Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 7

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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. metadata-eval 100.0%

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

   4. metadata-eval 100.0%

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

   5. metadata-eval 100.0%

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

   6. associate-*l/ 100.0%

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

   7. associate-/r/ 100.0%

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

   8. metadata-eval 100.0%

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

   9. metadata-eval 100.0%

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

   10. metadata-eval 100.0%

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

   11. associate-/r* 100.0%

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

   12. associate-*r/ 100.0%

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

   13. associate-/l* 100.0%

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

   14. associate-/r/ 100.0%

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

   15. metadata-eval 100.0%

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

   16. metadata-eval 100.0%

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

   17. metadata-eval 100.0%

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

   18. *-lft-identity 100.0%

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

   19. sub-neg 100.0%

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

   20. 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. Step-by-step derivation

   1. flip-+ 100.0%

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

   2. sub-neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. associate-/r/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. fma-neg 100.0%

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

   8. metadata-eval 100.0%

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

   9. +-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

   2. *-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.8999999999999999 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 98.3%

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

   5. Taylor expanded in x around inf 99.3%

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

   if -1.8999999999999999 < 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. neg-mul-1 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-commutative 99.3%

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

      5. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 3: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t_0 + \left(-1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -1.9) 1.0 (if (<= x 1.0) (+ t_0 (- -1.0 x)) (+ t_0 (/ 1.0 x))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1.9) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = t_0 + (-1.0 - x);
	} else {
		tmp = t_0 + (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-1.9d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = t_0 + ((-1.0d0) - x)
    else
        tmp = t_0 + (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1.9) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = t_0 + (-1.0 - x);
	} else {
		tmp = t_0 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -1.9:
		tmp = 1.0
	elif x <= 1.0:
		tmp = t_0 + (-1.0 - x)
	else:
		tmp = t_0 + (1.0 / x)
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.9)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = Float64(t_0 + Float64(-1.0 - x));
	else
		tmp = Float64(t_0 + Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -1.9)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = t_0 + (-1.0 - x);
	else
		tmp = t_0 + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.9], 1.0, If[LessEqual[x, 1.0], N[(t$95$0 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1.8999999999999999 < 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. neg-mul-1 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-commutative 99.3%

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

      5. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

   if 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 98.8%

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

4. Final simplification 99.3%

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

Alternative 4: 99.0% accurate, 1.0× 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 98.3%

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

   5. Taylor expanded in x around inf 99.3%

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

   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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.3%

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

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x / N[(x + 1.0), $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. Final simplification 100.0%

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

Alternative 6: 99.0% accurate, 2.1× 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 98.3%

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

   5. Taylor expanded in x around inf 99.3%

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

   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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. neg-mul-1 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-commutative 99.3%

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

      5. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around 0 99.3%

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

   8. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.3%

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

Alternative 7: 49.5% 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. metadata-eval 100.0%

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

   4. metadata-eval 100.0%

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

   5. metadata-eval 100.0%

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

   6. associate-*l/ 100.0%

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

   7. associate-/r/ 100.0%

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

   8. metadata-eval 100.0%

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

   9. metadata-eval 100.0%

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

   10. metadata-eval 100.0%

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

   11. associate-/r* 100.0%

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

   12. associate-*r/ 100.0%

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

   13. associate-/l* 100.0%

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

   14. associate-/r/ 100.0%

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

   15. metadata-eval 100.0%

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

   16. metadata-eval 100.0%

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

   17. metadata-eval 100.0%

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

   18. *-lft-identity 100.0%

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

   19. sub-neg 100.0%

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

   20. 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. Taylor expanded in x around 0 48.3%

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

   1. sub-neg 48.3%

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

   2. neg-mul-1 48.3%

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

   3. metadata-eval 48.3%

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

   4. +-commutative 48.3%

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

   5. unsub-neg 48.3%

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

6. Simplified 48.3%

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

7. Taylor expanded in x around 0 48.5%

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

8. Taylor expanded in x around 0 47.7%

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

9. Final simplification 47.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))