Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 74.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = 1.0;
	} else {
		tmp = (x / (1.0 + x)) + (-1.0 - x);
	}
	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
        tmp = (x / (1.0d0 + x)) + ((-1.0d0) - x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 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. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 97.0%

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

   10. Taylor expanded in x around inf 97.9%

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

   if -1.8999999999999999 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 71.0%

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

      1. sub-neg 71.0%

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

      2. neg-mul-1 71.0%

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

      3. metadata-eval 71.0%

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

      4. +-commutative 71.0%

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

      5. unsub-neg 71.0%

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

   7. Simplified 71.0%

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

4. Final simplification 78.2%

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

Alternative 3: 75.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = x / (1.0 + x);
	double tmp;
	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 / (1.0d0 + x)
    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 / (1.0 + x);
	double tmp;
	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 / (1.0 + x)
	tmp = 0
	if 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(1.0 + x))
	tmp = 0.0
	if (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 / (1.0 + x);
	tmp = 0.0;
	if (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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, 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 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 67.7%

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

      1. sub-neg 67.7%

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

      2. neg-mul-1 67.7%

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

      3. metadata-eval 67.7%

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

      4. +-commutative 67.7%

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

      5. unsub-neg 67.7%

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

   7. Simplified 67.7%

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

   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. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.6%

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

4. Final simplification 74.4%

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

Alternative 4: 74.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5) -1.0 (+ 1.0 (/ 1.0 (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (1.0 / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + (1.0d0 / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = -1.0
	else:
		tmp = 1.0 + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = -1.0;
	else
		tmp = 1.0 + (1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], -1.0, N[(1.0 + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 66.3%

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

   if 0.5 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 97.0%

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

   10. Taylor expanded in x around inf 97.9%

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

   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. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 72.2%

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

4. Final simplification 79.0%

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

Alternative 6: 74.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (1.0 / 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
        tmp = 1.0d0 + (1.0d0 / x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 66.3%

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

   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. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.6%

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

   6. Taylor expanded in x around inf 98.6%

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

4. Final simplification 73.1%

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

Alternative 7: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.5) -1.0 (/ x (+ x -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = x / (x + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = -1.0d0
    else
        tmp = x / (x + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -1.0;
	} else {
		tmp = x / (x + -1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = -1.0
	else:
		tmp = x / (x + -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = -1.0;
	else
		tmp = Float64(x / Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = -1.0;
	else
		tmp = x / (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], -1.0, N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 66.3%

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

   if 0.5 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 98.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 97.0%

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

   10. Taylor expanded in x around inf 97.9%

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

   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. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 72.2%

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

   6. Taylor expanded in x around 0 71.2%

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

4. Final simplification 78.3%

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

Alternative 9: 50.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. +-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. frac-add 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. *-commutative 100.0%

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

   6. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-/l/ 100.0%

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

   2. +-commutative 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. +-commutative 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. +-commutative 100.0%

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

   7. +-commutative 100.0%

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

   8. +-commutative 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 52.7%

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

10. Final simplification 52.7%

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

Reproduce

herbie shell --seed 2024066 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))