Asymptote C

Percentage Accurate: 53.9% → 99.8%

Time: 9.4s

Alternatives: 8

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: 99.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0)
   (/ (+ -3.0 (/ (- (+ -3.0 (/ -1.0 x)) x) (fma x x 0.0))) x)
   (* (fma x 3.0 1.0) (/ -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 + (((-3.0 + (-1.0 / x)) - x) / fma(x, x, 0.0))) / x;
	} else {
		tmp = fma(x, 3.0, 1.0) * (-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(Float64(Float64(-3.0 + Float64(-1.0 / x)) - x) / fma(x, x, 0.0))) / x);
	else
		tmp = Float64(fma(x, 3.0, 1.0) * Float64(-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[(N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(x * x + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * 3.0 + 1.0), $MachinePrecision] * N[(-1.0 / N[(x * x + -1.0), $MachinePrecision]), $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 6.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-lowering-+.f64 6.3

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

   4. Applied egg-rr 6.3%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-3 + \frac{\left(-3 + \frac{-1}{x}\right) - x}{\mathsf{fma}\left(x, x, 0\right)}}{x}} \]

   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 99.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. frac-add N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sub-neg N/A

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

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

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

      10. metadata-eval N/A

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

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

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

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

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

      13. distribute-neg-in N/A

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

      14. neg-mul-1 N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   4. Applied egg-rr 99.8%

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

   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. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 100.0%

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-frac N/A

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

      9. distribute-frac-neg2 N/A

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

      10. *-rgt-identity N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. associate-*r/ N/A

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

      20. *-rgt-identity N/A

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0)
   (/ (+ -3.0 (/ (- -3.0 x) (fma x x 0.0))) x)
   (* (fma x 3.0 1.0) (/ -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 + ((-3.0 - x) / fma(x, x, 0.0))) / x;
	} else {
		tmp = fma(x, 3.0, 1.0) * (-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(Float64(-3.0 - x) / fma(x, x, 0.0))) / x);
	else
		tmp = Float64(fma(x, 3.0, 1.0) * Float64(-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[(N[(-3.0 - x), $MachinePrecision] / N[(x * x + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * 3.0 + 1.0), $MachinePrecision] * N[(-1.0 / N[(x * x + -1.0), $MachinePrecision]), $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 6.6%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   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. Simplified 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 99.9%

      \[\leadsto \frac{\color{blue}{-3 + \frac{-3 - x}{\mathsf{fma}\left(x, x, 0\right)}}}{x} \]

   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 99.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. frac-add N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sub-neg N/A

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

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

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

      10. metadata-eval N/A

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

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

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

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

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

      13. distribute-neg-in N/A

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

      14. neg-mul-1 N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   4. Applied egg-rr 99.8%

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

   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. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 100.0%

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-frac N/A

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

      9. distribute-frac-neg2 N/A

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

      10. *-rgt-identity N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. associate-*r/ N/A

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

      20. *-rgt-identity N/A

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

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) (/ -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) * (-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) * Float64(-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[(-1.0 / N[(x * x + -1.0), $MachinePrecision]), $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 6.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. neg-mul-1 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 99.7

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

   5. Simplified 99.7%

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

   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 99.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. frac-add N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sub-neg N/A

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

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

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

      10. metadata-eval N/A

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

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

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

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

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

      13. distribute-neg-in N/A

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

      14. neg-mul-1 N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   4. Applied egg-rr 99.8%

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

   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. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 100.0%

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. distribute-neg-frac N/A

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

      9. distribute-frac-neg2 N/A

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

      10. *-rgt-identity N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. associate-*r/ N/A

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

      20. *-rgt-identity N/A

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

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

Alternative 4: 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 6.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. neg-mul-1 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 99.7

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

   5. Simplified 99.7%

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

   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 99.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. frac-add N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sub-neg N/A

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

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

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

      10. metadata-eval N/A

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

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

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

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

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

      13. distribute-neg-in N/A

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

      14. neg-mul-1 N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   4. Applied egg-rr 99.8%

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

   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. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 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 99.8%

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

Alternative 5: 99.5% 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 6.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 99.4

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

   5. Simplified 99.4%

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

   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 99.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. frac-add N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sub-neg N/A

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

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

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

      10. metadata-eval N/A

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

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

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

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

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

      13. distribute-neg-in N/A

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

      14. neg-mul-1 N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   4. Applied egg-rr 99.8%

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

   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. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 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 99.7%

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

Alternative 6: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.004:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.004)
   (/ -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))) <= 0.004) {
		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))) <= 0.004)
		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], 0.004], 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)))) < 0.0040000000000000001

   1. Initial program 7.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.5

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

   5. Simplified 98.5%

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

   if 0.0040000000000000001 < (-.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. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. sub-neg N/A

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

      2. flip-+ N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. associate-/r/ N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. un-div-inv N/A

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

      15. distribute-neg-frac2 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. +-commutative N/A

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

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

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

      19. +-commutative N/A

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

      20. distribute-neg-in N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.004:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot \mathsf{fma}\left(3, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.004:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.004)
   (/ -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))) <= 0.004) {
		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))) <= 0.004)
		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], 0.004], 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)))) < 0.0040000000000000001

   1. Initial program 7.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.5

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

   5. Simplified 98.5%

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

   if 0.0040000000000000001 < (-.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. accelerator-lowering-fma.f64 N/A

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

      3. +-lowering-+.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.004:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 53.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.6%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))