Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. 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. clear-num 100.0%

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

   2. frac-2neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. frac-add 100.0%

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

   5. *-un-lft-identity 100.0%

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

   6. +-commutative 100.0%

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

   7. distribute-neg-in 100.0%

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

   8. metadata-eval 100.0%

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

   9. sub-neg 100.0%

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

   10. +-commutative 100.0%

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

   11. +-commutative 100.0%

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

   12. +-commutative 100.0%

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

   13. distribute-neg-in 100.0%

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

   14. metadata-eval 100.0%

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

   15. sub-neg 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-+l- 100.0%

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

   2. sub-neg 100.0%

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

   3. *-commutative 100.0%

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

   4. mul-1-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. metadata-eval 100.0%

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

   7. sub-neg 100.0%

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

   8. div-sub 100.0%

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

   9. *-rgt-identity 100.0%

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

   10. associate-*r/ 100.0%

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

   11. rgt-mult-inverse 100.0%

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

   12. *-commutative 100.0%

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

   13. metadata-eval 100.0%

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

   14. sub-neg 100.0%

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

   15. div-sub 100.0%

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

   16. *-rgt-identity 100.0%

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

   17. associate-*r/ 99.9%

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

   18. rgt-mult-inverse 100.0%

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

8. Simplified 100.0%

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

   1. div-sub 100.0%

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

   2. sub-neg 100.0%

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

   3. *-commutative 100.0%

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

   4. div-inv 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. metadata-eval 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. +-commutative 100.0%

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

   9. sub-neg 100.0%

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

   10. associate-+l+ 100.0%

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

   11. distribute-neg-frac 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

10. Applied egg-rr 100.0%

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

12. Simplified 100.0%

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

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(-1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.6) (not (<= x 0.75)))
   (/ x (+ x (/ -1.0 x)))
   (+ (/ x (+ x 1.0)) (- -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.6) || !(x <= 0.75)) {
		tmp = x / (x + (-1.0 / x));
	} else {
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.6d0)) .or. (.not. (x <= 0.75d0))) then
        tmp = x / (x + ((-1.0d0) / x))
    else
        tmp = (x / (x + 1.0d0)) + ((-1.0d0) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.6) || !(x <= 0.75)) {
		tmp = x / (x + (-1.0 / x));
	} else {
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.6) or not (x <= 0.75):
		tmp = x / (x + (-1.0 / x))
	else:
		tmp = (x / (x + 1.0)) + (-1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.6) || !(x <= 0.75))
		tmp = Float64(x / Float64(x + Float64(-1.0 / x)));
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(-1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.6) || ~((x <= 0.75)))
		tmp = x / (x + (-1.0 / x));
	else
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.6], N[Not[LessEqual[x, 0.75]], $MachinePrecision]], N[(x / N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001 or 0.75 < 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. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-add 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. div-sub 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. rgt-mult-inverse 100.0%

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

      12. *-commutative 100.0%

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

      13. metadata-eval 100.0%

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

      14. sub-neg 100.0%

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

      15. div-sub 100.0%

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

      16. *-rgt-identity 100.0%

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

      17. associate-*r/ 99.8%

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

      18. rgt-mult-inverse 100.0%

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

   8. Simplified 100.0%

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

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. div-inv 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. metadata-eval 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-+l+ 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 98.5%

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

       1. mul-1-neg 98.5%

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

   15. Simplified 98.5%

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

   if -1.6000000000000001 < x < 0.75

   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 99.9%

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

      1. neg-mul-1 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(-1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} + t\_0\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;t\_0 + \left(-1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -1.0)
     (+ (/ 1.0 x) t_0)
     (if (<= x 0.75) (+ t_0 (- -1.0 x)) (/ x (+ x (/ -1.0 x)))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) + t_0;
	} else if (x <= 0.75) {
		tmp = t_0 + (-1.0 - x);
	} else {
		tmp = x / (x + (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-1.0d0)) then
        tmp = (1.0d0 / x) + t_0
    else if (x <= 0.75d0) then
        tmp = t_0 + ((-1.0d0) - x)
    else
        tmp = x / (x + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) + t_0;
	} else if (x <= 0.75) {
		tmp = t_0 + (-1.0 - x);
	} else {
		tmp = x / (x + (-1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) + t_0
	elif x <= 0.75:
		tmp = t_0 + (-1.0 - x)
	else:
		tmp = x / (x + (-1.0 / x))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) + t_0);
	elseif (x <= 0.75)
		tmp = Float64(t_0 + Float64(-1.0 - x));
	else
		tmp = Float64(x / Float64(x + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (1.0 / x) + t_0;
	elseif (x <= 0.75)
		tmp = t_0 + (-1.0 - x);
	else
		tmp = x / (x + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x, 0.75], N[(t$95$0 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(-1.0 / x), $MachinePrecision]), $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. +-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 98.3%

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

   if -1 < x < 0.75

   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 99.9%

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

      1. neg-mul-1 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if 0.75 < 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. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-add 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. div-sub 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. rgt-mult-inverse 100.0%

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

      12. *-commutative 100.0%

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

      13. metadata-eval 100.0%

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

      14. sub-neg 100.0%

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

      15. div-sub 100.0%

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

      16. *-rgt-identity 100.0%

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

      17. associate-*r/ 99.8%

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

      18. rgt-mult-inverse 100.0%

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

   8. Simplified 100.0%

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

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. div-inv 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. metadata-eval 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-+l+ 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 98.7%

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

       1. mul-1-neg 98.7%

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

   15. Simplified 98.7%

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

4. Final simplification 99.3%

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

Alternative 4: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.7 \lor \neg \left(x \leq 0.7\right):\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.7) (not (<= x 0.7))) (/ x (+ x (/ -1.0 x))) -1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.7) || !(x <= 0.7)) {
		tmp = x / (x + (-1.0 / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.7d0)) .or. (.not. (x <= 0.7d0))) then
        tmp = x / (x + ((-1.0d0) / x))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.7) || !(x <= 0.7)) {
		tmp = x / (x + (-1.0 / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.7) or not (x <= 0.7):
		tmp = x / (x + (-1.0 / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.7) || !(x <= 0.7))
		tmp = Float64(x / Float64(x + Float64(-1.0 / x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.7) || ~((x <= 0.7)))
		tmp = x / (x + (-1.0 / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.7], N[Not[LessEqual[x, 0.7]], $MachinePrecision]], N[(x / N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.69999999999999996 or 0.69999999999999996 < 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. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-add 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. div-sub 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. rgt-mult-inverse 100.0%

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

      12. *-commutative 100.0%

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

      13. metadata-eval 100.0%

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

      14. sub-neg 100.0%

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

      15. div-sub 100.0%

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

      16. *-rgt-identity 100.0%

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

      17. associate-*r/ 99.8%

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

      18. rgt-mult-inverse 100.0%

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

   8. Simplified 100.0%

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

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. div-inv 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. metadata-eval 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-+l+ 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 98.5%

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

       1. mul-1-neg 98.5%

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

   15. Simplified 98.5%

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

   if -0.69999999999999996 < x < 0.69999999999999996

   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 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.7 \lor \neg \left(x \leq 0.7\right):\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.9%

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

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 7: 50.9% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 54.2%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))