Asymptote C

Percentage Accurate: 54.4% → 99.8%

Time: 12.9s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{x + 1}{1 - x}\\ \mathbf{if}\;t\_0 \leq 0.0005:\\ \;\;\;\;\frac{\left(\frac{2}{x} + \frac{2}{{x}^{3}}\right) - \left(3 + \frac{2}{{x}^{2}}\right)}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (+ x 1.0) (- 1.0 x)))))
   (if (<= t_0 0.0005)
     (/
      (- (+ (/ 2.0 x) (/ 2.0 (pow x 3.0))) (+ 3.0 (/ 2.0 (pow x 2.0))))
      (+ x -1.0))
     t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x));
	double tmp;
	if (t_0 <= 0.0005) {
		tmp = (((2.0 / x) + (2.0 / pow(x, 3.0))) - (3.0 + (2.0 / pow(x, 2.0)))) / (x + -1.0);
	} else {
		tmp = t_0;
	}
	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)) + ((x + 1.0d0) / (1.0d0 - x))
    if (t_0 <= 0.0005d0) then
        tmp = (((2.0d0 / x) + (2.0d0 / (x ** 3.0d0))) - (3.0d0 + (2.0d0 / (x ** 2.0d0)))) / (x + (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x));
	double tmp;
	if (t_0 <= 0.0005) {
		tmp = (((2.0 / x) + (2.0 / Math.pow(x, 3.0))) - (3.0 + (2.0 / Math.pow(x, 2.0)))) / (x + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))
	tmp = 0
	if t_0 <= 0.0005:
		tmp = (((2.0 / x) + (2.0 / math.pow(x, 3.0))) - (3.0 + (2.0 / math.pow(x, 2.0)))) / (x + -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(x + 1.0) / Float64(1.0 - x)))
	tmp = 0.0
	if (t_0 <= 0.0005)
		tmp = Float64(Float64(Float64(Float64(2.0 / x) + Float64(2.0 / (x ^ 3.0))) - Float64(3.0 + Float64(2.0 / (x ^ 2.0)))) / Float64(x + -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0005)
		tmp = (((2.0 / x) + (2.0 / (x ^ 3.0))) - (3.0 + (2.0 / (x ^ 2.0)))) / (x + -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0005], N[(N[(N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(3.0 + N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.0000000000000001e-4

   1. Initial program 8.0%

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

      1. remove-double-neg 8.0%

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

      2. distribute-neg-frac 8.0%

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

      3. distribute-neg-in 8.0%

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

      4. sub-neg 8.0%

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

      5. distribute-frac-neg2 8.0%

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

      6. sub-neg 8.0%

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

      7. +-commutative 8.0%

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

      8. unsub-neg 8.0%

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

      9. metadata-eval 8.0%

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

      10. neg-sub0 8.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 8.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 8.0%

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

      13. +-commutative 8.0%

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

      14. unsub-neg 8.0%

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

   3. Simplified 8.0%

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

      1. clear-num 8.0%

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

      2. frac-sub 9.4%

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

      3. *-commutative 9.4%

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

      4. *-un-lft-identity 9.4%

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

      5. fma-neg 8.1%

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

      6. +-commutative 8.1%

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

      7. distribute-neg-in 8.1%

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

      8. metadata-eval 8.1%

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

      9. sub-neg 8.1%

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

   6. Applied egg-rr 8.1%

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

      1. associate-*r/ 8.1%

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

      2. *-commutative 8.1%

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

      3. associate-/l* 8.1%

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

      4. +-commutative 8.1%

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

      5. metadata-eval 8.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{\left(0 - -1\right)} + x}{-1 - x}} \]

      6. associate--r- 8.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{0 - \left(-1 - x\right)}}{-1 - x}} \]

      7. div-sub 8.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \color{blue}{\left(\frac{0}{-1 - x} - \frac{-1 - x}{-1 - x}\right)}} \]

      8. div0 8.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(\color{blue}{0} - \frac{-1 - x}{-1 - x}\right)} \]

      9. *-inverses 8.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(0 - \color{blue}{1}\right)} \]

      10. metadata-eval 8.1%

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

   8. Simplified 8.1%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. associate-*r/ 100.0%

          \[\leadsto \frac{\left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{{x}^{3}}\right) - \left(3 + 2 \cdot \frac{1}{{x}^{2}}\right)}{\left(1 - x\right) \cdot -1} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{\left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{{x}^{3}}\right) - \left(3 + 2 \cdot \frac{1}{{x}^{2}}\right)}{\left(1 - x\right) \cdot -1} \]

       3. associate-*r/ 100.0%

          \[\leadsto \frac{\left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}}\right) - \left(3 + 2 \cdot \frac{1}{{x}^{2}}\right)}{\left(1 - x\right) \cdot -1} \]

       4. metadata-eval 100.0%

          \[\leadsto \frac{\left(\frac{2}{x} + \frac{\color{blue}{2}}{{x}^{3}}\right) - \left(3 + 2 \cdot \frac{1}{{x}^{2}}\right)}{\left(1 - x\right) \cdot -1} \]

       5. associate-*r/ 100.0%

          \[\leadsto \frac{\left(\frac{2}{x} + \frac{2}{{x}^{3}}\right) - \left(3 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}\right)}{\left(1 - x\right) \cdot -1} \]

       6. metadata-eval 100.0%

          \[\leadsto \frac{\left(\frac{2}{x} + \frac{2}{{x}^{3}}\right) - \left(3 + \frac{\color{blue}{2}}{{x}^{2}}\right)}{\left(1 - x\right) \cdot -1} \]

   11. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\left(\frac{2}{x} + \frac{2}{{x}^{3}}\right) - \left(3 + \frac{2}{{x}^{2}}\right)}}{\left(1 - x\right) \cdot -1} \]

   if 5.0000000000000001e-4 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 0.0005:\\ \;\;\;\;\frac{\left(\frac{2}{x} + \frac{2}{{x}^{3}}\right) - \left(3 + \frac{2}{{x}^{2}}\right)}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{x + 1}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t\_0 + \frac{x + 1}{1 - x} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(x + 1\right) \cdot \frac{1}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (+ x 1.0) (- 1.0 x))) 5e-15)
     (+ (- (/ -3.0 x) (pow x -2.0)) (/ -3.0 (pow x 3.0)))
     (+ t_0 (* (+ x 1.0) (/ 1.0 (- 1.0 x)))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15) {
		tmp = ((-3.0 / x) - pow(x, -2.0)) + (-3.0 / pow(x, 3.0));
	} else {
		tmp = t_0 + ((x + 1.0) * (1.0 / (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 / (x + 1.0d0)
    if ((t_0 + ((x + 1.0d0) / (1.0d0 - x))) <= 5d-15) then
        tmp = (((-3.0d0) / x) - (x ** (-2.0d0))) + ((-3.0d0) / (x ** 3.0d0))
    else
        tmp = t_0 + ((x + 1.0d0) * (1.0d0 / (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 ((t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15) {
		tmp = ((-3.0 / x) - Math.pow(x, -2.0)) + (-3.0 / Math.pow(x, 3.0));
	} else {
		tmp = t_0 + ((x + 1.0) * (1.0 / (1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15:
		tmp = ((-3.0 / x) - math.pow(x, -2.0)) + (-3.0 / math.pow(x, 3.0))
	else:
		tmp = t_0 + ((x + 1.0) * (1.0 / (1.0 - x)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(x + 1.0) / Float64(1.0 - x))) <= 5e-15)
		tmp = Float64(Float64(Float64(-3.0 / x) - (x ^ -2.0)) + Float64(-3.0 / (x ^ 3.0)));
	else
		tmp = Float64(t_0 + Float64(Float64(x + 1.0) * Float64(1.0 / Float64(1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15)
		tmp = ((-3.0 / x) - (x ^ -2.0)) + (-3.0 / (x ^ 3.0));
	else
		tmp = t_0 + ((x + 1.0) * (1.0 / (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[N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-15], N[(N[(N[(-3.0 / x), $MachinePrecision] - N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] + N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 4.99999999999999999e-15

   1. Initial program 7.3%

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

      1. remove-double-neg 7.3%

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

      2. distribute-neg-frac 7.3%

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

      3. distribute-neg-in 7.3%

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

      4. sub-neg 7.3%

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

      5. distribute-frac-neg2 7.3%

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

      6. sub-neg 7.3%

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

      7. +-commutative 7.3%

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

      8. unsub-neg 7.3%

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

      9. metadata-eval 7.3%

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

      10. neg-sub0 7.3%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 7.3%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 7.3%

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

      13. +-commutative 7.3%

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

      14. unsub-neg 7.3%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

         \[\leadsto -\color{blue}{\left(\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right) + 3 \cdot \frac{1}{{x}^{3}}\right)} \]

      2. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(-\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right)} \]

      3. distribute-neg-in 99.5%

         \[\leadsto \color{blue}{\left(\left(-3 \cdot \frac{1}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)\right)} + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]

      4. associate-*r/ 99.9%

         \[\leadsto \left(\left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) + \left(-\frac{1}{{x}^{2}}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]

      5. metadata-eval 99.9%

         \[\leadsto \left(\left(-\frac{\color{blue}{3}}{x}\right) + \left(-\frac{1}{{x}^{2}}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]

      6. distribute-neg-frac 99.9%

         \[\leadsto \left(\color{blue}{\frac{-3}{x}} + \left(-\frac{1}{{x}^{2}}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]

      7. metadata-eval 99.9%

         \[\leadsto \left(\frac{\color{blue}{-3}}{x} + \left(-\frac{1}{{x}^{2}}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]

      8. distribute-neg-frac 99.9%

         \[\leadsto \left(\frac{-3}{x} + \color{blue}{\frac{-1}{{x}^{2}}}\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]

      9. metadata-eval 99.9%

         \[\leadsto \left(\frac{-3}{x} + \frac{\color{blue}{-1}}{{x}^{2}}\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]

      10. associate-*r/ 99.9%

          \[\leadsto \left(\frac{-3}{x} + \frac{-1}{{x}^{2}}\right) + \left(-\color{blue}{\frac{3 \cdot 1}{{x}^{3}}}\right) \]

      11. metadata-eval 99.9%

          \[\leadsto \left(\frac{-3}{x} + \frac{-1}{{x}^{2}}\right) + \left(-\frac{\color{blue}{3}}{{x}^{3}}\right) \]

      12. distribute-neg-frac 99.9%

          \[\leadsto \left(\frac{-3}{x} + \frac{-1}{{x}^{2}}\right) + \color{blue}{\frac{-3}{{x}^{3}}} \]

      13. metadata-eval 99.9%

          \[\leadsto \left(\frac{-3}{x} + \frac{-1}{{x}^{2}}\right) + \frac{\color{blue}{-3}}{{x}^{3}} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\frac{-3}{x} + \frac{-1}{{x}^{2}}\right) + \frac{-3}{{x}^{3}}} \]
   8. Step-by-step derivation

      1. div-inv 99.9%

         \[\leadsto \left(\frac{-3}{x} + \color{blue}{-1 \cdot \frac{1}{{x}^{2}}}\right) + \frac{-3}{{x}^{3}} \]

      2. pow-flip 99.9%

         \[\leadsto \left(\frac{-3}{x} + -1 \cdot \color{blue}{{x}^{\left(-2\right)}}\right) + \frac{-3}{{x}^{3}} \]

      3. metadata-eval 99.9%

         \[\leadsto \left(\frac{-3}{x} + -1 \cdot {x}^{\color{blue}{-2}}\right) + \frac{-3}{{x}^{3}} \]

   9. Applied egg-rr 99.9%

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

       1. neg-mul-1 99.9%

          \[\leadsto \left(\frac{-3}{x} + \color{blue}{\left(-{x}^{-2}\right)}\right) + \frac{-3}{{x}^{3}} \]

   11. Simplified 99.9%

       \[\leadsto \left(\frac{-3}{x} + \color{blue}{\left(-{x}^{-2}\right)}\right) + \frac{-3}{{x}^{3}} \]
   12. Step-by-step derivation

       1. unsub-neg 99.9%

          \[\leadsto \color{blue}{\left(\frac{-3}{x} - {x}^{-2}\right)} + \frac{-3}{{x}^{3}} \]

   13. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\left(\frac{-3}{x} - {x}^{-2}\right)} + \frac{-3}{{x}^{3}} \]

   if 4.99999999999999999e-15 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-neg-frac 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. unsub-neg 99.8%

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

      9. metadata-eval 99.8%

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

      10. neg-sub0 99.8%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 99.8%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 99.8%

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

      13. +-commutative 99.8%

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

      14. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{1}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t\_0 + \frac{x + 1}{1 - x} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2}{x} - \left(3 + \frac{2}{{x}^{2}}\right)}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(x + 1\right) \cdot \frac{1}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (+ x 1.0) (- 1.0 x))) 5e-15)
     (/ (- (/ 2.0 x) (+ 3.0 (/ 2.0 (pow x 2.0)))) (+ x -1.0))
     (+ t_0 (* (+ x 1.0) (/ 1.0 (- 1.0 x)))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15) {
		tmp = ((2.0 / x) - (3.0 + (2.0 / pow(x, 2.0)))) / (x + -1.0);
	} else {
		tmp = t_0 + ((x + 1.0) * (1.0 / (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 / (x + 1.0d0)
    if ((t_0 + ((x + 1.0d0) / (1.0d0 - x))) <= 5d-15) then
        tmp = ((2.0d0 / x) - (3.0d0 + (2.0d0 / (x ** 2.0d0)))) / (x + (-1.0d0))
    else
        tmp = t_0 + ((x + 1.0d0) * (1.0d0 / (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 ((t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15) {
		tmp = ((2.0 / x) - (3.0 + (2.0 / Math.pow(x, 2.0)))) / (x + -1.0);
	} else {
		tmp = t_0 + ((x + 1.0) * (1.0 / (1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15:
		tmp = ((2.0 / x) - (3.0 + (2.0 / math.pow(x, 2.0)))) / (x + -1.0)
	else:
		tmp = t_0 + ((x + 1.0) * (1.0 / (1.0 - x)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(x + 1.0) / Float64(1.0 - x))) <= 5e-15)
		tmp = Float64(Float64(Float64(2.0 / x) - Float64(3.0 + Float64(2.0 / (x ^ 2.0)))) / Float64(x + -1.0));
	else
		tmp = Float64(t_0 + Float64(Float64(x + 1.0) * Float64(1.0 / Float64(1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15)
		tmp = ((2.0 / x) - (3.0 + (2.0 / (x ^ 2.0)))) / (x + -1.0);
	else
		tmp = t_0 + ((x + 1.0) * (1.0 / (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[N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-15], N[(N[(N[(2.0 / x), $MachinePrecision] - N[(3.0 + N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 4.99999999999999999e-15

   1. Initial program 7.3%

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

      1. remove-double-neg 7.3%

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

      2. distribute-neg-frac 7.3%

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

      3. distribute-neg-in 7.3%

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

      4. sub-neg 7.3%

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

      5. distribute-frac-neg2 7.3%

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

      6. sub-neg 7.3%

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

      7. +-commutative 7.3%

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

      8. unsub-neg 7.3%

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

      9. metadata-eval 7.3%

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

      10. neg-sub0 7.3%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 7.3%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 7.3%

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

      13. +-commutative 7.3%

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

      14. unsub-neg 7.3%

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

   3. Simplified 7.3%

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

      1. clear-num 7.4%

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

      2. frac-sub 8.8%

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

      3. *-commutative 8.8%

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

      4. *-un-lft-identity 8.8%

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

      5. fma-neg 7.5%

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

      6. +-commutative 7.5%

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

      7. distribute-neg-in 7.5%

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

      8. metadata-eval 7.5%

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

      9. sub-neg 7.5%

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

   6. Applied egg-rr 7.5%

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

      1. associate-*r/ 7.5%

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

      2. *-commutative 7.5%

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

      3. associate-/l* 7.5%

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

      4. +-commutative 7.5%

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

      5. metadata-eval 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{\left(0 - -1\right)} + x}{-1 - x}} \]

      6. associate--r- 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{0 - \left(-1 - x\right)}}{-1 - x}} \]

      7. div-sub 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \color{blue}{\left(\frac{0}{-1 - x} - \frac{-1 - x}{-1 - x}\right)}} \]

      8. div0 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(\color{blue}{0} - \frac{-1 - x}{-1 - x}\right)} \]

      9. *-inverses 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(0 - \color{blue}{1}\right)} \]

      10. metadata-eval 7.5%

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

   8. Simplified 7.5%

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

   9. Taylor expanded in x around inf 99.9%

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

       1. associate-*r/ 99.9%

          \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{x}} - \left(3 + 2 \cdot \frac{1}{{x}^{2}}\right)}{\left(1 - x\right) \cdot -1} \]

       2. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\color{blue}{2}}{x} - \left(3 + 2 \cdot \frac{1}{{x}^{2}}\right)}{\left(1 - x\right) \cdot -1} \]

       3. associate-*r/ 99.9%

          \[\leadsto \frac{\frac{2}{x} - \left(3 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}\right)}{\left(1 - x\right) \cdot -1} \]

       4. metadata-eval 99.9%

          \[\leadsto \frac{\frac{2}{x} - \left(3 + \frac{\color{blue}{2}}{{x}^{2}}\right)}{\left(1 - x\right) \cdot -1} \]

   11. Simplified 99.9%

       \[\leadsto \frac{\color{blue}{\frac{2}{x} - \left(3 + \frac{2}{{x}^{2}}\right)}}{\left(1 - x\right) \cdot -1} \]

   if 4.99999999999999999e-15 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-neg-frac 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. unsub-neg 99.8%

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

      9. metadata-eval 99.8%

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

      10. neg-sub0 99.8%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 99.8%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 99.8%

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

      13. +-commutative 99.8%

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

      14. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2}{x} - \left(3 + \frac{2}{{x}^{2}}\right)}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{1}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t\_0 + \frac{x + 1}{1 - x} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2}{x} + -3}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(x + 1\right) \cdot \frac{1}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (+ x 1.0) (- 1.0 x))) 5e-15)
     (/ (+ (/ 2.0 x) -3.0) (+ x -1.0))
     (+ t_0 (* (+ x 1.0) (/ 1.0 (- 1.0 x)))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15) {
		tmp = ((2.0 / x) + -3.0) / (x + -1.0);
	} else {
		tmp = t_0 + ((x + 1.0) * (1.0 / (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 / (x + 1.0d0)
    if ((t_0 + ((x + 1.0d0) / (1.0d0 - x))) <= 5d-15) then
        tmp = ((2.0d0 / x) + (-3.0d0)) / (x + (-1.0d0))
    else
        tmp = t_0 + ((x + 1.0d0) * (1.0d0 / (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 ((t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15) {
		tmp = ((2.0 / x) + -3.0) / (x + -1.0);
	} else {
		tmp = t_0 + ((x + 1.0) * (1.0 / (1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15:
		tmp = ((2.0 / x) + -3.0) / (x + -1.0)
	else:
		tmp = t_0 + ((x + 1.0) * (1.0 / (1.0 - x)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(x + 1.0) / Float64(1.0 - x))) <= 5e-15)
		tmp = Float64(Float64(Float64(2.0 / x) + -3.0) / Float64(x + -1.0));
	else
		tmp = Float64(t_0 + Float64(Float64(x + 1.0) * Float64(1.0 / Float64(1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 + ((x + 1.0) / (1.0 - x))) <= 5e-15)
		tmp = ((2.0 / x) + -3.0) / (x + -1.0);
	else
		tmp = t_0 + ((x + 1.0) * (1.0 / (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[N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-15], N[(N[(N[(2.0 / x), $MachinePrecision] + -3.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 4.99999999999999999e-15

   1. Initial program 7.3%

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

      1. remove-double-neg 7.3%

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

      2. distribute-neg-frac 7.3%

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

      3. distribute-neg-in 7.3%

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

      4. sub-neg 7.3%

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

      5. distribute-frac-neg2 7.3%

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

      6. sub-neg 7.3%

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

      7. +-commutative 7.3%

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

      8. unsub-neg 7.3%

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

      9. metadata-eval 7.3%

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

      10. neg-sub0 7.3%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 7.3%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 7.3%

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

      13. +-commutative 7.3%

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

      14. unsub-neg 7.3%

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

   3. Simplified 7.3%

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

      1. clear-num 7.4%

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

      2. frac-sub 8.8%

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

      3. *-commutative 8.8%

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

      4. *-un-lft-identity 8.8%

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

      5. fma-neg 7.5%

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

      6. +-commutative 7.5%

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

      7. distribute-neg-in 7.5%

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

      8. metadata-eval 7.5%

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

      9. sub-neg 7.5%

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

   6. Applied egg-rr 7.5%

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

      1. associate-*r/ 7.5%

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

      2. *-commutative 7.5%

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

      3. associate-/l* 7.5%

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

      4. +-commutative 7.5%

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

      5. metadata-eval 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{\left(0 - -1\right)} + x}{-1 - x}} \]

      6. associate--r- 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{0 - \left(-1 - x\right)}}{-1 - x}} \]

      7. div-sub 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \color{blue}{\left(\frac{0}{-1 - x} - \frac{-1 - x}{-1 - x}\right)}} \]

      8. div0 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(\color{blue}{0} - \frac{-1 - x}{-1 - x}\right)} \]

      9. *-inverses 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(0 - \color{blue}{1}\right)} \]

      10. metadata-eval 7.5%

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

   8. Simplified 7.5%

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

   9. Taylor expanded in x around inf 99.7%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} - 3}}{\left(1 - x\right) \cdot -1} \]
   10. Step-by-step derivation

       1. sub-neg 99.7%

          \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} + \left(-3\right)}}{\left(1 - x\right) \cdot -1} \]

       2. associate-*r/ 99.7%

          \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{x}} + \left(-3\right)}{\left(1 - x\right) \cdot -1} \]

       3. metadata-eval 99.7%

          \[\leadsto \frac{\frac{\color{blue}{2}}{x} + \left(-3\right)}{\left(1 - x\right) \cdot -1} \]

       4. metadata-eval 99.7%

          \[\leadsto \frac{\frac{2}{x} + \color{blue}{-3}}{\left(1 - x\right) \cdot -1} \]

   11. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\frac{2}{x} + -3}}{\left(1 - x\right) \cdot -1} \]

   if 4.99999999999999999e-15 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-neg-frac 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. unsub-neg 99.8%

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

      9. metadata-eval 99.8%

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

      10. neg-sub0 99.8%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 99.8%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 99.8%

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

      13. +-commutative 99.8%

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

      14. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2}{x} + -3}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{1}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{x + 1}{1 - x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2}{x} + -3}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (+ x 1.0) (- 1.0 x)))))
   (if (<= t_0 5e-15) (/ (+ (/ 2.0 x) -3.0) (+ x -1.0)) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x));
	double tmp;
	if (t_0 <= 5e-15) {
		tmp = ((2.0 / x) + -3.0) / (x + -1.0);
	} else {
		tmp = t_0;
	}
	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)) + ((x + 1.0d0) / (1.0d0 - x))
    if (t_0 <= 5d-15) then
        tmp = ((2.0d0 / x) + (-3.0d0)) / (x + (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x));
	double tmp;
	if (t_0 <= 5e-15) {
		tmp = ((2.0 / x) + -3.0) / (x + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))
	tmp = 0
	if t_0 <= 5e-15:
		tmp = ((2.0 / x) + -3.0) / (x + -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(x + 1.0) / Float64(1.0 - x)))
	tmp = 0.0
	if (t_0 <= 5e-15)
		tmp = Float64(Float64(Float64(2.0 / x) + -3.0) / Float64(x + -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((x + 1.0) / (1.0 - x));
	tmp = 0.0;
	if (t_0 <= 5e-15)
		tmp = ((2.0 / x) + -3.0) / (x + -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-15], N[(N[(N[(2.0 / x), $MachinePrecision] + -3.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 4.99999999999999999e-15

   1. Initial program 7.3%

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

      1. remove-double-neg 7.3%

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

      2. distribute-neg-frac 7.3%

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

      3. distribute-neg-in 7.3%

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

      4. sub-neg 7.3%

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

      5. distribute-frac-neg2 7.3%

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

      6. sub-neg 7.3%

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

      7. +-commutative 7.3%

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

      8. unsub-neg 7.3%

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

      9. metadata-eval 7.3%

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

      10. neg-sub0 7.3%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 7.3%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 7.3%

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

      13. +-commutative 7.3%

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

      14. unsub-neg 7.3%

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

   3. Simplified 7.3%

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

      1. clear-num 7.4%

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

      2. frac-sub 8.8%

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

      3. *-commutative 8.8%

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

      4. *-un-lft-identity 8.8%

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

      5. fma-neg 7.5%

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

      6. +-commutative 7.5%

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

      7. distribute-neg-in 7.5%

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

      8. metadata-eval 7.5%

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

      9. sub-neg 7.5%

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

   6. Applied egg-rr 7.5%

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

      1. associate-*r/ 7.5%

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

      2. *-commutative 7.5%

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

      3. associate-/l* 7.5%

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

      4. +-commutative 7.5%

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

      5. metadata-eval 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{\left(0 - -1\right)} + x}{-1 - x}} \]

      6. associate--r- 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{0 - \left(-1 - x\right)}}{-1 - x}} \]

      7. div-sub 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \color{blue}{\left(\frac{0}{-1 - x} - \frac{-1 - x}{-1 - x}\right)}} \]

      8. div0 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(\color{blue}{0} - \frac{-1 - x}{-1 - x}\right)} \]

      9. *-inverses 7.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(0 - \color{blue}{1}\right)} \]

      10. metadata-eval 7.5%

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

   8. Simplified 7.5%

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

   9. Taylor expanded in x around inf 99.7%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} - 3}}{\left(1 - x\right) \cdot -1} \]
   10. Step-by-step derivation

       1. sub-neg 99.7%

          \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} + \left(-3\right)}}{\left(1 - x\right) \cdot -1} \]

       2. associate-*r/ 99.7%

          \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{x}} + \left(-3\right)}{\left(1 - x\right) \cdot -1} \]

       3. metadata-eval 99.7%

          \[\leadsto \frac{\frac{\color{blue}{2}}{x} + \left(-3\right)}{\left(1 - x\right) \cdot -1} \]

       4. metadata-eval 99.7%

          \[\leadsto \frac{\frac{2}{x} + \color{blue}{-3}}{\left(1 - x\right) \cdot -1} \]

   11. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\frac{2}{x} + -3}}{\left(1 - x\right) \cdot -1} \]

   if 4.99999999999999999e-15 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.8%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2}{x} + -3}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{x + 1}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + -1}{\frac{2}{x} - 3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ -3.0 x)
   (if (<= x 0.85)
     (+ 1.0 (* x (+ x 3.0)))
     (/ 1.0 (/ (+ x -1.0) (- (/ 2.0 x) 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 0.85) {
		tmp = 1.0 + (x * (x + 3.0));
	} else {
		tmp = 1.0 / ((x + -1.0) / ((2.0 / x) - 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 0.85d0) then
        tmp = 1.0d0 + (x * (x + 3.0d0))
    else
        tmp = 1.0d0 / ((x + (-1.0d0)) / ((2.0d0 / x) - 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 0.85) {
		tmp = 1.0 + (x * (x + 3.0));
	} else {
		tmp = 1.0 / ((x + -1.0) / ((2.0 / x) - 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 0.85:
		tmp = 1.0 + (x * (x + 3.0))
	else:
		tmp = 1.0 / ((x + -1.0) / ((2.0 / x) - 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-3.0 / x);
	elseif (x <= 0.85)
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(x + -1.0) / Float64(Float64(2.0 / x) - 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -3.0 / x;
	elseif (x <= 0.85)
		tmp = 1.0 + (x * (x + 3.0));
	else
		tmp = 1.0 / ((x + -1.0) / ((2.0 / x) - 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 0.85], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x + -1.0), $MachinePrecision] / N[(N[(2.0 / x), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 7.0%

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

      1. remove-double-neg 7.0%

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

      2. distribute-neg-frac 7.0%

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

      3. distribute-neg-in 7.0%

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

      4. sub-neg 7.0%

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

      5. distribute-frac-neg2 7.0%

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

      6. sub-neg 7.0%

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

      7. +-commutative 7.0%

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

      8. unsub-neg 7.0%

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

      9. metadata-eval 7.0%

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

      10. neg-sub0 7.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 7.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 7.0%

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

      13. +-commutative 7.0%

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

      14. unsub-neg 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 0.849999999999999978

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.6%

      \[\leadsto \color{blue}{1 + \left(3 \cdot x + {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 98.6%

         \[\leadsto 1 + \left(3 \cdot x + \color{blue}{x \cdot x}\right) \]

      2. distribute-rgt-out 98.6%

         \[\leadsto 1 + \color{blue}{x \cdot \left(3 + x\right)} \]

   7. Simplified 98.6%

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x\right)} \]

   if 0.849999999999999978 < x

   1. Initial program 9.0%

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

      1. remove-double-neg 9.0%

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

      2. distribute-neg-frac 9.0%

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

      3. distribute-neg-in 9.0%

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

      4. sub-neg 9.0%

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

      5. distribute-frac-neg2 9.0%

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

      6. sub-neg 9.0%

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

      7. +-commutative 9.0%

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

      8. unsub-neg 9.0%

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

      9. metadata-eval 9.0%

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

      10. neg-sub0 9.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 9.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 9.0%

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

      13. +-commutative 9.0%

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

      14. unsub-neg 9.0%

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

   3. Simplified 9.0%

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

      1. clear-num 9.0%

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

      2. frac-sub 10.8%

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

      3. *-commutative 10.8%

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

      4. *-un-lft-identity 10.8%

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

      5. fma-neg 9.3%

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

      6. +-commutative 9.3%

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

      7. distribute-neg-in 9.3%

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

      8. metadata-eval 9.3%

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

      9. sub-neg 9.3%

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

   6. Applied egg-rr 9.3%

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

      1. associate-*r/ 9.3%

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

      2. *-commutative 9.3%

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

      3. associate-/l* 9.3%

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

      4. +-commutative 9.3%

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

      5. metadata-eval 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{\left(0 - -1\right)} + x}{-1 - x}} \]

      6. associate--r- 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{0 - \left(-1 - x\right)}}{-1 - x}} \]

      7. div-sub 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \color{blue}{\left(\frac{0}{-1 - x} - \frac{-1 - x}{-1 - x}\right)}} \]

      8. div0 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(\color{blue}{0} - \frac{-1 - x}{-1 - x}\right)} \]

      9. *-inverses 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(0 - \color{blue}{1}\right)} \]

      10. metadata-eval 9.3%

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

   8. Simplified 9.3%

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

      1. clear-num 9.3%

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

      2. inv-pow 9.3%

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

      3. *-commutative 9.3%

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

      4. *-un-lft-identity 9.3%

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

      5. times-frac 9.3%

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

      6. metadata-eval 9.3%

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

   10. Applied egg-rr 9.3%

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

       1. unpow-1 9.3%

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

       2. mul-1-neg 9.3%

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

       3. distribute-neg-frac2 9.3%

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

       4. fma-define 10.8%

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

       5. *-lft-identity 10.8%

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

       6. associate-*l/ 9.9%

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

       7. distribute-neg-out 9.9%

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

       8. distribute-lft-neg-out 9.9%

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

       9. associate-*r* 9.7%

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

       10. sub-neg 9.7%

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

       11. metadata-eval 9.7%

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

       12. distribute-neg-in 9.7%

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

       13. +-commutative 9.7%

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

       14. *-rgt-identity 9.7%

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

       15. *-rgt-identity 9.7%

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

   12. Simplified 9.1%

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

   13. Taylor expanded in x around inf 99.1%

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

       1. associate-*r/ 99.1%

          \[\leadsto \frac{1}{\frac{1 - x}{3 - \color{blue}{\frac{2 \cdot 1}{x}}}} \]

       2. metadata-eval 99.1%

          \[\leadsto \frac{1}{\frac{1 - x}{3 - \frac{\color{blue}{2}}{x}}} \]

   15. Simplified 99.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + -1}{\frac{2}{x} - 3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{\frac{2}{x} + -3}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.85)))
   (/ (+ (/ 2.0 x) -3.0) (+ x -1.0))
   (+ 1.0 (* x (+ x 3.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.85)) {
		tmp = ((2.0 / x) + -3.0) / (x + -1.0);
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.85d0))) then
        tmp = ((2.0d0 / x) + (-3.0d0)) / (x + (-1.0d0))
    else
        tmp = 1.0d0 + (x * (x + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.85)) {
		tmp = ((2.0 / x) + -3.0) / (x + -1.0);
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.85):
		tmp = ((2.0 / x) + -3.0) / (x + -1.0)
	else:
		tmp = 1.0 + (x * (x + 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.85))
		tmp = Float64(Float64(Float64(2.0 / x) + -3.0) / Float64(x + -1.0));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.85)))
		tmp = ((2.0 / x) + -3.0) / (x + -1.0);
	else
		tmp = 1.0 + (x * (x + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.85]], $MachinePrecision]], N[(N[(N[(2.0 / x), $MachinePrecision] + -3.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.849999999999999978 < x

   1. Initial program 8.0%

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

      1. remove-double-neg 8.0%

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

      2. distribute-neg-frac 8.0%

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

      3. distribute-neg-in 8.0%

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

      4. sub-neg 8.0%

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

      5. distribute-frac-neg2 8.0%

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

      6. sub-neg 8.0%

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

      7. +-commutative 8.0%

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

      8. unsub-neg 8.0%

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

      9. metadata-eval 8.0%

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

      10. neg-sub0 8.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 8.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 8.0%

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

      13. +-commutative 8.0%

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

      14. unsub-neg 8.0%

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

   3. Simplified 8.0%

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

      1. clear-num 8.0%

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

      2. frac-sub 9.4%

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

      3. *-commutative 9.4%

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

      4. *-un-lft-identity 9.4%

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

      5. fma-neg 8.1%

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

      6. +-commutative 8.1%

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

      7. distribute-neg-in 8.1%

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

      8. metadata-eval 8.1%

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

      9. sub-neg 8.1%

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

   6. Applied egg-rr 8.1%

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

      1. associate-*r/ 8.1%

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

      2. *-commutative 8.1%

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

      3. associate-/l* 8.1%

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

      4. +-commutative 8.1%

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

      5. metadata-eval 8.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{\left(0 - -1\right)} + x}{-1 - x}} \]

      6. associate--r- 8.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{0 - \left(-1 - x\right)}}{-1 - x}} \]

      7. div-sub 8.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \color{blue}{\left(\frac{0}{-1 - x} - \frac{-1 - x}{-1 - x}\right)}} \]

      8. div0 8.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(\color{blue}{0} - \frac{-1 - x}{-1 - x}\right)} \]

      9. *-inverses 8.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(0 - \color{blue}{1}\right)} \]

      10. metadata-eval 8.1%

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

   8. Simplified 8.1%

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

   9. Taylor expanded in x around inf 99.4%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} - 3}}{\left(1 - x\right) \cdot -1} \]
   10. Step-by-step derivation

       1. sub-neg 99.4%

          \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} + \left(-3\right)}}{\left(1 - x\right) \cdot -1} \]

       2. associate-*r/ 99.4%

          \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{x}} + \left(-3\right)}{\left(1 - x\right) \cdot -1} \]

       3. metadata-eval 99.4%

          \[\leadsto \frac{\frac{\color{blue}{2}}{x} + \left(-3\right)}{\left(1 - x\right) \cdot -1} \]

       4. metadata-eval 99.4%

          \[\leadsto \frac{\frac{2}{x} + \color{blue}{-3}}{\left(1 - x\right) \cdot -1} \]

   11. Simplified 99.4%

       \[\leadsto \frac{\color{blue}{\frac{2}{x} + -3}}{\left(1 - x\right) \cdot -1} \]

   if -1 < x < 0.849999999999999978

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.6%

      \[\leadsto \color{blue}{1 + \left(3 \cdot x + {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 98.6%

         \[\leadsto 1 + \left(3 \cdot x + \color{blue}{x \cdot x}\right) \]

      2. distribute-rgt-out 98.6%

         \[\leadsto 1 + \color{blue}{x \cdot \left(3 + x\right)} \]

   7. Simplified 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{\frac{2}{x} + -3}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ -3.0 x) (+ 1.0 (* x (+ x 3.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-3.0d0) / x
    else
        tmp = 1.0d0 + (x * (x + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * (x + 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -3.0 / x
	else:
		tmp = 1.0 + (x * (x + 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -3.0 / x;
	else
		tmp = 1.0 + (x * (x + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-3.0 / x), $MachinePrecision], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.0%

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

      1. remove-double-neg 8.0%

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

      2. distribute-neg-frac 8.0%

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

      3. distribute-neg-in 8.0%

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

      4. sub-neg 8.0%

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

      5. distribute-frac-neg2 8.0%

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

      6. sub-neg 8.0%

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

      7. +-commutative 8.0%

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

      8. unsub-neg 8.0%

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

      9. metadata-eval 8.0%

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

      10. neg-sub0 8.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 8.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 8.0%

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

      13. +-commutative 8.0%

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

      14. unsub-neg 8.0%

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

   3. Simplified 8.0%

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

   5. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.6%

      \[\leadsto \color{blue}{1 + \left(3 \cdot x + {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 98.6%

         \[\leadsto 1 + \left(3 \cdot x + \color{blue}{x \cdot x}\right) \]

      2. distribute-rgt-out 98.6%

         \[\leadsto 1 + \color{blue}{x \cdot \left(3 + x\right)} \]

   7. Simplified 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot -0.3333333333333333 + 0.1111111111111111}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ -3.0 x)
   (if (<= x 1.0)
     (+ 1.0 (* x (+ x 3.0)))
     (/ 1.0 (+ (* x -0.3333333333333333) 0.1111111111111111)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * (x + 3.0));
	} else {
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-3.0d0) / x
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + (x * (x + 3.0d0))
    else
        tmp = 1.0d0 / ((x * (-0.3333333333333333d0)) + 0.1111111111111111d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -3.0 / x;
	} else if (x <= 1.0) {
		tmp = 1.0 + (x * (x + 3.0));
	} else {
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -3.0 / x
	elif x <= 1.0:
		tmp = 1.0 + (x * (x + 3.0))
	else:
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-3.0 / x);
	elseif (x <= 1.0)
		tmp = Float64(1.0 + Float64(x * Float64(x + 3.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(x * -0.3333333333333333) + 0.1111111111111111));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -3.0 / x;
	elseif (x <= 1.0)
		tmp = 1.0 + (x * (x + 3.0));
	else
		tmp = 1.0 / ((x * -0.3333333333333333) + 0.1111111111111111);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 + N[(x * N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * -0.3333333333333333), $MachinePrecision] + 0.1111111111111111), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 7.0%

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

      1. remove-double-neg 7.0%

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

      2. distribute-neg-frac 7.0%

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

      3. distribute-neg-in 7.0%

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

      4. sub-neg 7.0%

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

      5. distribute-frac-neg2 7.0%

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

      6. sub-neg 7.0%

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

      7. +-commutative 7.0%

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

      8. unsub-neg 7.0%

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

      9. metadata-eval 7.0%

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

      10. neg-sub0 7.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 7.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 7.0%

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

      13. +-commutative 7.0%

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

      14. unsub-neg 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.6%

      \[\leadsto \color{blue}{1 + \left(3 \cdot x + {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 98.6%

         \[\leadsto 1 + \left(3 \cdot x + \color{blue}{x \cdot x}\right) \]

      2. distribute-rgt-out 98.6%

         \[\leadsto 1 + \color{blue}{x \cdot \left(3 + x\right)} \]

   7. Simplified 98.6%

      \[\leadsto \color{blue}{1 + x \cdot \left(3 + x\right)} \]

   if 1 < x

   1. Initial program 9.0%

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

      1. remove-double-neg 9.0%

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

      2. distribute-neg-frac 9.0%

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

      3. distribute-neg-in 9.0%

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

      4. sub-neg 9.0%

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

      5. distribute-frac-neg2 9.0%

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

      6. sub-neg 9.0%

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

      7. +-commutative 9.0%

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

      8. unsub-neg 9.0%

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

      9. metadata-eval 9.0%

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

      10. neg-sub0 9.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 9.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 9.0%

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

      13. +-commutative 9.0%

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

      14. unsub-neg 9.0%

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

   3. Simplified 9.0%

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

      1. clear-num 9.0%

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

      2. frac-sub 10.8%

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

      3. *-commutative 10.8%

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

      4. *-un-lft-identity 10.8%

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

      5. fma-neg 9.3%

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

      6. +-commutative 9.3%

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

      7. distribute-neg-in 9.3%

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

      8. metadata-eval 9.3%

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

      9. sub-neg 9.3%

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

   6. Applied egg-rr 9.3%

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

      1. associate-*r/ 9.3%

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

      2. *-commutative 9.3%

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

      3. associate-/l* 9.3%

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

      4. +-commutative 9.3%

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

      5. metadata-eval 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{\left(0 - -1\right)} + x}{-1 - x}} \]

      6. associate--r- 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \frac{\color{blue}{0 - \left(-1 - x\right)}}{-1 - x}} \]

      7. div-sub 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \color{blue}{\left(\frac{0}{-1 - x} - \frac{-1 - x}{-1 - x}\right)}} \]

      8. div0 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(\color{blue}{0} - \frac{-1 - x}{-1 - x}\right)} \]

      9. *-inverses 9.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\left(1 - x\right) \cdot \left(0 - \color{blue}{1}\right)} \]

      10. metadata-eval 9.3%

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

   8. Simplified 9.3%

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

      1. clear-num 9.3%

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

      2. inv-pow 9.3%

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

      3. *-commutative 9.3%

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

      4. *-un-lft-identity 9.3%

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

      5. times-frac 9.3%

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

      6. metadata-eval 9.3%

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

   10. Applied egg-rr 9.3%

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

       1. unpow-1 9.3%

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

       2. mul-1-neg 9.3%

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

       3. distribute-neg-frac2 9.3%

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

       4. fma-define 10.8%

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

       5. *-lft-identity 10.8%

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

       6. associate-*l/ 9.9%

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

       7. distribute-neg-out 9.9%

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

       8. distribute-lft-neg-out 9.9%

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

       9. associate-*r* 9.7%

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

       10. sub-neg 9.7%

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

       11. metadata-eval 9.7%

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

       12. distribute-neg-in 9.7%

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

       13. +-commutative 9.7%

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

       14. *-rgt-identity 9.7%

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

       15. *-rgt-identity 9.7%

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

   12. Simplified 9.1%

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

   13. Taylor expanded in x around inf 98.9%

       \[\leadsto \frac{1}{\color{blue}{0.1111111111111111 + -0.3333333333333333 \cdot x}} \]
   14. Step-by-step derivation

       1. +-commutative 98.9%

          \[\leadsto \frac{1}{\color{blue}{-0.3333333333333333 \cdot x + 0.1111111111111111}} \]

       2. *-commutative 98.9%

          \[\leadsto \frac{1}{\color{blue}{x \cdot -0.3333333333333333} + 0.1111111111111111} \]

   15. Simplified 98.9%

       \[\leadsto \frac{1}{\color{blue}{x \cdot -0.3333333333333333 + 0.1111111111111111}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot -0.3333333333333333 + 0.1111111111111111}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ -3.0 x) (+ 1.0 (* x 3.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-3.0d0) / x
    else
        tmp = 1.0d0 + (x * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0 + (x * 3.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -3.0 / x
	else:
		tmp = 1.0 + (x * 3.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-3.0 / x);
	else
		tmp = Float64(1.0 + Float64(x * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -3.0 / x;
	else
		tmp = 1.0 + (x * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-3.0 / x), $MachinePrecision], N[(1.0 + N[(x * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.0%

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

      1. remove-double-neg 8.0%

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

      2. distribute-neg-frac 8.0%

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

      3. distribute-neg-in 8.0%

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

      4. sub-neg 8.0%

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

      5. distribute-frac-neg2 8.0%

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

      6. sub-neg 8.0%

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

      7. +-commutative 8.0%

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

      8. unsub-neg 8.0%

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

      9. metadata-eval 8.0%

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

      10. neg-sub0 8.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 8.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 8.0%

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

      13. +-commutative 8.0%

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

      14. unsub-neg 8.0%

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

   3. Simplified 8.0%

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

   5. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ -3.0 x) 1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / 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)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-3.0d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -3.0 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -3.0 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-3.0 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -3.0 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-3.0 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 8.0%

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

      1. remove-double-neg 8.0%

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

      2. distribute-neg-frac 8.0%

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

      3. distribute-neg-in 8.0%

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

      4. sub-neg 8.0%

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

      5. distribute-frac-neg2 8.0%

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

      6. sub-neg 8.0%

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

      7. +-commutative 8.0%

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

      8. unsub-neg 8.0%

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

      9. metadata-eval 8.0%

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

      10. neg-sub0 8.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 8.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 8.0%

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

      13. +-commutative 8.0%

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

      14. unsub-neg 8.0%

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

   3. Simplified 8.0%

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

   5. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

      11. associate-+l- 100.0%

          \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 97.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.5% accurate, 13.0× 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 56.1%

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

   1. remove-double-neg 56.1%

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

   2. distribute-neg-frac 56.1%

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

   3. distribute-neg-in 56.1%

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

   4. sub-neg 56.1%

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

   5. distribute-frac-neg2 56.1%

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

   6. sub-neg 56.1%

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

   7. +-commutative 56.1%

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

   8. unsub-neg 56.1%

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

   9. metadata-eval 56.1%

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

   10. neg-sub0 56.1%

       \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{0 - \left(x - 1\right)}} \]

   11. associate-+l- 56.1%

       \[\leadsto \frac{x}{x + 1} - \frac{-1 - x}{\color{blue}{\left(0 - x\right) + 1}} \]

   12. neg-sub0 56.1%

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

   13. +-commutative 56.1%

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

   14. unsub-neg 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in x around 0 52.7%

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

6. Final simplification 52.7%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))