Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 5

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. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.55 \lor \neg \left(x \leq 1.82\right):\\ \;\;\;\;\frac{1}{x} - \frac{x}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.55) (not (<= x 1.82)))
   (- (/ 1.0 x) (/ x (- -1.0 x)))
   (+ x (/ 1.0 (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.55) || !(x <= 1.82)) {
		tmp = (1.0 / 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 <= (-0.55d0)) .or. (.not. (x <= 1.82d0))) then
        tmp = (1.0d0 / 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 <= -0.55) || !(x <= 1.82)) {
		tmp = (1.0 / x) - (x / (-1.0 - x));
	} else {
		tmp = x + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.55) or not (x <= 1.82):
		tmp = (1.0 / x) - (x / (-1.0 - x))
	else:
		tmp = x + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.55) || !(x <= 1.82))
		tmp = Float64(Float64(1.0 / 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 <= -0.55) || ~((x <= 1.82)))
		tmp = (1.0 / 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, -0.55], N[Not[LessEqual[x, 1.82]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] - 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 < -0.55000000000000004 or 1.82000000000000006 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.8%

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

   if -0.55000000000000004 < x < 1.82000000000000006

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

   6. Simplified 99.3%

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

      1. sub-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. remove-double-neg 99.3%

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

   8. Applied egg-rr 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.55 \lor \neg \left(x \leq 1.82\right):\\ \;\;\;\;\frac{1}{x} - \frac{x}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \]

Alternative 3: 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. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.8%

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

   if -1 < x < 1.8999999999999999

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

   6. Simplified 99.3%

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

      1. sub-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. remove-double-neg 99.3%

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

   8. Applied egg-rr 99.3%

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

4. Final simplification 99.0%

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

Alternative 4: 98.9% 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. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.8%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around 0 99.2%

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

      1. sub-neg 99.2%

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

      2. neg-mul-1 99.2%

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

      3. metadata-eval 99.2%

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

      4. +-commutative 99.2%

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

      5. sub-neg 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.0%

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

Alternative 5: 49.7% 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. remove-double-neg 100.0%

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

   2. distribute-frac-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. sub-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. neg-mul-1 100.0%

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

   7. metadata-eval 100.0%

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

   8. *-commutative 100.0%

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

   9. associate-/l* 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. metadata-eval 100.0%

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

   13. associate-/l* 100.0%

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

   14. associate-*l/ 100.0%

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

   15. /-rgt-identity 100.0%

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

   16. distribute-rgt1-in 100.0%

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

   17. *-commutative 100.0%

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

   18. metadata-eval 100.0%

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

   19. neg-mul-1 100.0%

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

   20. unsub-neg 100.0%

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

   21. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 50.1%

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

   1. neg-mul-1 50.1%

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

6. Simplified 50.1%

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

7. Taylor expanded in x around 0 50.0%

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

   1. sub-neg 50.0%

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

   2. neg-mul-1 50.0%

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

   3. metadata-eval 50.0%

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

   4. +-commutative 50.0%

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

   5. sub-neg 50.0%

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

9. Simplified 50.0%

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

10. Taylor expanded in x around 0 49.3%

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

11. Final simplification 49.3%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023319 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))