Asymptote C

Percentage Accurate: 53.4% → 99.7%

Time: 9.0s

Alternatives: 9

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: 53.4% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 0:\\ \;\;\;\;\frac{-3 + \frac{-1 + \frac{-3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + -3 \cdot x}{\left(-1 - x\right) \cdot \left(1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ x (+ x 1.0)) (/ (+ x 1.0) (- 1.0 x))) 0.0)
   (/ (+ -3.0 (/ (+ -1.0 (/ -3.0 x)) x)) x)
   (/ (+ -1.0 (* -3.0 x)) (* (- -1.0 x) (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 0.0) {
		tmp = (-3.0 + ((-1.0 + (-3.0 / x)) / x)) / x;
	} else {
		tmp = (-1.0 + (-3.0 * x)) / ((-1.0 - x) * (1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x / (x + 1.0d0)) + ((x + 1.0d0) / (1.0d0 - x))) <= 0.0d0) then
        tmp = ((-3.0d0) + (((-1.0d0) + ((-3.0d0) / x)) / x)) / x
    else
        tmp = ((-1.0d0) + ((-3.0d0) * x)) / (((-1.0d0) - x) * (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 0.0) {
		tmp = (-3.0 + ((-1.0 + (-3.0 / x)) / x)) / x;
	} else {
		tmp = (-1.0 + (-3.0 * x)) / ((-1.0 - x) * (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x / (x + 1.0)) + ((x + 1.0) / (1.0 - x))) <= 0.0:
		tmp = (-3.0 + ((-1.0 + (-3.0 / x)) / x)) / x
	else:
		tmp = (-1.0 + (-3.0 * x)) / ((-1.0 - x) * (1.0 - x))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-3.0 + N[(N[(-1.0 + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-1.0 + N[(-3.0 * x), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 0.0

   1. Initial program 6.0%

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

      1. remove-double-neg 6.0%

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

      2. distribute-neg-frac 6.0%

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

      3. distribute-neg-in 6.0%

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

      4. sub-neg 6.0%

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

      5. distribute-frac-neg2 6.0%

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

      6. sub-neg 6.0%

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

      7. +-commutative 6.0%

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

      8. unsub-neg 6.0%

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

      9. metadata-eval 6.0%

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

      10. neg-sub0 6.0%

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

      11. associate-+l- 6.0%

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

      12. neg-sub0 6.0%

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

      13. +-commutative 6.0%

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

      14. unsub-neg 6.0%

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

   3. Simplified 6.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. metadata-eval 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r/ 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-neg-frac 100.0%

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

      11. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if 0.0 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 97.9%

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

      1. remove-double-neg 97.9%

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

      2. distribute-neg-frac 97.9%

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

      3. distribute-neg-in 97.9%

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

      4. sub-neg 97.9%

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

      5. distribute-frac-neg2 97.9%

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

      6. sub-neg 97.9%

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

      7. +-commutative 97.9%

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

      8. unsub-neg 97.9%

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

      9. metadata-eval 97.9%

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

      10. neg-sub0 97.9%

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

      11. associate-+l- 97.9%

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

      12. neg-sub0 97.9%

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

      13. +-commutative 97.9%

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

      14. unsub-neg 97.9%

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

   3. Simplified 97.9%

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

      1. frac-2neg 97.9%

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

      2. frac-sub 98.0%

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

      3. +-commutative 98.0%

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

      4. distribute-neg-in 98.0%

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

      5. metadata-eval 98.0%

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

      6. sub-neg 98.0%

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

      7. pow2 98.0%

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

      8. +-commutative 98.0%

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

      9. distribute-neg-in 98.0%

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

      10. metadata-eval 98.0%

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

      11. sub-neg 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{x + 1}{1 - x} \leq 0:\\ \;\;\;\;\frac{-3 + \frac{-1 + \frac{-3}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + -3 \cdot x}{\left(-1 - x\right) \cdot \left(1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (+ x 1.0) (- 1.0 x)))))
   (if (<= t_0 5e-5) (/ (+ -3.0 (/ (+ -1.0 (/ -3.0 x)) x)) x) 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-5) {
		tmp = (-3.0 + ((-1.0 + (-3.0 / x)) / x)) / x;
	} 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-5) then
        tmp = ((-3.0d0) + (((-1.0d0) + ((-3.0d0) / x)) / x)) / x
    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-5) {
		tmp = (-3.0 + ((-1.0 + (-3.0 / x)) / x)) / x;
	} 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-5:
		tmp = (-3.0 + ((-1.0 + (-3.0 / x)) / x)) / x
	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-5)
		tmp = Float64(Float64(-3.0 + Float64(Float64(-1.0 + Float64(-3.0 / x)) / x)) / x);
	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-5)
		tmp = (-3.0 + ((-1.0 + (-3.0 / x)) / x)) / x;
	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-5], N[(N[(-3.0 + N[(N[(-1.0 + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64)))) < 5.00000000000000024e-5

   1. Initial program 7.5%

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

      1. remove-double-neg 7.5%

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

      2. distribute-neg-frac 7.5%

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

      3. distribute-neg-in 7.5%

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

      4. sub-neg 7.5%

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

      5. distribute-frac-neg2 7.5%

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

      6. sub-neg 7.5%

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

      7. +-commutative 7.5%

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

      8. unsub-neg 7.5%

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

      9. metadata-eval 7.5%

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

      10. neg-sub0 7.5%

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

      11. associate-+l- 7.5%

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

      12. neg-sub0 7.5%

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

      13. +-commutative 7.5%

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

      14. unsub-neg 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. neg-mul-1 99.9%

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

      8. associate-*r/ 99.9%

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

      9. metadata-eval 99.9%

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

      10. distribute-neg-frac 99.9%

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

      11. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   if 5.00000000000000024e-5 < (-.f64 (/.f64 x (+.f64 x #s(literal 1 binary64))) (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))))

   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%

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

Alternative 3: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 + ((-1.0 + (-3.0 / x)) / x)) / 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) + (((-1.0d0) + ((-3.0d0) / x)) / x)) / 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 + ((-1.0 + (-3.0 / x)) / x)) / 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 + ((-1.0 + (-3.0 / x)) / x)) / 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(Float64(-3.0 + Float64(Float64(-1.0 + Float64(-3.0 / x)) / x)) / 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 + ((-1.0 + (-3.0 / x)) / x)) / 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[(N[(-3.0 + N[(N[(-1.0 + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 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 7.5%

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

      1. remove-double-neg 7.5%

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

      2. distribute-neg-frac 7.5%

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

      3. distribute-neg-in 7.5%

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

      4. sub-neg 7.5%

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

      5. distribute-frac-neg2 7.5%

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

      6. sub-neg 7.5%

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

      7. +-commutative 7.5%

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

      8. unsub-neg 7.5%

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

      9. metadata-eval 7.5%

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

      10. neg-sub0 7.5%

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

      11. associate-+l- 7.5%

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

      12. neg-sub0 7.5%

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

      13. +-commutative 7.5%

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

      14. unsub-neg 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. neg-mul-1 99.9%

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

      8. associate-*r/ 99.9%

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

      9. metadata-eval 99.9%

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

      10. distribute-neg-frac 99.9%

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

      11. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   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%

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

4. Final simplification 99.2%

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

Alternative 4: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{-3}{-1 - x} \cdot \frac{x}{1 - x} + \frac{-1}{\left(-1 - x\right) \cdot \left(1 - x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (* (/ -3.0 (- -1.0 x)) (/ x (- 1.0 x))) (/ -1.0 (* (- -1.0 x) (- 1.0 x)))))

C

double code(double x) {
	return ((-3.0 / (-1.0 - x)) * (x / (1.0 - x))) + (-1.0 / ((-1.0 - x) * (1.0 - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-3.0d0) / ((-1.0d0) - x)) * (x / (1.0d0 - x))) + ((-1.0d0) / (((-1.0d0) - x) * (1.0d0 - x)))
end function

Java

public static double code(double x) {
	return ((-3.0 / (-1.0 - x)) * (x / (1.0 - x))) + (-1.0 / ((-1.0 - x) * (1.0 - x)));
}

Python

def code(x):
	return ((-3.0 / (-1.0 - x)) * (x / (1.0 - x))) + (-1.0 / ((-1.0 - x) * (1.0 - x)))

Julia

function code(x)
	return Float64(Float64(Float64(-3.0 / Float64(-1.0 - x)) * Float64(x / Float64(1.0 - x))) + Float64(-1.0 / Float64(Float64(-1.0 - x) * Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = ((-3.0 / (-1.0 - x)) * (x / (1.0 - x))) + (-1.0 / ((-1.0 - x) * (1.0 - x)));
end

Wolfram

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

Derivation

1. Initial program 55.2%

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

   1. remove-double-neg 55.2%

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

   2. distribute-neg-frac 55.2%

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

   3. distribute-neg-in 55.2%

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

   4. sub-neg 55.2%

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

   5. distribute-frac-neg2 55.2%

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

   6. sub-neg 55.2%

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

   7. +-commutative 55.2%

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

   8. unsub-neg 55.2%

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

   9. metadata-eval 55.2%

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

   10. neg-sub0 55.2%

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

   11. associate-+l- 55.2%

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

   12. neg-sub0 55.2%

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

   13. +-commutative 55.2%

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

   14. unsub-neg 55.2%

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

3. Simplified 55.2%

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

   1. frac-2neg 55.2%

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

   2. frac-sub 53.6%

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

   3. +-commutative 53.6%

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

   4. distribute-neg-in 53.6%

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

   5. metadata-eval 53.6%

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

   6. sub-neg 53.6%

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

   7. pow2 53.6%

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

   8. +-commutative 53.6%

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

   9. distribute-neg-in 53.6%

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

   10. metadata-eval 53.6%

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

   11. sub-neg 53.6%

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

6. Applied egg-rr 53.6%

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

7. Taylor expanded in x around 0 78.4%

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

   1. div-sub 78.4%

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

   2. times-frac 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

    \[\leadsto \frac{-3}{-1 - x} \cdot \frac{x}{1 - x} + \frac{-1}{\left(-1 - x\right) \cdot \left(1 - x\right)} \]
11. Add Preprocessing

Alternative 5: 98.6% 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 7.5%

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

      1. remove-double-neg 7.5%

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

      2. distribute-neg-frac 7.5%

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

      3. distribute-neg-in 7.5%

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

      4. sub-neg 7.5%

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

      5. distribute-frac-neg2 7.5%

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

      6. sub-neg 7.5%

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

      7. +-commutative 7.5%

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

      8. unsub-neg 7.5%

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

      9. metadata-eval 7.5%

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

      10. neg-sub0 7.5%

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

      11. associate-+l- 7.5%

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

      12. neg-sub0 7.5%

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

      13. +-commutative 7.5%

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

      14. unsub-neg 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in x around inf 98.9%

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

   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%

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

4. Final simplification 98.7%

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

Alternative 6: 99.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-3.0 + (-1.0 / x)) / 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) + ((-1.0d0) / x)) / 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 + (-1.0 / x)) / 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 + (-1.0 / x)) / 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(Float64(-3.0 + Float64(-1.0 / x)) / 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 + (-1.0 / x)) / 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[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / 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 7.5%

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

      1. remove-double-neg 7.5%

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

      2. distribute-neg-frac 7.5%

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

      3. distribute-neg-in 7.5%

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

      4. sub-neg 7.5%

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

      5. distribute-frac-neg2 7.5%

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

      6. sub-neg 7.5%

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

      7. +-commutative 7.5%

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

      8. unsub-neg 7.5%

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

      9. metadata-eval 7.5%

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

      10. neg-sub0 7.5%

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

      11. associate-+l- 7.5%

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

      12. neg-sub0 7.5%

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

      13. +-commutative 7.5%

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

      14. unsub-neg 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. neg-mul-1 99.6%

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

      3. distribute-neg-in 99.6%

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

      4. metadata-eval 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   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%

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

4. Final simplification 99.1%

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

Alternative 7: 98.4% 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 7.5%

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

      1. remove-double-neg 7.5%

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

      2. distribute-neg-frac 7.5%

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

      3. distribute-neg-in 7.5%

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

      4. sub-neg 7.5%

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

      5. distribute-frac-neg2 7.5%

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

      6. sub-neg 7.5%

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

      7. +-commutative 7.5%

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

      8. unsub-neg 7.5%

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

      9. metadata-eval 7.5%

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

      10. neg-sub0 7.5%

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

      11. associate-+l- 7.5%

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

      12. neg-sub0 7.5%

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

      13. +-commutative 7.5%

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

      14. unsub-neg 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in x around inf 98.9%

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

   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 98.5%

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

4. Final simplification 98.7%

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

Alternative 8: 98.0% 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 7.5%

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

      1. remove-double-neg 7.5%

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

      2. distribute-neg-frac 7.5%

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

      3. distribute-neg-in 7.5%

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

      4. sub-neg 7.5%

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

      5. distribute-frac-neg2 7.5%

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

      6. sub-neg 7.5%

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

      7. +-commutative 7.5%

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

      8. unsub-neg 7.5%

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

      9. metadata-eval 7.5%

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

      10. neg-sub0 7.5%

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

      11. associate-+l- 7.5%

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

      12. neg-sub0 7.5%

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

      13. +-commutative 7.5%

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

      14. unsub-neg 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in x around inf 98.9%

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

   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 98.0%

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

4. Final simplification 98.4%

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

Alternative 9: 49.9% 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 55.2%

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

   1. remove-double-neg 55.2%

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

   2. distribute-neg-frac 55.2%

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

   3. distribute-neg-in 55.2%

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

   4. sub-neg 55.2%

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

   5. distribute-frac-neg2 55.2%

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

   6. sub-neg 55.2%

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

   7. +-commutative 55.2%

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

   8. unsub-neg 55.2%

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

   9. metadata-eval 55.2%

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

   10. neg-sub0 55.2%

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

   11. associate-+l- 55.2%

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

   12. neg-sub0 55.2%

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

   13. +-commutative 55.2%

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

   14. unsub-neg 55.2%

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

3. Simplified 55.2%

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

5. Taylor expanded in x around 0 52.4%

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

6. Final simplification 52.4%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))