Asymptote C

Percentage Accurate: 53.9% → 99.6%

Time: 8.9s

Alternatives: 11

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 2e-11)
     (* (/ (+ 3.0 (/ 1.0 x)) x) (+ -1.0 (/ -1.0 (* x x))))
     (fma (/ -1.0 (- 1.0 x)) (- -1.0 x) t_0))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-11) {
		tmp = ((3.0 + (1.0 / x)) / x) * (-1.0 + (-1.0 / (x * x)));
	} else {
		tmp = fma((-1.0 / (1.0 - x)), (-1.0 - x), t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 2e-11)
		tmp = Float64(Float64(Float64(3.0 + Float64(1.0 / x)) / x) * Float64(-1.0 + Float64(-1.0 / Float64(x * x))));
	else
		tmp = fma(Float64(-1.0 / Float64(1.0 - x)), Float64(-1.0 - x), t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-11], N[(N[(N[(3.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(-1.0 + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision] + t$95$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 6.7%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. unpow2 N/A

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

   5. Applied rewrites 100.0%

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

   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 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 100.0

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 100.0

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. sub-neg N/A

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

      18. *-rgt-identity N/A

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

      19. lower--.f64 100.0

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-neg.f64 N/A

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

      23. unsub-neg N/A

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

      24. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 2e-11)
     (/ (+ -3.0 (/ (+ -1.0 (/ -3.0 x)) x)) x)
     (fma (/ -1.0 (- 1.0 x)) (- -1.0 x) t_0))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-11) {
		tmp = (-3.0 + ((-1.0 + (-3.0 / x)) / x)) / x;
	} else {
		tmp = fma((-1.0 / (1.0 - x)), (-1.0 - x), t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 2e-11)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-1.0 + Float64(-3.0 / x)) / x)) / x);
	else
		tmp = fma(Float64(-1.0 / Float64(1.0 - x)), Float64(-1.0 - x), t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-11], N[(N[(-3.0 + N[(N[(-1.0 + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision] + t$95$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 6.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.9%

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

   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 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 100.0

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 100.0

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. sub-neg N/A

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

      18. *-rgt-identity N/A

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

      19. lower--.f64 100.0

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-neg.f64 N/A

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

      23. unsub-neg N/A

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

      24. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 2e-11)
     (/ (+ -3.0 (/ -1.0 x)) x)
     (fma (/ -1.0 (- 1.0 x)) (- -1.0 x) t_0))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-11) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = fma((-1.0 / (1.0 - x)), (-1.0 - x), t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 2e-11)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = fma(Float64(-1.0 / Float64(1.0 - x)), Float64(-1.0 - x), t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + 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[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision] + t$95$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 6.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. 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. lower-+.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. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 100.0

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 100.0

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. sub-neg N/A

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

      18. *-rgt-identity N/A

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

      19. lower--.f64 100.0

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-neg.f64 N/A

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

      23. unsub-neg N/A

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

      24. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

4. Final simplification 99.9%

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

Alternative 4: 99.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 2e-11) (/ (+ -3.0 (/ -1.0 x)) x) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-11) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + 1.0d0)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 2d-11) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-11) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 2e-11:
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 2e-11)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 2e-11)
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-11], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 1.99999999999999988e-11

   1. Initial program 6.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{3 + \frac{1}{x}}{x}} \]
   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. 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. lower-+.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. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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 100.0%

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

4. Final simplification 99.8%

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

Alternative 5: 98.4% accurate, 0.5× speedup

Math

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

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 (fma (fma x x 1.0) 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, fma(fma(x, x, 1.0), 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 = fma(x, fma(fma(x, x, 1.0), 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[(x * N[(N[(x * x + 1.0), $MachinePrecision] * 3.0 + x), $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 6.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. 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. lower-+.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. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. lower-*.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. lower-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. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 99.3%

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

4. Final simplification 99.5%

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

Alternative 6: 98.2% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 2e-11)
   (/ -3.0 x)
   (fma x (fma (fma x x 1.0) 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, fma(fma(x, x, 1.0), 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 = fma(x, fma(fma(x, x, 1.0), 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[(x * N[(N[(x * x + 1.0), $MachinePrecision] * 3.0 + x), $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 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 99.1

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

   5. Applied rewrites 99.1%

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

   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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. lower-*.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. lower-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. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 99.3%

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

4. Final simplification 99.2%

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

Alternative 7: 98.2% 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 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 99.1

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

   5. Applied rewrites 99.1%

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

   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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-commutative N/A

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

      8. lower-*.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. lower-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. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.2%

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

Alternative 8: 98.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 2e-11)
   (/ -3.0 x)
   (fma x x (fma 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, fma(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, x, fma(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 * x + N[(x * 3.0 + 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 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 99.1

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

   5. Applied rewrites 99.1%

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

   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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

   7. Applied rewrites 99.0%

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

   9. Applied rewrites 99.0%

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

4. Final simplification 99.1%

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

Alternative 9: 98.2% 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 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 99.1

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

   5. Applied rewrites 99.1%

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

   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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 99.1%

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

Alternative 10: 50.4% 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 54.8%

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

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

   3. lower-+.f64 52.2

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

5. Applied rewrites 52.2%

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

6. Final simplification 52.2%

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

Alternative 11: 50.5% 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 54.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.0%

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

Reproduce

herbie shell --seed 2024238 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))