.9

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

   9. Simplified 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3 + \frac{-3 - x}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right) - x, x, \frac{x + 1}{1 - x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right) - x, x, \frac{x + 1}{1 - x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.005)
   (/ (+ -3.0 (/ 2.0 x)) (+ x -1.0))
   (fma (- (fma x x 1.0) x) x (/ (+ x 1.0) (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.005) {
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0);
	} else {
		tmp = fma((fma(x, x, 1.0) - x), 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(-1.0 - x) / Float64(x + -1.0))) <= 0.005)
		tmp = Float64(Float64(-3.0 + Float64(2.0 / x)) / Float64(x + -1.0));
	else
		tmp = fma(Float64(fma(x, x, 1.0) - x), x, Float64(Float64(x + 1.0) / Float64(1.0 - x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(-3.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x + 1.0), $MachinePrecision] - x), $MachinePrecision] * x + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 0.0050000000000000001

   1. Initial program 9.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. distribute-neg-frac N/A

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

      5. frac-add N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. clear-num N/A

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

      11. times-frac N/A

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

      12. *-commutative N/A

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

      13. difference-of-sqr-1 N/A

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

      14. distribute-rgt1-in N/A

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

      15. *-rgt-identity N/A

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

      16. +-commutative N/A

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

   4. Applied egg-rr 9.6%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 98.9

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

   7. Simplified 98.9%

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

   if 0.0050000000000000001 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. flip-+ N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. metadata-eval 100.0

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

   4. Applied egg-rr 100.0%

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. clear-num N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. flip-+ N/A

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

      12. /-lowering-/.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

      1. distribute-rgt-out-- N/A

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

      2. unpow2 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 98.9

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

   9. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right) - x, x, \frac{x + 1}{1 - x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.005)
   (/ (+ -3.0 (/ 2.0 x)) (+ x -1.0))
   (* (fma x x 1.0) (fma 3.0 x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.005) {
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0);
	} else {
		tmp = fma(x, x, 1.0) * fma(3.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 0.005)
		tmp = Float64(Float64(-3.0 + Float64(2.0 / x)) / Float64(x + -1.0));
	else
		tmp = Float64(fma(x, x, 1.0) * fma(3.0, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(-3.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(3.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 0.0050000000000000001

   1. Initial program 9.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. distribute-neg-frac N/A

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

      5. frac-add N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. clear-num N/A

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

      11. times-frac N/A

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

      12. *-commutative N/A

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

      13. difference-of-sqr-1 N/A

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

      14. distribute-rgt1-in N/A

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

      15. *-rgt-identity N/A

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

      16. +-commutative N/A

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

   4. Applied egg-rr 9.6%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 98.9

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

   7. Simplified 98.9%

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

   if 0.0050000000000000001 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.005)
   (/ (+ -3.0 (/ -1.0 x)) x)
   (* (fma x x 1.0) (fma 3.0 x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.005) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = fma(x, x, 1.0) * fma(3.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 0.005)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = Float64(fma(x, x, 1.0) * fma(3.0, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(3.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 0.0050000000000000001

   1. Initial program 9.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

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

   if 0.0050000000000000001 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.005)
   (/ -3.0 x)
   (* (fma x x 1.0) (fma 3.0 x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.005) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(x, x, 1.0) * fma(3.0, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 0.005)
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(fma(x, x, 1.0) * fma(3.0, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(-3.0 / x), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(3.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 0.0050000000000000001

   1. Initial program 9.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 97.2

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

   5. Simplified 97.2%

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

   if 0.0050000000000000001 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.005)
   (/ -3.0 x)
   (fma x (+ x 3.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.005) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(x, (x + 3.0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 0.005)
		tmp = Float64(-3.0 / x);
	else
		tmp = fma(x, Float64(x + 3.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(-3.0 / x), $MachinePrecision], N[(x * N[(x + 3.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 0.0050000000000000001

   1. Initial program 9.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 97.2

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

   5. Simplified 97.2%

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

   if 0.0050000000000000001 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-lowering-+.f64 98.8

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

   5. Simplified 98.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.005:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.6% accurate, 35.0× 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 51.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 47.4%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))