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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. flip-- 100.0%

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

   2. associate-/r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. fma-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. +-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

   2. *-lft-identity 100.0%

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

   3. +-commutative 100.0%

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

5. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. clear-num 100.0%

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

   3. frac-add 73.8%

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

   4. *-un-lft-identity 73.8%

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

   5. metadata-eval 73.8%

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

   6. fma-neg 73.8%

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

   7. metadata-eval 73.8%

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

   8. flip-- 74.6%

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

   9. *-commutative 74.6%

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

   10. *-un-lft-identity 74.6%

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

   11. sub-neg 74.6%

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

   12. metadata-eval 74.6%

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

   13. metadata-eval 74.6%

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

   14. fma-neg 74.6%

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

   15. metadata-eval 74.6%

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

   16. flip-- 100.0%

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

   17. sub-neg 100.0%

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

   18. metadata-eval 100.0%

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

7. Applied egg-rr 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+l+ 100.0%

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

   3. +-commutative 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 100.0%

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

11. Taylor expanded in x around 0 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{1 + x} + \left(-1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.95)
   1.0
   (if (<= x 1.0) (+ (/ x (+ 1.0 x)) (- -1.0 x)) (+ 1.0 (/ (/ 2.0 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.95) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = (x / (1.0 + x)) + (-1.0 - x);
	} else {
		tmp = 1.0 + ((2.0 / x) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.95d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = (x / (1.0d0 + x)) + ((-1.0d0) - x)
    else
        tmp = 1.0d0 + ((2.0d0 / x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.95) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = (x / (1.0 + x)) + (-1.0 - x);
	} else {
		tmp = 1.0 + ((2.0 / x) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.95:
		tmp = 1.0
	elif x <= 1.0:
		tmp = (x / (1.0 + x)) + (-1.0 - x)
	else:
		tmp = 1.0 + ((2.0 / x) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.95)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(-1.0 - x));
	else
		tmp = Float64(1.0 + Float64(Float64(2.0 / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.95)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = (x / (1.0 + x)) + (-1.0 - x);
	else
		tmp = 1.0 + ((2.0 / x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.95], 1.0, If[LessEqual[x, 1.0], N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.94999999999999996

   1. Initial program 100.0%

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

      1. flip-- 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1.94999999999999996 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. metadata-eval 98.5%

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

      4. +-commutative 98.5%

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

      5. unsub-neg 98.5%

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

   4. Simplified 98.5%

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

   if 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.8%

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

      1. associate-*r/ 98.8%

         \[\leadsto 1 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}} \]

      2. metadata-eval 98.8%

         \[\leadsto 1 + \frac{\color{blue}{2}}{{x}^{2}} \]

      3. unpow2 98.8%

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

      4. associate-/r* 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{1 + x} + \left(-1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{2}{x}}{x}\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-1 - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -1.0 - (x * x);
	} else {
		tmp = 1.0 + ((2.0 / x) / x);
	}
	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) - (x * x)
    else
        tmp = 1.0d0 + ((2.0d0 / x) / x)
    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 - (x * x);
	} else {
		tmp = 1.0 + ((2.0 / x) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.0:
		tmp = -1.0 - (x * x)
	else:
		tmp = 1.0 + ((2.0 / x) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = Float64(-1.0 - Float64(x * x));
	else
		tmp = Float64(1.0 + Float64(Float64(2.0 / x) / x));
	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 - (x * x);
	else
		tmp = 1.0 + ((2.0 / x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], N[(-1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. flip-- 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

   3. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{-1 \cdot {x}^{2} - 1} \]
   4. Step-by-step derivation

      1. sub-neg 98.5%

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

      2. metadata-eval 98.5%

         \[\leadsto -1 \cdot {x}^{2} + \color{blue}{-1} \]

      3. +-commutative 98.5%

         \[\leadsto \color{blue}{-1 + -1 \cdot {x}^{2}} \]

      4. mul-1-neg 98.5%

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

      5. unsub-neg 98.5%

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

      6. unpow2 98.5%

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

   5. Simplified 98.5%

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

   if 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.8%

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

      1. associate-*r/ 98.8%

         \[\leadsto 1 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}} \]

      2. metadata-eval 98.8%

         \[\leadsto 1 + \frac{\color{blue}{2}}{{x}^{2}} \]

      3. unpow2 98.8%

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

      4. associate-/r* 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-1 - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{2}{x}}{x}\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 5: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -1.0 - (x * x);
	} 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) - (x * x)
    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 - (x * x);
	} 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 - (x * x)
	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 = Float64(-1.0 - Float64(x * x));
	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 - (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], N[(-1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], 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. flip-- 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 99.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

   3. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{-1 \cdot {x}^{2} - 1} \]
   4. Step-by-step derivation

      1. sub-neg 98.5%

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

      2. metadata-eval 98.5%

         \[\leadsto -1 \cdot {x}^{2} + \color{blue}{-1} \]

      3. +-commutative 98.5%

         \[\leadsto \color{blue}{-1 + -1 \cdot {x}^{2}} \]

      4. mul-1-neg 98.5%

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

      5. unsub-neg 98.5%

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

      6. unpow2 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 98.8%

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

Alternative 6: 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. flip-- 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 99.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. flip-- 99.9%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. +-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. clear-num 99.9%

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

      2. clear-num 99.9%

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

      3. frac-add 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. metadata-eval 99.9%

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

      6. fma-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. flip-- 99.9%

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

      9. *-commutative 99.9%

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

      10. *-un-lft-identity 99.9%

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

      11. sub-neg 99.9%

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

      12. metadata-eval 99.9%

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

      13. metadata-eval 99.9%

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

      14. fma-neg 99.9%

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

      15. metadata-eval 99.9%

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

      16. flip-- 99.9%

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

      17. sub-neg 99.9%

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

      18. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 98.5%

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

4. Final simplification 98.8%

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

Alternative 7: 50.0% 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. flip-- 100.0%

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

   2. associate-/r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. fma-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. +-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

   2. *-lft-identity 100.0%

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

   3. +-commutative 100.0%

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

5. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. clear-num 100.0%

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

   3. frac-add 73.8%

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

   4. *-un-lft-identity 73.8%

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

   5. metadata-eval 73.8%

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

   6. fma-neg 73.8%

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

   7. metadata-eval 73.8%

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

   8. flip-- 74.6%

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

   9. *-commutative 74.6%

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

   10. *-un-lft-identity 74.6%

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

   11. sub-neg 74.6%

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

   12. metadata-eval 74.6%

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

   13. metadata-eval 74.6%

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

   14. fma-neg 74.6%

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

   15. metadata-eval 74.6%

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

   16. flip-- 100.0%

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

   17. sub-neg 100.0%

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

   18. metadata-eval 100.0%

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

7. Applied egg-rr 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+l+ 100.0%

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

   3. +-commutative 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 51.9%

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

11. Final simplification 51.9%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023189 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))