Asymptote C

Percentage Accurate: 54.2% → 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: 54.2% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-7}:\\ \;\;\;\;\frac{-3 + \frac{-1 + \frac{-3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 1e-7)
   (/ (+ -3.0 (/ (+ -1.0 (/ -3.0 x)) x)) x)
   (/ (fma -3.0 x -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))) <= 1e-7) {
		tmp = (-3.0 + ((-1.0 + (-3.0 / x)) / x)) / 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(-1.0 - x) / Float64(x + -1.0))) <= 1e-7)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-1.0 + Float64(-3.0 / x)) / x)) / 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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(-3.0 + N[(N[(-1.0 + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-3.0 * x + -1.0), $MachinePrecision] / N[(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)))) < 9.9999999999999995e-8

   1. Initial program 7.0%

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

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

   if 9.9999999999999995e-8 < (-.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. frac-sub N/A

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

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

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

      3. metadata-eval N/A

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

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

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

      5. --lowering--.f64 N/A

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

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

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

      7. sub-neg N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

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

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

      16. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   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. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 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 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 10^{-7}:\\ \;\;\;\;\frac{-3 + \frac{-1 + \frac{-3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 5e-13)
   (/ (+ -3.0 (/ -1.0 x)) x)
   (/ (fma -3.0 x -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))) <= 5e-13) {
		tmp = (-3.0 + (-1.0 / x)) / 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(-1.0 - x) / Float64(x + -1.0))) <= 5e-13)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / 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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-13], N[(N[(-3.0 + N[(-1.0 / x), $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)))) < 4.9999999999999999e-13

   1. Initial program 6.1%

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

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

   5. Simplified 100.0%

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

   if 4.9999999999999999e-13 < (-.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 99.2%

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

      1. frac-sub N/A

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

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

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

      3. metadata-eval N/A

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

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

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

      5. --lowering--.f64 N/A

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

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

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

      7. sub-neg N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

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

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

      16. metadata-eval 99.2

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

   4. Applied egg-rr 99.2%

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

   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. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 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 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, -1\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ 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)) + ((-1.0 - x) / (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(-1.0 - x) / 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[(-1.0 - x), $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 5.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 99.7

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

   5. Simplified 99.7%

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

   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 98.7%

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

      1. frac-sub N/A

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

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

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

      3. metadata-eval N/A

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

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

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

      5. --lowering--.f64 N/A

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

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

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

      7. sub-neg N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

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

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

      16. metadata-eval 98.8

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

   4. Applied egg-rr 98.8%

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

   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. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 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 99.9%

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

Alternative 4: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0001:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(3, x, 1\right), 3\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0001)
   (/ -3.0 x)
   (fma x (fma x (fma 3.0 x 1.0) 3.0) 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))) <= 0.0001) {
		tmp = -3.0 / x;
	} else {
		tmp = fma(x, fma(x, fma(3.0, x, 1.0), 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))) <= 0.0001)
		tmp = Float64(-3.0 / x);
	else
		tmp = fma(x, fma(x, fma(3.0, x, 1.0), 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], 0.0001], N[(-3.0 / x), $MachinePrecision], N[(x * N[(x * N[(3.0 * x + 1.0), $MachinePrecision] + 3.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 1.00000000000000005e-4

   1. Initial program 7.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.6

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

   5. Simplified 98.6%

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

   if 1.00000000000000005e-4 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 99.1

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

   5. Simplified 99.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0001:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(3, x, 1\right), 3\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0001:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0001)
   (/ -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))) <= 0.0001) {
		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))) <= 0.0001)
		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], 0.0001], 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)))) < 1.00000000000000005e-4

   1. Initial program 7.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.6

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

   5. Simplified 98.6%

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

   if 1.00000000000000005e-4 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

      1. frac-sub N/A

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

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

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

      3. metadata-eval N/A

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

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

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

      5. --lowering--.f64 N/A

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

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

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

      7. sub-neg N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

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

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

      16. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

   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. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 100.0%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

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

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

      4. distribute-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 N/A

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

      14. accelerator-lowering-fma.f64 100.0

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. accelerator-lowering-fma.f64 99.0

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

   12. Simplified 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0001:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0001:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0))) 0.0001)
   (/ -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))) <= 0.0001) {
		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))) <= 0.0001)
		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], 0.0001], N[(-3.0 / x), $MachinePrecision], N[(x * N[(x + 3.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 1.00000000000000005e-4

   1. Initial program 7.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.6

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

   5. Simplified 98.6%

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

   if 1.00000000000000005e-4 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-lowering-+.f64 98.6

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

   5. Simplified 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0001:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x + 3, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.0%

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
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 53.8

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

5. Simplified 53.8%

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

6. Final simplification 53.8%

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

Alternative 8: 50.8% 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 57.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 53.2%

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

Reproduce

herbie shell --seed 2024194 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))