Asymptote C

Percentage Accurate: 54.2% → 99.6%

Time: 8.0s

Alternatives: 10

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: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, x + 3, 3\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 2e-11) (/ (fma (/ 1.0 (* x x)) (+ x 3.0) 3.0) (- 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 <= 2e-11) {
		tmp = fma((1.0 / (x * x)), (x + 3.0), 3.0) / -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 <= 2e-11)
		tmp = Float64(fma(Float64(1.0 / Float64(x * x)), Float64(x + 3.0), 3.0) / Float64(-x));
	else
		tmp = t_0;
	end
	return 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, 2e-11], N[(N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x + 3.0), $MachinePrecision] + 3.0), $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)))) < 1.99999999999999988e-11

   1. Initial program 8.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 9.4%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 98.7%

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

   if 1.99999999999999988e-11 < (-.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 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, x + 3, 3\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 2e-11) (/ (+ -3.0 (/ 2.0 x)) (+ x -1.0)) 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 <= 2e-11) {
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0);
	} 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 <= 2d-11) then
        tmp = ((-3.0d0) + (2.0d0 / x)) / (x + (-1.0d0))
    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 <= 2e-11) {
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0);
	} 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 <= 2e-11:
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0)
	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 <= 2e-11)
		tmp = Float64(Float64(-3.0 + Float64(2.0 / x)) / Float64(x + -1.0));
	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 <= 2e-11)
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0);
	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, 2e-11], N[(N[(-3.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $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)))) < 1.99999999999999988e-11

   1. Initial program 8.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 9.4%

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

   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 98.6

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

   7. Simplified 98.6%

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

   if 1.99999999999999988e-11 < (-.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 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right), x \cdot 3, \mathsf{fma}\left(x, x, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 2e-11)
   (/ (+ -3.0 (/ 2.0 x)) (+ x -1.0))
   (fma (fma x x 1.0) (* x 3.0) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 2e-11) {
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0);
	} else {
		tmp = fma(fma(x, x, 1.0), (x * 3.0), 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))) <= 2e-11)
		tmp = Float64(Float64(-3.0 + Float64(2.0 / x)) / Float64(x + -1.0));
	else
		tmp = fma(fma(x, x, 1.0), Float64(x * 3.0), 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], 2e-11], N[(N[(-3.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(x * 3.0), $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)))) < 1.99999999999999988e-11

   1. Initial program 8.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 9.4%

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

   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 98.6

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

   7. Simplified 98.6%

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

   if 1.99999999999999988e-11 < (-.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. 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 99.4

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

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, x, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. distribute-lft-in N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. accelerator-lowering-fma.f64 99.4

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

   7. Applied egg-rr 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right), x \cdot 3, \mathsf{fma}\left(x, x, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 2e-11)
   (/ (+ -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))) <= 2e-11) {
		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))) <= 2e-11)
		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], 2e-11], 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)))) < 1.99999999999999988e-11

   1. Initial program 8.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 9.4%

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

   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 98.6

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

   7. Simplified 98.6%

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

   if 1.99999999999999988e-11 < (-.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. 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 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 2e-11)
   (/ (+ -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))) <= 2e-11) {
		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))) <= 2e-11)
		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], 2e-11], 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)))) < 1.99999999999999988e-11

   1. Initial program 8.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 98.6

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

   5. Simplified 98.6%

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

   if 1.99999999999999988e-11 < (-.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. 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 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.3333333333333333, 0.1111111111111111\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 2e-11)
   (/ 1.0 (fma x -0.3333333333333333 0.1111111111111111))
   (* (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))) <= 2e-11) {
		tmp = 1.0 / fma(x, -0.3333333333333333, 0.1111111111111111);
	} 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))) <= 2e-11)
		tmp = Float64(1.0 / fma(x, -0.3333333333333333, 0.1111111111111111));
	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], 2e-11], N[(1.0 / N[(x * -0.3333333333333333 + 0.1111111111111111), $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)))) < 1.99999999999999988e-11

   1. Initial program 8.3%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 9.4%

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

      1. clear-num N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

      13. unsub-neg N/A

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

      14. --lowering--.f64 8.3

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

   6. Applied egg-rr 8.3%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} - \frac{1}{3}\right)}} \]
   8. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{1}{9} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{x \cdot \left(\frac{1}{9} \cdot \frac{1}{x} + \color{blue}{\frac{-1}{3}}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{-1}{3} + \frac{1}{9} \cdot \frac{1}{x}\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{-1}{3} + x \cdot \left(\frac{1}{9} \cdot \frac{1}{x}\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \frac{-1}{3} + x \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{1}{9}\right)}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{1}{x \cdot \frac{-1}{3} + \color{blue}{\left(x \cdot \frac{1}{x}\right) \cdot \frac{1}{9}}} \]

      7. rgt-mult-inverse N/A

         \[\leadsto \frac{1}{x \cdot \frac{-1}{3} + \color{blue}{1} \cdot \frac{1}{9}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{x \cdot \frac{-1}{3} + \color{blue}{\frac{1}{9}}} \]

      9. accelerator-lowering-fma.f64 98.1

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

   9. Simplified 98.1%

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

   if 1.99999999999999988e-11 < (-.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. 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 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, -0.3333333333333333, 0.1111111111111111\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 2e-11)
   (/ -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))) <= 2e-11) {
		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))) <= 2e-11)
		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], 2e-11], 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)))) < 1.99999999999999988e-11

   1. Initial program 8.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.1

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

   5. Simplified 98.1%

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

   if 1.99999999999999988e-11 < (-.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. 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 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 2e-11)
   (/ -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))) <= 2e-11) {
		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))) <= 2e-11)
		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], 2e-11], 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)))) < 1.99999999999999988e-11

   1. Initial program 8.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.1

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

   5. Simplified 98.1%

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

   if 1.99999999999999988e-11 < (-.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. 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 99.2

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

   5. Simplified 99.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x + 3, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (+ x 3.0) 1.0))

C

double code(double x) {
	return fma(x, (x + 3.0), 1.0);
}

Julia

function code(x)
	return fma(x, Float64(x + 3.0), 1.0)
end

Wolfram

code[x_] := N[(x * N[(x + 3.0), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
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 50.8

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

5. Simplified 50.8%

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

6. Final simplification 50.8%

   \[\leadsto \mathsf{fma}\left(x, x + 3, 1\right) \]
7. Add Preprocessing

Alternative 10: 50.9% 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 54.1%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.3%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))