3frac (problem 3.3.3)

Percentage Accurate: 85.2% → 99.4%

Time: 9.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x\right)\\ t_1 := \frac{1}{1 + x}\\ t_2 := \left(t_1 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0 + \left(1 + x\right) \cdot \left(x - 2\right)}{\left(1 + x\right) \cdot t_0}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{2}{x} \cdot \frac{\left(1 - x\right) - x \cdot -0.5}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 x)))
        (t_1 (/ 1.0 (+ 1.0 x)))
        (t_2 (+ (- t_1 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_2 -1e-8)
     (/ (+ t_0 (* (+ 1.0 x) (- x 2.0))) (* (+ 1.0 x) t_0))
     (if (<= t_2 5e-26)
       (+ (/ 2.0 (pow x 5.0)) (/ 2.0 (pow x 3.0)))
       (+ t_1 (* (/ 2.0 x) (/ (- (- 1.0 x) (* x -0.5)) (+ x -1.0))))))))

C

double code(double x) {
	double t_0 = x * (1.0 - x);
	double t_1 = 1.0 / (1.0 + x);
	double t_2 = (t_1 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_2 <= -1e-8) {
		tmp = (t_0 + ((1.0 + x) * (x - 2.0))) / ((1.0 + x) * t_0);
	} else if (t_2 <= 5e-26) {
		tmp = (2.0 / pow(x, 5.0)) + (2.0 / pow(x, 3.0));
	} else {
		tmp = t_1 + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (1.0d0 - x)
    t_1 = 1.0d0 / (1.0d0 + x)
    t_2 = (t_1 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_2 <= (-1d-8)) then
        tmp = (t_0 + ((1.0d0 + x) * (x - 2.0d0))) / ((1.0d0 + x) * t_0)
    else if (t_2 <= 5d-26) then
        tmp = (2.0d0 / (x ** 5.0d0)) + (2.0d0 / (x ** 3.0d0))
    else
        tmp = t_1 + ((2.0d0 / x) * (((1.0d0 - x) - (x * (-0.5d0))) / (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (1.0 - x);
	double t_1 = 1.0 / (1.0 + x);
	double t_2 = (t_1 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_2 <= -1e-8) {
		tmp = (t_0 + ((1.0 + x) * (x - 2.0))) / ((1.0 + x) * t_0);
	} else if (t_2 <= 5e-26) {
		tmp = (2.0 / Math.pow(x, 5.0)) + (2.0 / Math.pow(x, 3.0));
	} else {
		tmp = t_1 + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (1.0 - x)
	t_1 = 1.0 / (1.0 + x)
	t_2 = (t_1 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_2 <= -1e-8:
		tmp = (t_0 + ((1.0 + x) * (x - 2.0))) / ((1.0 + x) * t_0)
	elif t_2 <= 5e-26:
		tmp = (2.0 / math.pow(x, 5.0)) + (2.0 / math.pow(x, 3.0))
	else:
		tmp = t_1 + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(1.0 - x))
	t_1 = Float64(1.0 / Float64(1.0 + x))
	t_2 = Float64(Float64(t_1 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_2 <= -1e-8)
		tmp = Float64(Float64(t_0 + Float64(Float64(1.0 + x) * Float64(x - 2.0))) / Float64(Float64(1.0 + x) * t_0));
	elseif (t_2 <= 5e-26)
		tmp = Float64(Float64(2.0 / (x ^ 5.0)) + Float64(2.0 / (x ^ 3.0)));
	else
		tmp = Float64(t_1 + Float64(Float64(2.0 / x) * Float64(Float64(Float64(1.0 - x) - Float64(x * -0.5)) / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (1.0 - x);
	t_1 = 1.0 / (1.0 + x);
	t_2 = (t_1 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_2 <= -1e-8)
		tmp = (t_0 + ((1.0 + x) * (x - 2.0))) / ((1.0 + x) * t_0);
	elseif (t_2 <= 5e-26)
		tmp = (2.0 / (x ^ 5.0)) + (2.0 / (x ^ 3.0));
	else
		tmp = t_1 + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-8], N[(N[(t$95$0 + N[(N[(1.0 + x), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-26], N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(2.0 / x), $MachinePrecision] * N[(N[(N[(1.0 - x), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -1e-8

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. sub-neg 99.3%

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

      3. neg-mul-1 99.3%

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

      4. metadata-eval 99.3%

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

      5. cancel-sign-sub-inv 99.3%

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

      6. +-commutative 99.3%

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

      7. *-lft-identity 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. frac-2neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. frac-sub 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. metadata-eval 99.3%

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

      7. sub-neg 99.3%

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

      8. *-commutative 99.3%

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

      9. neg-mul-1 99.3%

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

      10. +-commutative 99.3%

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

      11. distribute-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

      13. sub-neg 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. unsub-neg 99.3%

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

   8. Simplified 99.3%

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

      1. frac-sub 99.9%

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

      2. *-un-lft-identity 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. +-commutative 99.9%

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

       2. +-commutative 99.9%

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

   12. Simplified 99.9%

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

   if -1e-8 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 5.00000000000000019e-26

   1. Initial program 59.9%

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

      1. associate-+l- 59.9%

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

      2. sub-neg 59.9%

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

      3. neg-mul-1 59.9%

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

      4. metadata-eval 59.9%

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

      5. cancel-sign-sub-inv 59.9%

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

      6. +-commutative 59.9%

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

      7. *-lft-identity 59.9%

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

      8. sub-neg 59.9%

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

      9. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in x around inf 98.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 98.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + 2 \cdot \frac{1}{{x}^{3}} \]

      2. metadata-eval 98.4%

         \[\leadsto \frac{\color{blue}{2}}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]

      3. associate-*r/ 98.4%

         \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} \]

      4. metadata-eval 98.4%

         \[\leadsto \frac{2}{{x}^{5}} + \frac{\color{blue}{2}}{{x}^{3}} \]

   6. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}} \]

   if 5.00000000000000019e-26 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. metadata-eval 99.9%

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

      5. cancel-sign-sub-inv 99.9%

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

      6. +-commutative 99.9%

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

      7. *-lft-identity 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-sub 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. div-inv 99.9%

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

      5. metadata-eval 99.9%

         \[\leadsto \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot \color{blue}{0.5}\right) \cdot 1}{\frac{x}{2} \cdot \left(x + -1\right)} \]

      6. div-inv 99.9%

         \[\leadsto \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(x + -1\right)} \]

      7. metadata-eval 99.9%

         \[\leadsto \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\left(x \cdot \color{blue}{0.5}\right) \cdot \left(x + -1\right)} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\left(x \cdot 0.5\right) \cdot \left(x + -1\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 100.0%

         \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{x \cdot 0.5}}{x + -1}} \]

      2. *-rgt-identity 100.0%

         \[\leadsto \frac{1}{1 + x} - \frac{\frac{\left(x + -1\right) - \color{blue}{x \cdot 0.5}}{x \cdot 0.5}}{x + -1} \]

      3. associate--l+ 99.9%

         \[\leadsto \frac{1}{1 + x} - \frac{\frac{\color{blue}{x + \left(-1 - x \cdot 0.5\right)}}{x \cdot 0.5}}{x + -1} \]

   7. Simplified 99.9%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\frac{x + \left(-1 - x \cdot 0.5\right)}{x \cdot 0.5}}{x + -1}} \]
   8. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(-\frac{\frac{x + \left(-1 - x \cdot 0.5\right)}{x \cdot 0.5}}{x + -1}\right)} \]

      2. associate-/l/ 99.9%

         \[\leadsto \frac{1}{1 + x} + \left(-\color{blue}{\frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}}\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(-\frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}\right)} \]
   10. Step-by-step derivation

       1. sub-neg 99.9%

          \[\leadsto \color{blue}{\frac{1}{1 + x} - \frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}} \]

       2. *-lft-identity 99.9%

          \[\leadsto \frac{1}{1 + x} - \color{blue}{1 \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}} \]

       3. +-commutative 99.9%

          \[\leadsto \frac{1}{\color{blue}{x + 1}} - 1 \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)} \]

       4. associate-*r/ 99.9%

          \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1 \cdot \left(x + \left(-1 - x \cdot 0.5\right)\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}} \]

       5. *-commutative 99.9%

          \[\leadsto \frac{1}{x + 1} - \frac{1 \cdot \left(x + \left(-1 - x \cdot 0.5\right)\right)}{\color{blue}{\left(x \cdot 0.5\right) \cdot \left(x + -1\right)}} \]

       6. times-frac 99.9%

          \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1}{x \cdot 0.5} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1}} \]

       7. *-commutative 99.9%

          \[\leadsto \frac{1}{x + 1} - \frac{1}{\color{blue}{0.5 \cdot x}} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1} \]

       8. associate-/r* 99.9%

          \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{\frac{1}{0.5}}{x}} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1} \]

       9. metadata-eval 99.9%

          \[\leadsto \frac{1}{x + 1} - \frac{\color{blue}{2}}{x} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1} \]

       10. associate-+r- 100.0%

           \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\color{blue}{\left(x + -1\right) - x \cdot 0.5}}{x + -1} \]

       11. sub-neg 100.0%

           \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\color{blue}{\left(x + -1\right) + \left(-x \cdot 0.5\right)}}{x + -1} \]

       12. distribute-rgt-neg-in 100.0%

           \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\left(x + -1\right) + \color{blue}{x \cdot \left(-0.5\right)}}{x + -1} \]

       13. metadata-eval 100.0%

           \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\left(x + -1\right) + x \cdot \color{blue}{-0.5}}{x + -1} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\left(x + -1\right) + x \cdot -0.5}{x + -1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(1 + x\right) \cdot \left(x - 2\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{2}{x} \cdot \frac{\left(1 - x\right) - x \cdot -0.5}{x + -1}\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x\right)\\ t_1 := \frac{1}{1 + x}\\ t_2 := \left(t_1 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\frac{t_0 + \left(1 + x\right) \cdot \left(x - 2\right)}{\left(1 + x\right) \cdot t_0}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{2}{x} \cdot \frac{\left(1 - x\right) - x \cdot -0.5}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 x)))
        (t_1 (/ 1.0 (+ 1.0 x)))
        (t_2 (+ (- t_1 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_2 -1e-13)
     (/ (+ t_0 (* (+ 1.0 x) (- x 2.0))) (* (+ 1.0 x) t_0))
     (if (<= t_2 5e-26)
       (/ 2.0 (pow x 3.0))
       (+ t_1 (* (/ 2.0 x) (/ (- (- 1.0 x) (* x -0.5)) (+ x -1.0))))))))

C

double code(double x) {
	double t_0 = x * (1.0 - x);
	double t_1 = 1.0 / (1.0 + x);
	double t_2 = (t_1 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_2 <= -1e-13) {
		tmp = (t_0 + ((1.0 + x) * (x - 2.0))) / ((1.0 + x) * t_0);
	} else if (t_2 <= 5e-26) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = t_1 + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (1.0d0 - x)
    t_1 = 1.0d0 / (1.0d0 + x)
    t_2 = (t_1 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_2 <= (-1d-13)) then
        tmp = (t_0 + ((1.0d0 + x) * (x - 2.0d0))) / ((1.0d0 + x) * t_0)
    else if (t_2 <= 5d-26) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = t_1 + ((2.0d0 / x) * (((1.0d0 - x) - (x * (-0.5d0))) / (x + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (1.0 - x);
	double t_1 = 1.0 / (1.0 + x);
	double t_2 = (t_1 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_2 <= -1e-13) {
		tmp = (t_0 + ((1.0 + x) * (x - 2.0))) / ((1.0 + x) * t_0);
	} else if (t_2 <= 5e-26) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = t_1 + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (1.0 - x)
	t_1 = 1.0 / (1.0 + x)
	t_2 = (t_1 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_2 <= -1e-13:
		tmp = (t_0 + ((1.0 + x) * (x - 2.0))) / ((1.0 + x) * t_0)
	elif t_2 <= 5e-26:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = t_1 + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(1.0 - x))
	t_1 = Float64(1.0 / Float64(1.0 + x))
	t_2 = Float64(Float64(t_1 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_2 <= -1e-13)
		tmp = Float64(Float64(t_0 + Float64(Float64(1.0 + x) * Float64(x - 2.0))) / Float64(Float64(1.0 + x) * t_0));
	elseif (t_2 <= 5e-26)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(t_1 + Float64(Float64(2.0 / x) * Float64(Float64(Float64(1.0 - x) - Float64(x * -0.5)) / Float64(x + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (1.0 - x);
	t_1 = 1.0 / (1.0 + x);
	t_2 = (t_1 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_2 <= -1e-13)
		tmp = (t_0 + ((1.0 + x) * (x - 2.0))) / ((1.0 + x) * t_0);
	elseif (t_2 <= 5e-26)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = t_1 + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-13], N[(N[(t$95$0 + N[(N[(1.0 + x), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-26], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(2.0 / x), $MachinePrecision] * N[(N[(N[(1.0 - x), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -1e-13

   1. Initial program 98.5%

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

      1. associate-+l- 98.5%

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

      2. sub-neg 98.5%

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

      3. neg-mul-1 98.5%

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

      4. metadata-eval 98.5%

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

      5. cancel-sign-sub-inv 98.5%

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

      6. +-commutative 98.5%

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

      7. *-lft-identity 98.5%

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

      8. sub-neg 98.5%

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

      9. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. frac-2neg 98.5%

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

      2. metadata-eval 98.5%

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

      3. frac-sub 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-neg-in 98.5%

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

      6. metadata-eval 98.5%

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

      7. sub-neg 98.5%

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

      8. *-commutative 98.5%

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

      9. neg-mul-1 98.5%

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

      10. +-commutative 98.5%

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

      11. distribute-neg-in 98.5%

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

      12. metadata-eval 98.5%

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

      13. sub-neg 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. neg-mul-1 98.5%

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

      2. unsub-neg 98.5%

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

   8. Simplified 98.5%

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

      1. frac-sub 99.9%

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

      2. *-un-lft-identity 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. +-commutative 99.9%

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

       2. +-commutative 99.9%

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

   12. Simplified 99.9%

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

   if -1e-13 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 5.00000000000000019e-26

   1. Initial program 59.9%

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

      1. associate-+l- 59.9%

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

      2. sub-neg 59.9%

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

      3. neg-mul-1 59.9%

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

      4. metadata-eval 59.9%

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

      5. cancel-sign-sub-inv 59.9%

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

      6. +-commutative 59.9%

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

      7. *-lft-identity 59.9%

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

      8. sub-neg 59.9%

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

      9. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in x around inf 98.4%

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

   if 5.00000000000000019e-26 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. neg-mul-1 99.9%

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

      4. metadata-eval 99.9%

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

      5. cancel-sign-sub-inv 99.9%

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

      6. +-commutative 99.9%

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

      7. *-lft-identity 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-sub 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. div-inv 99.9%

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

      5. metadata-eval 99.9%

         \[\leadsto \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot \color{blue}{0.5}\right) \cdot 1}{\frac{x}{2} \cdot \left(x + -1\right)} \]

      6. div-inv 99.9%

         \[\leadsto \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(x + -1\right)} \]

      7. metadata-eval 99.9%

         \[\leadsto \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\left(x \cdot \color{blue}{0.5}\right) \cdot \left(x + -1\right)} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\left(x \cdot 0.5\right) \cdot \left(x + -1\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 100.0%

         \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{x \cdot 0.5}}{x + -1}} \]

      2. *-rgt-identity 100.0%

         \[\leadsto \frac{1}{1 + x} - \frac{\frac{\left(x + -1\right) - \color{blue}{x \cdot 0.5}}{x \cdot 0.5}}{x + -1} \]

      3. associate--l+ 99.9%

         \[\leadsto \frac{1}{1 + x} - \frac{\frac{\color{blue}{x + \left(-1 - x \cdot 0.5\right)}}{x \cdot 0.5}}{x + -1} \]

   7. Simplified 99.9%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\frac{x + \left(-1 - x \cdot 0.5\right)}{x \cdot 0.5}}{x + -1}} \]
   8. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(-\frac{\frac{x + \left(-1 - x \cdot 0.5\right)}{x \cdot 0.5}}{x + -1}\right)} \]

      2. associate-/l/ 99.9%

         \[\leadsto \frac{1}{1 + x} + \left(-\color{blue}{\frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}}\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(-\frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}\right)} \]
   10. Step-by-step derivation

       1. sub-neg 99.9%

          \[\leadsto \color{blue}{\frac{1}{1 + x} - \frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}} \]

       2. *-lft-identity 99.9%

          \[\leadsto \frac{1}{1 + x} - \color{blue}{1 \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}} \]

       3. +-commutative 99.9%

          \[\leadsto \frac{1}{\color{blue}{x + 1}} - 1 \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)} \]

       4. associate-*r/ 99.9%

          \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1 \cdot \left(x + \left(-1 - x \cdot 0.5\right)\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}} \]

       5. *-commutative 99.9%

          \[\leadsto \frac{1}{x + 1} - \frac{1 \cdot \left(x + \left(-1 - x \cdot 0.5\right)\right)}{\color{blue}{\left(x \cdot 0.5\right) \cdot \left(x + -1\right)}} \]

       6. times-frac 99.9%

          \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1}{x \cdot 0.5} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1}} \]

       7. *-commutative 99.9%

          \[\leadsto \frac{1}{x + 1} - \frac{1}{\color{blue}{0.5 \cdot x}} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1} \]

       8. associate-/r* 99.9%

          \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{\frac{1}{0.5}}{x}} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1} \]

       9. metadata-eval 99.9%

          \[\leadsto \frac{1}{x + 1} - \frac{\color{blue}{2}}{x} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1} \]

       10. associate-+r- 100.0%

           \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\color{blue}{\left(x + -1\right) - x \cdot 0.5}}{x + -1} \]

       11. sub-neg 100.0%

           \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\color{blue}{\left(x + -1\right) + \left(-x \cdot 0.5\right)}}{x + -1} \]

       12. distribute-rgt-neg-in 100.0%

           \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\left(x + -1\right) + \color{blue}{x \cdot \left(-0.5\right)}}{x + -1} \]

       13. metadata-eval 100.0%

           \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\left(x + -1\right) + x \cdot \color{blue}{-0.5}}{x + -1} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\left(x + -1\right) + x \cdot -0.5}{x + -1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(1 + x\right) \cdot \left(x - 2\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{2}{x} \cdot \frac{\left(1 - x\right) - x \cdot -0.5}{x + -1}\\ \end{array} \]

Alternative 3: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + x} + \frac{2}{x} \cdot \frac{\left(1 - x\right) - x \cdot -0.5}{x + -1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ 1.0 (+ 1.0 x)) (* (/ 2.0 x) (/ (- (- 1.0 x) (* x -0.5)) (+ x -1.0)))))

C

double code(double x) {
	return (1.0 / (1.0 + x)) + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (1.0d0 + x)) + ((2.0d0 / x) * (((1.0d0 - x) - (x * (-0.5d0))) / (x + (-1.0d0))))
end function

Java

public static double code(double x) {
	return (1.0 / (1.0 + x)) + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)));
}

Python

def code(x):
	return (1.0 / (1.0 + x)) + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(Float64(2.0 / x) * Float64(Float64(Float64(1.0 - x) - Float64(x * -0.5)) / Float64(x + -1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (1.0 + x)) + ((2.0 / x) * (((1.0 - x) - (x * -0.5)) / (x + -1.0)));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x), $MachinePrecision] * N[(N[(N[(1.0 - x), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. associate-+l- 78.7%

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

   2. sub-neg 78.7%

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

   3. neg-mul-1 78.7%

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

   4. metadata-eval 78.7%

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

   5. cancel-sign-sub-inv 78.7%

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

   6. +-commutative 78.7%

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

   7. *-lft-identity 78.7%

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

   8. sub-neg 78.7%

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

   9. metadata-eval 78.7%

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

3. Simplified 78.7%

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

   1. clear-num 78.7%

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

   2. frac-sub 55.8%

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

   3. *-un-lft-identity 55.8%

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

   4. div-inv 55.8%

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

   5. metadata-eval 55.8%

      \[\leadsto \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot \color{blue}{0.5}\right) \cdot 1}{\frac{x}{2} \cdot \left(x + -1\right)} \]

   6. div-inv 55.8%

      \[\leadsto \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(x + -1\right)} \]

   7. metadata-eval 55.8%

      \[\leadsto \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\left(x \cdot \color{blue}{0.5}\right) \cdot \left(x + -1\right)} \]

5. Applied egg-rr 55.8%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\left(x \cdot 0.5\right) \cdot \left(x + -1\right)}} \]
6. Step-by-step derivation

   1. associate-/r* 78.7%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{x \cdot 0.5}}{x + -1}} \]

   2. *-rgt-identity 78.7%

      \[\leadsto \frac{1}{1 + x} - \frac{\frac{\left(x + -1\right) - \color{blue}{x \cdot 0.5}}{x \cdot 0.5}}{x + -1} \]

   3. associate--l+ 78.7%

      \[\leadsto \frac{1}{1 + x} - \frac{\frac{\color{blue}{x + \left(-1 - x \cdot 0.5\right)}}{x \cdot 0.5}}{x + -1} \]

7. Simplified 78.7%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\frac{x + \left(-1 - x \cdot 0.5\right)}{x \cdot 0.5}}{x + -1}} \]
8. Step-by-step derivation

   1. sub-neg 78.7%

      \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(-\frac{\frac{x + \left(-1 - x \cdot 0.5\right)}{x \cdot 0.5}}{x + -1}\right)} \]

   2. associate-/l/ 55.8%

      \[\leadsto \frac{1}{1 + x} + \left(-\color{blue}{\frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}}\right) \]

9. Applied egg-rr 55.8%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(-\frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}\right)} \]
10. Step-by-step derivation

    1. sub-neg 55.8%

       \[\leadsto \color{blue}{\frac{1}{1 + x} - \frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}} \]

    2. *-lft-identity 55.8%

       \[\leadsto \frac{1}{1 + x} - \color{blue}{1 \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}} \]

    3. +-commutative 55.8%

       \[\leadsto \frac{1}{\color{blue}{x + 1}} - 1 \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)} \]

    4. associate-*r/ 55.8%

       \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1 \cdot \left(x + \left(-1 - x \cdot 0.5\right)\right)}{\left(x + -1\right) \cdot \left(x \cdot 0.5\right)}} \]

    5. *-commutative 55.8%

       \[\leadsto \frac{1}{x + 1} - \frac{1 \cdot \left(x + \left(-1 - x \cdot 0.5\right)\right)}{\color{blue}{\left(x \cdot 0.5\right) \cdot \left(x + -1\right)}} \]

    6. times-frac 78.8%

       \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1}{x \cdot 0.5} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1}} \]

    7. *-commutative 78.8%

       \[\leadsto \frac{1}{x + 1} - \frac{1}{\color{blue}{0.5 \cdot x}} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1} \]

    8. associate-/r* 78.8%

       \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{\frac{1}{0.5}}{x}} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1} \]

    9. metadata-eval 78.8%

       \[\leadsto \frac{1}{x + 1} - \frac{\color{blue}{2}}{x} \cdot \frac{x + \left(-1 - x \cdot 0.5\right)}{x + -1} \]

    10. associate-+r- 78.8%

        \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\color{blue}{\left(x + -1\right) - x \cdot 0.5}}{x + -1} \]

    11. sub-neg 78.8%

        \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\color{blue}{\left(x + -1\right) + \left(-x \cdot 0.5\right)}}{x + -1} \]

    12. distribute-rgt-neg-in 78.8%

        \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\left(x + -1\right) + \color{blue}{x \cdot \left(-0.5\right)}}{x + -1} \]

    13. metadata-eval 78.8%

        \[\leadsto \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\left(x + -1\right) + x \cdot \color{blue}{-0.5}}{x + -1} \]

11. Simplified 78.8%

    \[\leadsto \color{blue}{\frac{1}{x + 1} - \frac{2}{x} \cdot \frac{\left(x + -1\right) + x \cdot -0.5}{x + -1}} \]

12. Final simplification 78.8%

    \[\leadsto \frac{1}{1 + x} + \frac{2}{x} \cdot \frac{\left(1 - x\right) - x \cdot -0.5}{x + -1} \]

Alternative 4: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))))

C

double code(double x) {
	return ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	return ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
}

Python

def code(x):
	return ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

2. Final simplification 78.7%

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

Alternative 5: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + x} + \frac{\frac{2 - x}{x}}{x + -1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ 1.0 (+ 1.0 x)) (/ (/ (- 2.0 x) x) (+ x -1.0))))

C

double code(double x) {
	return (1.0 / (1.0 + x)) + (((2.0 - x) / x) / (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (1.0d0 + x)) + (((2.0d0 - x) / x) / (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / (1.0 + x)) + (((2.0 - x) / x) / (x + -1.0));
}

Python

def code(x):
	return (1.0 / (1.0 + x)) + (((2.0 - x) / x) / (x + -1.0))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(Float64(Float64(2.0 - x) / x) / Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (1.0 + x)) + (((2.0 - x) / x) / (x + -1.0));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 - x), $MachinePrecision] / x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. associate-+l- 78.7%

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

   2. sub-neg 78.7%

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

   3. neg-mul-1 78.7%

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

   4. metadata-eval 78.7%

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

   5. cancel-sign-sub-inv 78.7%

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

   6. +-commutative 78.7%

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

   7. *-lft-identity 78.7%

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

   8. sub-neg 78.7%

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

   9. metadata-eval 78.7%

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

3. Simplified 78.7%

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

   1. frac-sub 55.7%

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

   2. associate-/r* 78.4%

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

   3. *-rgt-identity 78.4%

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

   4. distribute-rgt-in 78.4%

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

   5. metadata-eval 78.4%

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

   6. metadata-eval 78.4%

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

   7. fma-def 78.4%

      \[\leadsto \frac{1}{1 + x} - \frac{\frac{\color{blue}{\mathsf{fma}\left(x, 2, -2\right)} - x}{x}}{x + -1} \]

   8. metadata-eval 78.4%

      \[\leadsto \frac{1}{1 + x} - \frac{\frac{\mathsf{fma}\left(x, 2, \color{blue}{-2}\right) - x}{x}}{x + -1} \]

5. Applied egg-rr 78.4%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 2, -2\right) - x}{x}}{x + -1}} \]

6. Taylor expanded in x around 0 78.7%

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

7. Final simplification 78.7%

   \[\leadsto \frac{1}{1 + x} + \frac{\frac{2 - x}{x}}{x + -1} \]

Alternative 6: 76.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.39\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.39))) (/ -1.0 (* x x)) (- (- x) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.39)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -x - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.39d0))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = -x - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.39)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -x - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.39):
		tmp = -1.0 / (x * x)
	else:
		tmp = -x - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.39))
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(Float64(-x) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.39)))
		tmp = -1.0 / (x * x);
	else
		tmp = -x - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.39]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[((-x) - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.39000000000000001 < x

   1. Initial program 60.5%

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

      1. associate-+l- 60.5%

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

      2. sub-neg 60.5%

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

      3. neg-mul-1 60.5%

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

      4. metadata-eval 60.5%

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

      5. cancel-sign-sub-inv 60.5%

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

      6. +-commutative 60.5%

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

      7. *-lft-identity 60.5%

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

      8. sub-neg 60.5%

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

      9. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in x around inf 58.6%

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

   5. Taylor expanded in x around inf 48.3%

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 48.3%

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

   7. Simplified 48.3%

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

   if -1 < x < 0.39000000000000001

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.8%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. neg-mul-1 98.8%

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

      2. associate-*r/ 98.8%

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

      3. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.39\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) - \frac{2}{x}\\ \end{array} \]

Alternative 7: 76.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.38\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.38))) (/ -1.0 (* x x)) (/ -2.0 x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.38)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.38d0))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = (-2.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.38)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.38):
		tmp = -1.0 / (x * x)
	else:
		tmp = -2.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.38))
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(-2.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.38)))
		tmp = -1.0 / (x * x);
	else
		tmp = -2.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.38]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-2.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.38 < x

   1. Initial program 60.5%

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

      1. associate-+l- 60.5%

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

      2. sub-neg 60.5%

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

      3. neg-mul-1 60.5%

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

      4. metadata-eval 60.5%

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

      5. cancel-sign-sub-inv 60.5%

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

      6. +-commutative 60.5%

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

      7. *-lft-identity 60.5%

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

      8. sub-neg 60.5%

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

      9. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in x around inf 58.6%

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

   5. Taylor expanded in x around inf 48.3%

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 48.3%

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

   7. Simplified 48.3%

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

   if -1 < x < 0.38

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. *-lft-identity 100.0%

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

      8. sub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.38\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 8: 83.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 1 - \left(\frac{2}{x} - -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (- (/ 2.0 x) -1.0)))

C

double code(double x) {
	return 1.0 - ((2.0 / x) - -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - ((2.0d0 / x) - (-1.0d0))
end function

Java

public static double code(double x) {
	return 1.0 - ((2.0 / x) - -1.0);
}

Python

def code(x):
	return 1.0 - ((2.0 / x) - -1.0)

Julia

function code(x)
	return Float64(1.0 - Float64(Float64(2.0 / x) - -1.0))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - ((2.0 / x) - -1.0);
end

Wolfram

code[x_] := N[(1.0 - N[(N[(2.0 / x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. associate-+l- 78.7%

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

   2. sub-neg 78.7%

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

   3. neg-mul-1 78.7%

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

   4. metadata-eval 78.7%

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

   5. cancel-sign-sub-inv 78.7%

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

   6. +-commutative 78.7%

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

   7. *-lft-identity 78.7%

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

   8. sub-neg 78.7%

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

   9. metadata-eval 78.7%

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

3. Simplified 78.7%

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

4. Taylor expanded in x around 0 47.4%

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

5. Taylor expanded in x around 0 76.8%

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

6. Final simplification 76.8%

   \[\leadsto 1 - \left(\frac{2}{x} - -1\right) \]

Alternative 9: 52.4% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ -2.0 x))

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

function code(x)
	return Float64(-2.0 / x)
end

MATLAB

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

Wolfram

code[x_] := N[(-2.0 / x), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. associate-+l- 78.7%

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

   2. sub-neg 78.7%

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

   3. neg-mul-1 78.7%

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

   4. metadata-eval 78.7%

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

   5. cancel-sign-sub-inv 78.7%

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

   6. +-commutative 78.7%

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

   7. *-lft-identity 78.7%

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

   8. sub-neg 78.7%

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

   9. metadata-eval 78.7%

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

3. Simplified 78.7%

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

4. Taylor expanded in x around 0 48.1%

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

5. Final simplification 48.1%

   \[\leadsto \frac{-2}{x} \]

Alternative 10: 3.3% accurate, 15.0× 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 78.7%

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

   1. associate-+l- 78.7%

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

   2. sub-neg 78.7%

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

   3. neg-mul-1 78.7%

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

   4. metadata-eval 78.7%

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

   5. cancel-sign-sub-inv 78.7%

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

   6. +-commutative 78.7%

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

   7. *-lft-identity 78.7%

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

   8. sub-neg 78.7%

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

   9. metadata-eval 78.7%

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

3. Simplified 78.7%

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

4. Taylor expanded in x around 0 47.4%

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

5. Taylor expanded in x around inf 3.2%

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

6. Final simplification 3.2%

   \[\leadsto -1 \]

Developer target: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot \left(x \cdot x - 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* x (- (* x x) 1.0))))

C

double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (x * ((x * x) - 1.0d0))
end function

Java

public static double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Python

def code(x):
	return 2.0 / (x * ((x * x) - 1.0))

Julia

function code(x)
	return Float64(2.0 / Float64(x * Float64(Float64(x * x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (x * ((x * x) - 1.0));
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023257 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))