Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x))))
   (if (or (<= x -1.0) (not (<= x 1.0))) (/ x t_0) (/ (/ 1.0 x) t_0))))

C

double code(double x) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = x / t_0;
	} else {
		tmp = (1.0 / x) / 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)
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = x / t_0
    else
        tmp = (1.0d0 / x) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x + (-1.0 / x);
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = x / t_0;
	} else {
		tmp = (1.0 / x) / t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x + (-1.0 / x)
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = x / t_0
	else:
		tmp = (1.0 / x) / t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(x + Float64(-1.0 / x))
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(x / t_0);
	else
		tmp = Float64(Float64(1.0 / x) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x + (-1.0 / x);
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = x / t_0;
	else
		tmp = (1.0 / x) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x / t$95$0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. associate-*r/ 52.2%

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

      3. associate-/r/ 52.0%

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

      4. +-commutative 52.0%

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

      5. associate-+l+ 52.0%

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

      6. +-commutative 52.0%

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

      7. *-commutative 52.0%

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

      8. metadata-eval 52.0%

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

      9. sub-neg 52.0%

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

      10. difference-of-sqr-1 52.0%

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

      11. fma-neg 52.0%

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

      12. metadata-eval 52.0%

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

   8. Applied egg-rr 52.0%

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

      1. *-lft-identity 52.0%

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

      2. associate-/r/ 52.2%

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

      3. +-commutative 52.2%

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

      4. associate-+r+ 52.2%

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

      5. metadata-eval 52.2%

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

      6. sub-neg 52.2%

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

      7. sub-neg 52.2%

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

      8. metadata-eval 52.2%

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

      9. associate-+r+ 52.2%

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

      10. *-lft-identity 52.2%

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

      11. associate-*l/ 52.2%

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

      12. distribute-lft-in 52.2%

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

      13. lft-mult-inverse 52.2%

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

      14. *-rgt-identity 52.2%

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

      15. associate-+l+ 52.2%

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

      16. metadata-eval 52.2%

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

      17. metadata-eval 52.2%

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

      18. fma-neg 52.2%

          \[\leadsto \frac{x + \left(\frac{1}{x} + 0\right)}{\frac{\color{blue}{x \cdot x - 1}}{x}} \]

      19. unpow2 52.2%

          \[\leadsto \frac{x + \left(\frac{1}{x} + 0\right)}{\frac{\color{blue}{{x}^{2}} - 1}{x}} \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf 99.8%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. associate-*r/ 99.9%

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

      3. associate-/r/ 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+l+ 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. difference-of-sqr-1 99.7%

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

      11. fma-neg 99.7%

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

      12. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. *-lft-identity 99.7%

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

      2. associate-/r/ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. metadata-eval 99.9%

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

      6. sub-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

      9. associate-+r+ 99.9%

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

      10. *-lft-identity 99.9%

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

      11. associate-*l/ 100.0%

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

      12. distribute-lft-in 100.0%

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

      13. lft-mult-inverse 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-+l+ 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. fma-neg 100.0%

          \[\leadsto \frac{x + \left(\frac{1}{x} + 0\right)}{\frac{\color{blue}{x \cdot x - 1}}{x}} \]

      19. unpow2 100.0%

          \[\leadsto \frac{x + \left(\frac{1}{x} + 0\right)}{\frac{\color{blue}{{x}^{2}} - 1}{x}} \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0 98.3%

       \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{x + \frac{-1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.75 \lor \neg \left(x \leq 1.65\right):\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.75) (not (<= x 1.65)))
   (/ x (+ x (/ -1.0 x)))
   (+ x (/ 1.0 (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.75) || !(x <= 1.65)) {
		tmp = x / (x + (-1.0 / x));
	} else {
		tmp = x + (1.0 / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.75d0)) .or. (.not. (x <= 1.65d0))) then
        tmp = x / (x + ((-1.0d0) / x))
    else
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.75) || !(x <= 1.65)) {
		tmp = x / (x + (-1.0 / x));
	} else {
		tmp = x + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.75) or not (x <= 1.65):
		tmp = x / (x + (-1.0 / x))
	else:
		tmp = x + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.75) || !(x <= 1.65))
		tmp = Float64(x / Float64(x + Float64(-1.0 / x)));
	else
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.75) || ~((x <= 1.65)))
		tmp = x / (x + (-1.0 / x));
	else
		tmp = x + (1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.75], N[Not[LessEqual[x, 1.65]], $MachinePrecision]], N[(x / N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.75 or 1.6499999999999999 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{1 + x} + \frac{1}{x + -1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. associate-*r/ 52.2%

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

      3. associate-/r/ 52.0%

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

      4. +-commutative 52.0%

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

      5. associate-+l+ 52.0%

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

      6. +-commutative 52.0%

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

      7. *-commutative 52.0%

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

      8. metadata-eval 52.0%

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

      9. sub-neg 52.0%

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

      10. difference-of-sqr-1 52.0%

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

      11. fma-neg 52.0%

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

      12. metadata-eval 52.0%

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

   8. Applied egg-rr 52.0%

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

      1. *-lft-identity 52.0%

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

      2. associate-/r/ 52.2%

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

      3. +-commutative 52.2%

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

      4. associate-+r+ 52.2%

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

      5. metadata-eval 52.2%

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

      6. sub-neg 52.2%

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

      7. sub-neg 52.2%

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

      8. metadata-eval 52.2%

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

      9. associate-+r+ 52.2%

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

      10. *-lft-identity 52.2%

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

      11. associate-*l/ 52.2%

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

      12. distribute-lft-in 52.2%

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

      13. lft-mult-inverse 52.2%

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

      14. *-rgt-identity 52.2%

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

      15. associate-+l+ 52.2%

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

      16. metadata-eval 52.2%

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

      17. metadata-eval 52.2%

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

      18. fma-neg 52.2%

          \[\leadsto \frac{x + \left(\frac{1}{x} + 0\right)}{\frac{\color{blue}{x \cdot x - 1}}{x}} \]

      19. unpow2 52.2%

          \[\leadsto \frac{x + \left(\frac{1}{x} + 0\right)}{\frac{\color{blue}{{x}^{2}} - 1}{x}} \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf 99.8%

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

   if -0.75 < x < 1.6499999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.3%

      \[\leadsto \color{blue}{x} + \frac{1}{x + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.75 \lor \neg \left(x \leq 1.65\right):\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.9) (+ x (/ 1.0 (+ x -1.0))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.9) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.9d0) then
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.9) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.9:
		tmp = x + (1.0 / (x + -1.0))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.9)
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.9)
		tmp = x + (1.0 / (x + -1.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.9], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.8999999999999999 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.2%

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

   6. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{1} \]

   if -1 < x < 1.8999999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.3%

      \[\leadsto \color{blue}{x} + \frac{1}{x + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.0) 1.0 (if (<= x 1.0) -1.0 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.0:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.2%

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

   6. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{1} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.3%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.1% accurate, 11.0× 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 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 46.9%

   \[\leadsto \color{blue}{-1} \]

6. Final simplification 46.9%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))