Asymptote C

Percentage Accurate: 53.4% → 99.7%

Time: 6.7s

Alternatives: 9

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: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{1 + x}\\ \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 10^{-5}:\\ \;\;\;\;\frac{-3 - \frac{\frac{3}{x} - -1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{1 + x}, \frac{x}{t\_0}, \frac{-1 - x}{x - 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- x 1.0) (+ 1.0 x))))
   (if (<= (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- x 1.0))) 1e-5)
     (/ (- -3.0 (/ (- (/ 3.0 x) -1.0) x)) x)
     (fma (/ t_0 (+ 1.0 x)) (/ x t_0) (/ (- -1.0 x) (- x 1.0))))))

C

double code(double x) {
	double t_0 = (x - 1.0) / (1.0 + x);
	double tmp;
	if (((x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))) <= 1e-5) {
		tmp = (-3.0 - (((3.0 / x) - -1.0) / x)) / x;
	} else {
		tmp = fma((t_0 / (1.0 + x)), (x / t_0), ((-1.0 - x) / (x - 1.0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x - 1.0) / Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(x / Float64(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0))) <= 1e-5)
		tmp = Float64(Float64(-3.0 - Float64(Float64(Float64(3.0 / x) - -1.0) / x)) / x);
	else
		tmp = fma(Float64(t_0 / Float64(1.0 + x)), Float64(x / t_0), Float64(Float64(-1.0 - x) / Float64(x - 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-5], N[(N[(-3.0 - N[(N[(N[(3.0 / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(x / t$95$0), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(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)))) < 1.00000000000000008e-5

   1. Initial program 8.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.7%

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

   if 1.00000000000000008e-5 < (-.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. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-rgt-identity N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{-3 - \frac{\frac{3}{x} - -1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-3, x, 2\right), x, 1\right)}{\mathsf{fma}\left(x, x, -1\right)}}{x - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- x 1.0))) 3e-14)
   (/ (- -3.0 (/ (- (/ 3.0 x) -1.0) x)) x)
   (/ (/ (fma (fma -3.0 x 2.0) x 1.0) (fma x x -1.0)) (- x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))) <= 3e-14) {
		tmp = (-3.0 - (((3.0 / x) - -1.0) / x)) / x;
	} else {
		tmp = (fma(fma(-3.0, x, 2.0), x, 1.0) / fma(x, x, -1.0)) / (x - 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0))) <= 3e-14)
		tmp = Float64(Float64(-3.0 - Float64(Float64(Float64(3.0 / x) - -1.0) / x)) / x);
	else
		tmp = Float64(Float64(fma(fma(-3.0, x, 2.0), x, 1.0) / fma(x, x, -1.0)) / Float64(x - 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3e-14], N[(N[(-3.0 - N[(N[(N[(3.0 / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(-3.0 * x + 2.0), $MachinePrecision] * x + 1.0), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(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)))) < 2.9999999999999998e-14

   1. Initial program 7.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.7%

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

   if 2.9999999999999998e-14 < (-.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 99.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-rgt-identity N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift--.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. associate--r+ N/A

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

      12. lift--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. +-commutative N/A

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

      15. flip-+ N/A

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

      16. lift--.f64 N/A

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

      17. frac-sub N/A

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 99.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-3, x, 2\right)}, x, 1\right)}{\mathsf{fma}\left(x, x, -1\right)}}{x - 1} \]

   11. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{-3 - \frac{\frac{3}{x} - -1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-3, x, 2\right), x, 1\right)}{\mathsf{fma}\left(x, x, -1\right)}}{x - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x} - \frac{1 + x}{x - 1}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{-3 - \frac{\frac{3}{x} - -1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- x 1.0)))))
   (if (<= t_0 1e-5) (/ (- -3.0 (/ (- (/ 3.0 x) -1.0) x)) x) t_0)))

C

double code(double x) {
	double t_0 = (x / (1.0 + x)) - ((1.0 + x) / (x - 1.0));
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = (-3.0 - (((3.0 / x) - -1.0) / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (1.0d0 + x)) - ((1.0d0 + x) / (x - 1.0d0))
    if (t_0 <= 1d-5) then
        tmp = ((-3.0d0) - (((3.0d0 / x) - (-1.0d0)) / x)) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (1.0 + x)) - ((1.0 + x) / (x - 1.0));
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = (-3.0 - (((3.0 / x) - -1.0) / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))
	tmp = 0
	if t_0 <= 1e-5:
		tmp = (-3.0 - (((3.0 / x) - -1.0) / x)) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0)))
	tmp = 0.0
	if (t_0 <= 1e-5)
		tmp = Float64(Float64(-3.0 - Float64(Float64(Float64(3.0 / x) - -1.0) / x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (1.0 + x)) - ((1.0 + x) / (x - 1.0));
	tmp = 0.0;
	if (t_0 <= 1e-5)
		tmp = (-3.0 - (((3.0 / x) - -1.0) / x)) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-5], N[(N[(-3.0 - N[(N[(N[(3.0 / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 1.00000000000000008e-5

   1. Initial program 8.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.7%

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

   if 1.00000000000000008e-5 < (-.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. Recombined 2 regimes into one program.

4. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x} - \frac{1 + x}{x - 1}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{\frac{2}{x} - 3}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- x 1.0)))))
   (if (<= t_0 1e-5) (/ (- (/ 2.0 x) 3.0) (- x 1.0)) t_0)))

C

double code(double x) {
	double t_0 = (x / (1.0 + x)) - ((1.0 + x) / (x - 1.0));
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = ((2.0 / x) - 3.0) / (x - 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (1.0d0 + x)) - ((1.0d0 + x) / (x - 1.0d0))
    if (t_0 <= 1d-5) then
        tmp = ((2.0d0 / x) - 3.0d0) / (x - 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (1.0 + x)) - ((1.0 + x) / (x - 1.0));
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = ((2.0 / x) - 3.0) / (x - 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))
	tmp = 0
	if t_0 <= 1e-5:
		tmp = ((2.0 / x) - 3.0) / (x - 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0)))
	tmp = 0.0
	if (t_0 <= 1e-5)
		tmp = Float64(Float64(Float64(2.0 / x) - 3.0) / Float64(x - 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (1.0 + x)) - ((1.0 + x) / (x - 1.0));
	tmp = 0.0;
	if (t_0 <= 1e-5)
		tmp = ((2.0 / x) - 3.0) / (x - 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-5], N[(N[(N[(2.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 1.00000000000000008e-5

   1. Initial program 8.5%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-rgt-identity N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 N/A

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

   4. Applied rewrites 9.6%

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

   6. Applied rewrites 10.1%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} - 3}}{x - 1} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 99.3

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

   9. Applied rewrites 99.3%

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

   if 1.00000000000000008e-5 < (-.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. Recombined 2 regimes into one program.

4. Final simplification 99.6%

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

Alternative 5: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 0.005:\\ \;\;\;\;\frac{\frac{2}{x} - 3}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- x 1.0))) 0.005)
   (/ (- (/ 2.0 x) 3.0) (- x 1.0))
   (* (fma x 3.0 1.0) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))) <= 0.005) {
		tmp = ((2.0 / x) - 3.0) / (x - 1.0);
	} 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(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0))) <= 0.005)
		tmp = Float64(Float64(Float64(2.0 / x) - 3.0) / Float64(x - 1.0));
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(N[(2.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $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)))) < 0.0050000000000000001

   1. Initial program 9.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-rgt-identity N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 N/A

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

   4. Applied rewrites 10.3%

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

   6. Applied rewrites 10.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} - 3}}{x - 1} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if 0.0050000000000000001 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 0.005:\\ \;\;\;\;\frac{\frac{2}{x} - 3}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 0.005:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- x 1.0))) 0.005)
   (/ (- (/ -1.0 x) 3.0) x)
   (* (fma x 3.0 1.0) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))) <= 0.005) {
		tmp = ((-1.0 / x) - 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(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0))) <= 0.005)
		tmp = Float64(Float64(Float64(-1.0 / x) - 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[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(N[(-1.0 / x), $MachinePrecision] - 3.0), $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)))) < 0.0050000000000000001

   1. Initial program 9.1%

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

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

      3. neg-mul-1 N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 0.0050000000000000001 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 0.005:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 0.005:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- x 1.0))) 0.005)
   (/ -3.0 x)
   (* (fma x 3.0 1.0) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if (((x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))) <= 0.005) {
		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(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0))) <= 0.005)
		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[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], 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)))) < 0.0050000000000000001

   1. Initial program 9.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 0.0050000000000000001 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt1-in N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 0.005:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 0.005:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3 + x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ 1.0 x)) (/ (+ 1.0 x) (- x 1.0))) 0.005)
   (/ -3.0 x)
   (fma (+ 3.0 x) x 1.0)))

C

double code(double x) {
	double tmp;
	if (((x / (1.0 + x)) - ((1.0 + x) / (x - 1.0))) <= 0.005) {
		tmp = -3.0 / x;
	} else {
		tmp = fma((3.0 + x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x / Float64(1.0 + x)) - Float64(Float64(1.0 + x) / Float64(x - 1.0))) <= 0.005)
		tmp = Float64(-3.0 / x);
	else
		tmp = fma(Float64(3.0 + x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(-3.0 / x), $MachinePrecision], N[(N[(3.0 + x), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 0.0050000000000000001

   1. Initial program 9.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 0.0050000000000000001 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \leq 0.005:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3 + x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.8% 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 55.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.5%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))