Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 8

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: 98.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (or (<= x -1.0) (not (<= x 1.0)))
     (+ (/ 1.0 x) t_0)
     (- (+ -1.0 t_0) x))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (1.0 / x) + t_0;
	} else {
		tmp = (-1.0 + t_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)) .or. (.not. (x <= 1.0d0))) then
        tmp = (1.0d0 / x) + t_0
    else
        tmp = ((-1.0d0) + t_0) - 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) || !(x <= 1.0)) {
		tmp = (1.0 / x) + t_0;
	} else {
		tmp = (-1.0 + t_0) - x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (1.0 / x) + t_0
	else:
		tmp = (-1.0 + t_0) - x
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(1.0 / x) + t_0);
	else
		tmp = Float64(Float64(-1.0 + t_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) || ~((x <= 1.0)))
		tmp = (1.0 / x) + t_0;
	else
		tmp = (-1.0 + t_0) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(-1.0 + t$95$0), $MachinePrecision] - x), $MachinePrecision]]]

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 99.5%

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

   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 98.7%

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

      1. neg-mul-1 98.7%

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

      2. sub-neg 98.7%

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

      3. metadata-eval 98.7%

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

      4. +-commutative 98.7%

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

      5. sub-neg 98.7%

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

   7. Simplified 98.7%

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

      1. associate-+r- 98.7%

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

      2. +-commutative 98.7%

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

   9. Applied egg-rr 98.7%

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

4. Final simplification 99.1%

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

Alternative 3: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.55 \lor \neg \left(x \leq 1.85\right):\\ \;\;\;\;\frac{1}{x} + \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.55) (not (<= x 1.85)))
   (+ (/ 1.0 x) (/ x (+ x 1.0)))
   (+ x (/ 1.0 (+ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.55) || !(x <= 1.85)) {
		tmp = (1.0 / x) + (x / (x + 1.0));
	} else {
		tmp = x + (1.0 / (-1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.55d0)) .or. (.not. (x <= 1.85d0))) then
        tmp = (1.0d0 / x) + (x / (x + 1.0d0))
    else
        tmp = x + (1.0d0 / ((-1.0d0) + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.55) || !(x <= 1.85)) {
		tmp = (1.0 / x) + (x / (x + 1.0));
	} else {
		tmp = x + (1.0 / (-1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.55) or not (x <= 1.85):
		tmp = (1.0 / x) + (x / (x + 1.0))
	else:
		tmp = x + (1.0 / (-1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.55) || !(x <= 1.85))
		tmp = Float64(Float64(1.0 / x) + Float64(x / Float64(x + 1.0)));
	else
		tmp = Float64(x + Float64(1.0 / Float64(-1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.55) || ~((x <= 1.85)))
		tmp = (1.0 / x) + (x / (x + 1.0));
	else
		tmp = x + (1.0 / (-1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.55], N[Not[LessEqual[x, 1.85]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.55000000000000004 or 1.8500000000000001 < 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 99.5%

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

   if -0.55000000000000004 < x < 1.8500000000000001

   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 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.55 \lor \neg \left(x \leq 1.85\right):\\ \;\;\;\;\frac{1}{x} + \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.8999999999999999 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. 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 99.5%

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

   if -1 < x < 1.8999999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. 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 98.7%

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

4. Final simplification 99.1%

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

Alternative 5: 98.8% 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 99.5%

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

   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 98.7%

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

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 7: 49.6% 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 54.7%

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

Alternative 8: 10.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

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. clear-num 100.0%

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

   2. associate-/r/ 99.9%

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

   3. +-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

8. Applied egg-rr 11.0%

   \[\leadsto \color{blue}{-2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))