Asymptote C

Percentage Accurate: 53.7% → 99.7%

Time: 7.0s

Alternatives: 5

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: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 0.0

   1. Initial program 6.5%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   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 98.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. sub-neg N/A

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

      21. lower-fma.f64 N/A

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

      22. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(x, -3.0, -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(-1.0 - x) / Float64(x + -1.0))) <= 0.0)
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(fma(x, -3.0, -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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-3.0 / x), $MachinePrecision], N[(N[(x * -3.0 + -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 7.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 0.0 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 98.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. sub-neg N/A

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

      21. lower-fma.f64 N/A

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

      22. metadata-eval 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.4%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))