Asymptote C

Percentage Accurate: 54.3% → 99.8%

Time: 17.7s

Alternatives: 11

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: 54.3% 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 0.0001:\\ \;\;\;\;\frac{-3}{x} - \left(\frac{3}{{x}^{3}} + \left({x}^{-2} + {x}^{-4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 0.0001)
     (- (/ -3.0 x) (+ (/ 3.0 (pow x 3.0)) (+ (pow x -2.0) (pow x -4.0))))
     t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = (-3.0 / x) - ((3.0 / pow(x, 3.0)) + (pow(x, -2.0) + pow(x, -4.0)));
	} 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)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 0.0001d0) then
        tmp = ((-3.0d0) / x) - ((3.0d0 / (x ** 3.0d0)) + ((x ** (-2.0d0)) + (x ** (-4.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = (-3.0 / x) - ((3.0 / Math.pow(x, 3.0)) + (Math.pow(x, -2.0) + Math.pow(x, -4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 0.0001:
		tmp = (-3.0 / x) - ((3.0 / math.pow(x, 3.0)) + (math.pow(x, -2.0) + math.pow(x, -4.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 0.0001)
		tmp = Float64(Float64(-3.0 / x) - Float64(Float64(3.0 / (x ^ 3.0)) + Float64((x ^ -2.0) + (x ^ -4.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 0.0001)
		tmp = (-3.0 / x) - ((3.0 / (x ^ 3.0)) + ((x ^ -2.0) + (x ^ -4.0)));
	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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0001], N[(N[(-3.0 / x), $MachinePrecision] - N[(N[(3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, -2.0], $MachinePrecision] + N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]), $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))) < 1.00000000000000005e-4

   1. Initial program 9.6%

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

      1. clear-num 9.5%

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

      2. associate-/r/ 9.4%

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

   4. Applied egg-rr 9.4%

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

   5. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto -\left(3 \cdot \frac{1}{{x}^{3}} + \color{blue}{\left(\left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right) + 3 \cdot \frac{1}{x}\right)}\right) \]

      2. associate-+r+ 99.1%

         \[\leadsto -\color{blue}{\left(\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) + 3 \cdot \frac{1}{x}\right)} \]

      3. +-commutative 99.1%

         \[\leadsto -\color{blue}{\left(3 \cdot \frac{1}{x} + \left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)\right)} \]

      4. distribute-neg-in 99.1%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) + \left(-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)\right)} \]

      5. unsub-neg 99.1%

         \[\leadsto \color{blue}{\left(-3 \cdot \frac{1}{x}\right) - \left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)} \]

      6. associate-*r/ 99.5%

         \[\leadsto \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      7. metadata-eval 99.5%

         \[\leadsto \left(-\frac{\color{blue}{3}}{x}\right) - \left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      8. distribute-neg-frac 99.5%

         \[\leadsto \color{blue}{\frac{-3}{x}} - \left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      9. metadata-eval 99.5%

         \[\leadsto \frac{\color{blue}{-3}}{x} - \left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      10. associate-*r/ 99.5%

          \[\leadsto \frac{-3}{x} - \left(\color{blue}{\frac{3 \cdot 1}{{x}^{3}}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

      11. metadata-eval 99.5%

          \[\leadsto \frac{-3}{x} - \left(\frac{\color{blue}{3}}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right) \]

   7. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{-3}{x} - \left(\frac{3}{{x}^{3}} + \left({x}^{-2} + {x}^{-4}\right)\right)} \]

   if 1.00000000000000005e-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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0001:\\ \;\;\;\;\frac{-3}{x} - \left(\frac{3}{{x}^{3}} + \left({x}^{-2} + {x}^{-4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 8.0%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. metadata-eval 98.4%

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

      2. unpow2 98.4%

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

      3. frac-times 98.4%

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

   5. Applied egg-rr 98.9%

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

      1. distribute-rgt-out 98.9%

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

      2. associate-*l/ 99.3%

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

      3. *-un-lft-identity 99.3%

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

   7. Applied egg-rr 99.3%

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

   if 0.0 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. frac-2neg 99.4%

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

      4. frac-sub 99.3%

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

      5. sub-neg 99.3%

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

      6. metadata-eval 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. *-rgt-identity 99.3%

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

      11. sub-neg 99.3%

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

      12. *-rgt-identity 99.3%

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

      13. +-commutative 99.3%

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

      14. +-commutative 99.3%

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

      15. distribute-neg-in 99.3%

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

      16. metadata-eval 99.3%

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

      17. +-commutative 99.3%

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

      18. sub-neg 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. metadata-eval 99.3%

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

      2. remove-double-neg 99.3%

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

      3. distribute-neg-in 99.3%

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

      4. distribute-neg-in 99.3%

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

      5. metadata-eval 99.3%

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

      6. remove-double-neg 99.3%

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

      7. unsub-neg 99.3%

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

   8. Simplified 99.3%

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

      1. prod-diff 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. distribute-lft-neg-in 99.9%

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

       2. distribute-neg-in 99.9%

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

       3. metadata-eval 99.9%

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

       4. sub-neg 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 0.0001:\\ \;\;\;\;3 \cdot \frac{-1}{{x}^{3}} + \frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 0.0001)
     (+ (* 3.0 (/ -1.0 (pow x 3.0))) (/ (- (/ -1.0 x) 3.0) x))
     t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = (3.0 * (-1.0 / pow(x, 3.0))) + (((-1.0 / x) - 3.0) / 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)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 0.0001d0) then
        tmp = (3.0d0 * ((-1.0d0) / (x ** 3.0d0))) + ((((-1.0d0) / x) - 3.0d0) / 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)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = (3.0 * (-1.0 / Math.pow(x, 3.0))) + (((-1.0 / x) - 3.0) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 0.0001:
		tmp = (3.0 * (-1.0 / math.pow(x, 3.0))) + (((-1.0 / x) - 3.0) / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 0.0001)
		tmp = Float64(Float64(3.0 * Float64(-1.0 / (x ^ 3.0))) + Float64(Float64(Float64(-1.0 / x) - 3.0) / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 0.0001)
		tmp = (3.0 * (-1.0 / (x ^ 3.0))) + (((-1.0 / x) - 3.0) / 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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0001], N[(N[(3.0 * N[(-1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / x), $MachinePrecision] - 3.0), $MachinePrecision] / x), $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))) < 1.00000000000000005e-4

   1. Initial program 9.6%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. metadata-eval 97.9%

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

      2. unpow2 97.9%

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

      3. frac-times 97.9%

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

   5. Applied egg-rr 98.8%

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

      1. distribute-rgt-out 98.8%

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

      2. associate-*l/ 99.2%

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

      3. *-un-lft-identity 99.2%

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

   7. Applied egg-rr 99.2%

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

   if 1.00000000000000005e-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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0001:\\ \;\;\;\;3 \cdot \frac{-1}{{x}^{3}} + \frac{\frac{-1}{x} - 3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= (+ t_0 (/ (- -1.0 x) (+ x -1.0))) 2e-10)
     (/ (+ -3.0 (/ -1.0 x)) x)
     (+ t_0 (* (+ x 1.0) (/ -1.0 (+ x -1.0)))))))

C

double code(double x) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-10) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)));
	}
	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)
    if ((t_0 + (((-1.0d0) - x) / (x + (-1.0d0)))) <= 2d-10) then
        tmp = ((-3.0d0) + ((-1.0d0) / x)) / x
    else
        tmp = t_0 + ((x + 1.0d0) * ((-1.0d0) / (x + (-1.0d0))))
    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 + ((-1.0 - x) / (x + -1.0))) <= 2e-10) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (x + 1.0)
	tmp = 0
	if (t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-10:
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(-1.0 - x) / Float64(x + -1.0))) <= 2e-10)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = Float64(t_0 + Float64(Float64(x + 1.0) * Float64(-1.0 / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if ((t_0 + ((-1.0 - x) / (x + -1.0))) <= 2e-10)
		tmp = (-3.0 + (-1.0 / x)) / x;
	else
		tmp = t_0 + ((x + 1.0) * (-1.0 / (x + -1.0)));
	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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-10], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] * N[(-1.0 / N[(x + -1.0), $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))) < 2.00000000000000007e-10

   1. Initial program 8.4%

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

      1. clear-num 8.3%

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

      2. associate-/r/ 8.2%

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

   4. Applied egg-rr 8.2%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. distribute-neg-in 98.4%

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

      2. exp-to-pow 49.4%

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

      3. *-commutative 49.4%

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

      4. exp-neg 49.4%

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

      5. distribute-lft-neg-in 49.4%

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

      6. metadata-eval 49.4%

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

      7. log-pow 98.4%

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

      8. rem-exp-log 98.4%

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

      9. unsub-neg 98.4%

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

      10. associate-*r/ 98.8%

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

      11. metadata-eval 98.8%

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

      12. distribute-neg-frac 98.8%

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

      13. metadata-eval 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{-3}{x} - {x}^{-2}} \]
   8. Step-by-step derivation

      1. metadata-eval 98.8%

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

      2. pow-div 98.8%

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

      3. inv-pow 98.8%

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

      4. pow1 98.8%

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

      5. sub-div 98.8%

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

   9. Applied egg-rr 98.8%

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

   if 2.00000000000000007e-10 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 99.3%

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

Alternative 5: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 2e-10) (/ (+ -3.0 (/ -1.0 x)) x) t_0)))

C

double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-10) {
		tmp = (-3.0 + (-1.0 / 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)) + (((-1.0d0) - x) / (x + (-1.0d0)))
    if (t_0 <= 2d-10) then
        tmp = ((-3.0d0) + ((-1.0d0) / 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)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 2e-10) {
		tmp = (-3.0 + (-1.0 / x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0))
	tmp = 0
	if t_0 <= 2e-10:
		tmp = (-3.0 + (-1.0 / x)) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 2e-10)
		tmp = Float64(Float64(-3.0 + Float64(-1.0 / x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	tmp = 0.0;
	if (t_0 <= 2e-10)
		tmp = (-3.0 + (-1.0 / 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[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-10], N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $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))) < 2.00000000000000007e-10

   1. Initial program 8.4%

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

      1. clear-num 8.3%

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

      2. associate-/r/ 8.2%

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

   4. Applied egg-rr 8.2%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. distribute-neg-in 98.4%

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

      2. exp-to-pow 49.4%

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

      3. *-commutative 49.4%

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

      4. exp-neg 49.4%

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

      5. distribute-lft-neg-in 49.4%

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

      6. metadata-eval 49.4%

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

      7. log-pow 98.4%

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

      8. rem-exp-log 98.4%

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

      9. unsub-neg 98.4%

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

      10. associate-*r/ 98.8%

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

      11. metadata-eval 98.8%

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

      12. distribute-neg-frac 98.8%

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

      13. metadata-eval 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{-3}{x} - {x}^{-2}} \]
   8. Step-by-step derivation

      1. metadata-eval 98.8%

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

      2. pow-div 98.8%

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

      3. inv-pow 98.8%

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

      4. pow1 98.8%

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

      5. sub-div 98.8%

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

   9. Applied egg-rr 98.8%

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

   if 2.00000000000000007e-10 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 99.7%

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

4. Final simplification 99.3%

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

Alternative 6: 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 9.6%

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

   3. Taylor expanded in x around inf 97.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. unpow2 99.7%

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

      2. distribute-rgt-out 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 98.5%

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

Alternative 7: 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 9.6%

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

      1. clear-num 9.5%

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

      2. associate-/r/ 9.4%

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

   4. Applied egg-rr 9.4%

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

   5. Taylor expanded in x around inf 97.9%

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

      1. distribute-neg-in 97.9%

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

      2. exp-to-pow 48.6%

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

      3. *-commutative 48.6%

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

      4. exp-neg 48.6%

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

      5. distribute-lft-neg-in 48.6%

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

      6. metadata-eval 48.6%

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

      7. log-pow 97.9%

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

      8. rem-exp-log 97.9%

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

      9. unsub-neg 97.9%

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

      10. associate-*r/ 98.3%

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

      11. metadata-eval 98.3%

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

      12. distribute-neg-frac 98.3%

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

      13. metadata-eval 98.3%

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

   7. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{-3}{x} - {x}^{-2}} \]
   8. Step-by-step derivation

      1. metadata-eval 98.3%

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

      2. pow-div 98.3%

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

      3. inv-pow 98.3%

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

      4. pow1 98.3%

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

      5. sub-div 98.3%

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

   9. Applied egg-rr 98.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. unpow2 99.7%

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

      2. distribute-rgt-out 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.0%

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

Alternative 8: 99.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 8.7%

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

   3. Taylor expanded in x around inf 98.5%

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

      1. metadata-eval 98.5%

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

      2. unpow2 98.5%

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

      3. frac-times 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in x around 0 98.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. unpow2 99.7%

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

      2. distribute-rgt-out 99.7%

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

   5. Simplified 99.7%

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

   if 1 < x

   1. Initial program 10.5%

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

      1. clear-num 10.2%

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

      2. associate-/r/ 10.7%

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

   4. Applied egg-rr 10.7%

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

   5. Taylor expanded in x around inf 97.2%

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

      1. distribute-neg-in 97.2%

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

      2. exp-to-pow 97.2%

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

      3. *-commutative 97.2%

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

      4. exp-neg 97.2%

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

      5. distribute-lft-neg-in 97.2%

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

      6. metadata-eval 97.2%

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

      7. log-pow 97.2%

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

      8. rem-exp-log 97.2%

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

      9. unsub-neg 97.2%

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

      10. associate-*r/ 97.6%

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

      11. metadata-eval 97.6%

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

      12. distribute-neg-frac 97.6%

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

      13. metadata-eval 97.6%

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

   7. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{-3}{x} - {x}^{-2}} \]
   8. Step-by-step derivation

      1. metadata-eval 97.6%

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

      2. pow-div 97.6%

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

      3. inv-pow 97.6%

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

      4. pow1 97.6%

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

      5. sub-div 97.7%

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

   9. Applied egg-rr 97.7%

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

4. Final simplification 99.0%

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

Alternative 9: 98.5% 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 9.6%

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

   3. Taylor expanded in x around inf 97.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.5%

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

4. Final simplification 98.4%

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

Alternative 10: 97.9% 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 9.6%

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

   3. Taylor expanded in x around inf 97.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.5%

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

4. Final simplification 97.3%

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

Alternative 11: 50.9% 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 57.6%

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

3. Taylor expanded in x around 0 53.6%

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

4. Final simplification 53.6%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))