Asymptote C

Percentage Accurate: 54.5% → 99.8%

Time: 8.2s

Alternatives: 9

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: 54.5% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\frac{-3 - {x}^{-1}}{x} - 1}{x} + -3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{x \cdot x - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 5e-14)
   (/ (+ (/ (- (/ (- -3.0 (pow x -1.0)) x) 1.0) x) -3.0) x)
   (/ (fma -3.0 x -1.0) (- (* x x) 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 5e-14) {
		tmp = (((((-3.0 - pow(x, -1.0)) / x) - 1.0) / x) + -3.0) / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / ((x * x) - 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 5e-14)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-3.0 - (x ^ -1.0)) / x) - 1.0) / x) + -3.0) / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / Float64(Float64(x * x) - 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-14], N[(N[(N[(N[(N[(N[(-3.0 - N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision] + -3.0), $MachinePrecision] / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - 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)))) < 5.0000000000000002e-14

   1. Initial program 6.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.6%

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

   if 5.0000000000000002e-14 < (-.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 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. difference-of-squares-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. difference-of-squares-rev N/A

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

      18. difference-of-sqr--1-rev N/A

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

      19. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-in N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. rgt-mult-inverse N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. distribute-lft-neg-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. rgt-mult-inverse N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

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

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

      3. difference-of-sqr-1-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\frac{-3 - {x}^{-1}}{x} - 1}{x} + -3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{x \cdot x - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{-3 - {x}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{x \cdot x - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 5e-14)
   (/ (- -3.0 (pow x -1.0)) x)
   (/ (fma -3.0 x -1.0) (- (* x x) 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 5e-14) {
		tmp = (-3.0 - pow(x, -1.0)) / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / ((x * x) - 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 5e-14)
		tmp = Float64(Float64(-3.0 - (x ^ -1.0)) / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / Float64(Float64(x * x) - 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-14], N[(N[(-3.0 - N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - 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)))) < 5.0000000000000002e-14

   1. Initial program 6.8%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 5.0000000000000002e-14 < (-.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 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. difference-of-squares-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. difference-of-squares-rev N/A

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

      18. difference-of-sqr--1-rev N/A

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

      19. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-in N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. rgt-mult-inverse N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. distribute-lft-neg-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. rgt-mult-inverse N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

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

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

      3. difference-of-sqr-1-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{-3 - {x}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{x \cdot x - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{-3 - {x}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 5e-14)
   (/ (- -3.0 (pow x -1.0)) x)
   (/ (fma -3.0 x -1.0) (fma x x -1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 5e-14) {
		tmp = (-3.0 - pow(x, -1.0)) / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / fma(x, x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 5e-14)
		tmp = Float64(Float64(-3.0 - (x ^ -1.0)) / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / 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[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-14], N[(N[(-3.0 - N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $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)))) < 5.0000000000000002e-14

   1. Initial program 6.8%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 5.0000000000000002e-14 < (-.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 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. difference-of-squares-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. difference-of-squares-rev N/A

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

      18. difference-of-sqr--1-rev N/A

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

      19. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-in N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. rgt-mult-inverse N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. distribute-lft-neg-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. rgt-mult-inverse N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{-3 - {x}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{-3 - \frac{1 - \frac{-3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{x \cdot x - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 5e-14)
   (/ (- -3.0 (/ (- 1.0 (/ -3.0 x)) x)) x)
   (/ (fma -3.0 x -1.0) (- (* x x) 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 5e-14) {
		tmp = (-3.0 - ((1.0 - (-3.0 / x)) / x)) / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / ((x * x) - 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 5e-14)
		tmp = Float64(Float64(-3.0 - Float64(Float64(1.0 - Float64(-3.0 / x)) / x)) / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / Float64(Float64(x * x) - 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-14], N[(N[(-3.0 - N[(N[(1.0 - N[(-3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - 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)))) < 5.0000000000000002e-14

   1. Initial program 6.8%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. Applied rewrites 99.6%

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

   if 5.0000000000000002e-14 < (-.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 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. difference-of-squares-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. difference-of-squares-rev N/A

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

      18. difference-of-sqr--1-rev N/A

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

      19. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-in N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. rgt-mult-inverse N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. distribute-lft-neg-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. rgt-mult-inverse N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

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

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

      3. difference-of-sqr-1-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.0)
   (/ -3.0 x)
   (/ (fma -3.0 x -1.0) (fma x x -1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.0) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(-3.0, x, -1.0) / fma(x, x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 0.0)
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(fma(-3.0, x, -1.0) / 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[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-3.0 / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $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)))) < 0.0

   1. Initial program 6.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.0 < (-.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 98.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. difference-of-squares-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. difference-of-squares-rev N/A

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

      18. difference-of-sqr--1-rev N/A

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

      19. lower-fma.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-in N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. rgt-mult-inverse N/A

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

      6. fp-cancel-sub-sign N/A

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

      7. distribute-lft-neg-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. rgt-mult-inverse N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 6: 98.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- 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)) - ((x + 1.0) / (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(x + 1.0) / 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[(x + 1.0), $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 8.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 98.1

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

   5. Applied rewrites 98.1%

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

   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 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

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

      8. unpow2 N/A

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

      9. fp-cancel-sign-sub N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. fp-cancel-sign-sub N/A

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

      14. unpow2 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 7: 98.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(-3.0 / x), $MachinePrecision], N[(N[(3.0 + x), $MachinePrecision] * x + 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 8.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 98.1

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

   5. Applied rewrites 98.1%

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

   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 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 8: 51.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.1%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-+.f64 45.6

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

5. Applied rewrites 45.6%

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

Alternative 9: 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 49.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 45.4%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))