Asymptote C

Percentage Accurate: 54.6% → 99.9%

Time: 7.0s

Alternatives: 8

Speedup: 1.0×

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: 54.6% 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: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-2 \cdot x}{1 + x} - 1}{x - 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (/ (* -2.0 x) (+ 1.0 x)) 1.0) (- x 1.0)))

C

double code(double x) {
	return (((-2.0 * x) / (1.0 + x)) - 1.0) / (x - 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((-2.0d0) * x) / (1.0d0 + x)) - 1.0d0) / (x - 1.0d0)
end function

Java

public static double code(double x) {
	return (((-2.0 * x) / (1.0 + x)) - 1.0) / (x - 1.0);
}

Python

def code(x):
	return (((-2.0 * x) / (1.0 + x)) - 1.0) / (x - 1.0)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(-2.0 * x) / Float64(1.0 + x)) - 1.0) / Float64(x - 1.0))
end

MATLAB

function tmp = code(x)
	tmp = (((-2.0 * x) / (1.0 + x)) - 1.0) / (x - 1.0);
end

Wolfram

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

Derivation

1. Initial program 58.1%

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

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

   3. lift-+.f64 N/A

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

   4. div-add N/A

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

   5. associate--r+ N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. lift-+.f64 N/A

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

   9. associate-/r/ N/A

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

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

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

4. Applied rewrites 62.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lift-fma.f64 N/A

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

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

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

   9. lift--.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. associate-/r* N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow-1 N/A

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

   15. associate-/r/ N/A

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

   16. lift-fma.f64 N/A

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

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

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

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

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

6. Applied rewrites 65.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{\color{blue}{-2} \cdot x}{1 + x} - 1}{x - 1} \]
10. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.0)
   (/ (- -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))) <= 0.0) {
		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))) <= 0.0)
		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], 0.0], 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)))) < 0.0

   1. Initial program 9.7%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. 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 98.2

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

   5. Applied rewrites 98.2%

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

   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 98.3%

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

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

      3. lift-+.f64 N/A

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

      4. div-add N/A

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r/ N/A

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

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

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-1 N/A

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

      9. associate-*l/ N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\mathsf{fma}\left(x, x, -1\right)} \]
   8. 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-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-rgt-neg-in N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. lft-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 100.0

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

   9. 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 99.1%

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

Alternative 3: 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 9.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 96.8

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

   5. Applied rewrites 96.8%

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

   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 98.3%

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

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

      3. lift-+.f64 N/A

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

      4. div-add N/A

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r/ N/A

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

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

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-1 N/A

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

      9. associate-*l/ N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\mathsf{fma}\left(x, x, -1\right)} \]
   8. 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-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-rgt-neg-in N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. lft-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 100.0

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

   9. 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 4: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.6:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right), 3 \cdot x, \mathsf{fma}\left(x, x, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.6)
   (/ -3.0 x)
   (fma (fma x x 1.0) (* 3.0 x) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.6) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(fma(x, x, 1.0), (3.0 * x), fma(x, x, 1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0))) <= 0.6)
		tmp = Float64(-3.0 / x);
	else
		tmp = fma(fma(x, x, 1.0), Float64(3.0 * x), fma(x, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(-3.0 / x), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] * N[(3.0 * x), $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)))) < 0.599999999999999978

   1. Initial program 12.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if 0.599999999999999978 < (-.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

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

      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 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

Alternative 5: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.6:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.6)
   (/ -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))) <= 0.6) {
		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))) <= 0.6)
		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], 0.6], 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)))) < 0.599999999999999978

   1. Initial program 12.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if 0.599999999999999978 < (-.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

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

      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 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 6: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.6:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3 + x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.6)
   (/ -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))) <= 0.6) {
		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))) <= 0.6)
		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], 0.6], 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)))) < 0.599999999999999978

   1. Initial program 12.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if 0.599999999999999978 < (-.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. *-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 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.1%

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

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

   3. lift-+.f64 N/A

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

   4. div-add N/A

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

   5. associate--r+ N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. lift-+.f64 N/A

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

   9. associate-/r/ N/A

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

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

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

4. Applied rewrites 62.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow-1 N/A

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

   9. associate-*l/ N/A

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

   10. sub-div N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 62.7%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\mathsf{fma}\left(x, x, -1\right)} \]
8. 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-in N/A

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

   8. distribute-rgt-in N/A

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

   9. distribute-rgt-neg-in N/A

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

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

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

   11. metadata-eval N/A

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

   12. distribute-rgt-neg-out N/A

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

   13. lft-mult-inverse N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 80.9

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

9. Applied rewrites 80.9%

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

    1. lift-/.f64 N/A

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

    2. lift-fma.f64 N/A

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

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

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

    4. +-commutative N/A

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

    5. *-lft-identity N/A

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

    6. lift--.f64 N/A

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

    7. associate-/r* N/A

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

    8. lower-/.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. *-lft-identity N/A

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

    11. lift-+.f64 99.4

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

11. Applied rewrites 99.4%

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

Alternative 8: 51.2% 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 58.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.8%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))