Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{x}\\ \frac{1 + \frac{t\_0}{x + -1}}{t\_0} \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = 1.0 + (1.0 / x);
	return (1.0 + (t_0 / (x + -1.0))) / t_0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 + (1.0d0 / x)
    code = (1.0d0 + (t_0 / (x + (-1.0d0)))) / t_0
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (1.0 / x);
	return (1.0 + (t_0 / (x + -1.0))) / t_0;
}

Python

def code(x):
	t_0 = 1.0 + (1.0 / x)
	return (1.0 + (t_0 / (x + -1.0))) / t_0

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(1.0 / x))
	return Float64(Float64(1.0 + Float64(t_0 / Float64(x + -1.0))) / t_0)
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 + (1.0 / x);
	tmp = (1.0 + (t_0 / (x + -1.0))) / t_0;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(t$95$0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $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. +-commutative 100.0%

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

   2. frac-2neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. clear-num 100.0%

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

   5. frac-add 100.0%

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

   6. fma-define 100.0%

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

   7. +-commutative 100.0%

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

   8. distribute-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. metadata-eval 100.0%

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

   14. sub-neg 100.0%

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

6. Applied egg-rr 100.0%

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

   1. fma-undefine 100.0%

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

   2. *-commutative 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. +-commutative 100.0%

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

   5. associate-+l- 100.0%

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

   6. sub-neg 100.0%

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

   7. *-commutative 100.0%

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

   8. mul-1-neg 100.0%

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

   9. remove-double-neg 100.0%

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

   10. +-commutative 100.0%

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

   11. metadata-eval 100.0%

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

   12. sub-neg 100.0%

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

   13. div-sub 100.0%

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

   14. *-rgt-identity 100.0%

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

   15. associate-*r/ 100.0%

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

   16. rgt-mult-inverse 100.0%

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

   17. +-commutative 100.0%

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

   18. metadata-eval 100.0%

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

   19. sub-neg 100.0%

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

   20. div-sub 100.0%

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

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. associate--r+ 100.0%

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

   2. div-sub 100.0%

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

   3. div-inv 100.0%

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

   4. cancel-sign-sub-inv 100.0%

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

   5. metadata-eval 100.0%

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

   6. *-un-lft-identity 100.0%

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

   7. div-inv 100.0%

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

   8. cancel-sign-sub-inv 100.0%

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

   9. metadata-eval 100.0%

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

   10. *-un-lft-identity 100.0%

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

   11. div-inv 100.0%

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

   12. cancel-sign-sub-inv 100.0%

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

   13. metadata-eval 100.0%

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

   14. *-un-lft-identity 100.0%

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

10. Applied egg-rr 100.0%

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

    1. associate-/r* 100.0%

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

    2. *-inverses 100.0%

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

    3. associate-/r* 100.0%

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

    4. sub-div 100.0%

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

12. Applied egg-rr 100.0%

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

13. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 + x)
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (1.0d0 / x) + t_0
    else
        tmp = t_0 + ((-1.0d0) - x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 + N[(-1.0 - x), $MachinePrecision]), $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 97.8%

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

   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 98.4%

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

      1. sub-neg 98.4%

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

      2. metadata-eval 98.4%

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

      3. neg-mul-1 98.4%

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

      4. +-commutative 98.4%

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

      5. unsub-neg 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 98.2%

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

Alternative 3: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.8999999999999999 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 98.5%

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

   if -1.8999999999999999 < 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 97.9%

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

      1. sub-neg 97.9%

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

      2. metadata-eval 97.9%

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

      3. neg-mul-1 97.9%

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

      4. +-commutative 97.9%

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

      5. unsub-neg 97.9%

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

   7. Simplified 97.9%

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

Alternative 4: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.92:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.92) (+ x (/ 1.0 (+ x -1.0))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.92) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.92d0) then
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.92) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.92:
		tmp = x + (1.0 / (x + -1.0))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.92)
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.92)
		tmp = x + (1.0 / (x + -1.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.92], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.9199999999999999 < 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 97.7%

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

   if -1 < x < 1.9199999999999999

   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 98.4%

      \[\leadsto \color{blue}{x} + \frac{1}{x + -1} \]
3. Recombined 2 regimes into one program.
4. 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 97.7%

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

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. clear-num 100.0%

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

      5. frac-add 100.0%

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

      6. fma-define 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. *-commutative 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+l- 100.0%

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

      6. sub-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. mul-1-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. div-sub 100.0%

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

      14. *-rgt-identity 100.0%

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

      15. associate-*r/ 100.0%

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

      16. rgt-mult-inverse 100.0%

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

      17. +-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. sub-neg 100.0%

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

      20. div-sub 100.0%

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

   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. Taylor expanded in x around 0 98.4%

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

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 7: 49.7% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. +-commutative 100.0%

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

   2. frac-2neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. clear-num 100.0%

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

   5. frac-add 100.0%

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

   6. fma-define 100.0%

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

   7. +-commutative 100.0%

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

   8. distribute-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. metadata-eval 100.0%

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

   14. sub-neg 100.0%

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

6. Applied egg-rr 100.0%

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

   1. fma-undefine 100.0%

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

   2. *-commutative 100.0%

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

   3. *-rgt-identity 100.0%

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

   4. +-commutative 100.0%

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

   5. associate-+l- 100.0%

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

   6. sub-neg 100.0%

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

   7. *-commutative 100.0%

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

   8. mul-1-neg 100.0%

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

   9. remove-double-neg 100.0%

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

   10. +-commutative 100.0%

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

   11. metadata-eval 100.0%

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

   12. sub-neg 100.0%

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

   13. div-sub 100.0%

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

   14. *-rgt-identity 100.0%

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

   15. associate-*r/ 100.0%

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

   16. rgt-mult-inverse 100.0%

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

   17. +-commutative 100.0%

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

   18. metadata-eval 100.0%

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

   19. sub-neg 100.0%

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

   20. div-sub 100.0%

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

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. Taylor expanded in x around 0 55.6%

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

Reproduce

herbie shell --seed 2024121 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))