3frac (problem 3.3.3)

Percentage Accurate: 84.6% → 98.7%

Time: 6.9s

Alternatives: 8

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: 84.6% 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: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -20:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 10^{-31}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{2 - x}{x \cdot x - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 x))) (t_1 (+ (- t_0 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -20.0)
     t_1
     (if (<= t_1 1e-31)
       (/ 2.0 (pow x 3.0))
       (+ t_0 (/ (- 2.0 x) (- (* x x) x)))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -20.0) {
		tmp = t_1;
	} else if (t_1 <= 1e-31) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = t_0 + ((2.0 - x) / ((x * x) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + x)
    t_1 = (t_0 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_1 <= (-20.0d0)) then
        tmp = t_1
    else if (t_1 <= 1d-31) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = t_0 + ((2.0d0 - x) / ((x * x) - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -20.0) {
		tmp = t_1;
	} else if (t_1 <= 1e-31) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = t_0 + ((2.0 - x) / ((x * x) - x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + x)
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_1 <= -20.0:
		tmp = t_1
	elif t_1 <= 1e-31:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = t_0 + ((2.0 - x) / ((x * x) - x))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + x))
	t_1 = Float64(Float64(t_0 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -20.0)
		tmp = t_1;
	elseif (t_1 <= 1e-31)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(t_0 + Float64(Float64(2.0 - x) / Float64(Float64(x * x) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (1.0 + x);
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= -20.0)
		tmp = t_1;
	elseif (t_1 <= 1e-31)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = t_0 + ((2.0 - x) / ((x * x) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20.0], t$95$1, If[LessEqual[t$95$1, 1e-31], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(2.0 - x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - x), $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))) < -20

   1. Initial program 100.0%

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

   if -20 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1e-31

   1. Initial program 71.3%

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

      1. associate-+l- 71.3%

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

      2. sub-neg 71.3%

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

      3. neg-mul-1 71.3%

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

      4. metadata-eval 71.3%

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

      5. cancel-sign-sub-inv 71.3%

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

      6. +-commutative 71.3%

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

      7. *-lft-identity 71.3%

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

      8. sub-neg 71.3%

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

      9. metadata-eval 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in x around inf 97.9%

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

   if 1e-31 < (+.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. frac-2neg 99.9%

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

      2. frac-2neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. frac-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. metadata-eval 99.9%

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

      9. sub-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-neg-in 99.9%

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

      12. metadata-eval 99.9%

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

      13. sub-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. cancel-sign-sub 99.9%

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

      2. *-commutative 99.9%

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

      3. neg-mul-1 99.9%

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

      4. unsub-neg 99.9%

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

      5. sub-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-lft-in 99.9%

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

      8. sqr-neg 99.9%

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

      9. unpow2 99.9%

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

      10. *-rgt-identity 99.9%

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

      11. sub-neg 99.9%

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

      12. unpow2 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -20:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-31}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{2 - x}{x \cdot x - x}\\ \end{array} \]

Alternative 2: 84.6% 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 86.1%

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

2. Final simplification 86.1%

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

Alternative 3: 76.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0 / (x * x);
	} else if (x <= 1.0) {
		tmp = -x - (2.0 / x);
	} else {
		tmp = 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 * x)
    else if (x <= 1.0d0) then
        tmp = -x - (2.0d0 / x)
    else
        tmp = 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 * x);
	} else if (x <= 1.0) {
		tmp = -x - (2.0 / x);
	} else {
		tmp = 1.0 / (x * x);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 75.2%

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

      1. associate-+l- 75.2%

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

      2. sub-neg 75.2%

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

      3. neg-mul-1 75.2%

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

      4. metadata-eval 75.2%

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

      5. cancel-sign-sub-inv 75.2%

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

      6. +-commutative 75.2%

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

      7. *-lft-identity 75.2%

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

      8. sub-neg 75.2%

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

      9. metadata-eval 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around inf 75.3%

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

      1. flip-+ 24.1%

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

      2. sub-neg 24.1%

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

      3. metadata-eval 24.1%

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

      4. distribute-neg-in 24.1%

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

      5. +-commutative 24.1%

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

      6. associate-/r/ 18.6%

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

      7. metadata-eval 18.6%

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

      8. +-commutative 18.6%

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

      9. distribute-neg-in 18.6%

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

      10. metadata-eval 18.6%

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

      11. sub-neg 18.6%

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

   6. Applied egg-rr 18.6%

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

      1. associate-*l/ 19.1%

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

      2. *-lft-identity 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in x around inf 58.3%

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

       1. unpow2 58.3%

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

   11. Simplified 58.3%

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

   if -1 < x < 1

   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 99.2%

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

   5. Taylor expanded in x around 0 99.2%

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

      1. neg-mul-1 99.2%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

   if 1 < x

   1. Initial program 66.9%

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

      1. associate-+l- 66.9%

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

      2. sub-neg 66.9%

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

      3. neg-mul-1 66.9%

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

      4. metadata-eval 66.9%

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

      5. cancel-sign-sub-inv 66.9%

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

      6. +-commutative 66.9%

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

      7. *-lft-identity 66.9%

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

      8. sub-neg 66.9%

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

      9. metadata-eval 66.9%

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

   3. Simplified 66.9%

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

      1. flip-+ 18.4%

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

      2. sub-neg 18.4%

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

      3. metadata-eval 18.4%

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

      4. distribute-neg-in 18.4%

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

      5. +-commutative 18.4%

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

      6. associate-/r/ 18.4%

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

      7. metadata-eval 18.4%

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

      8. +-commutative 18.4%

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

      9. distribute-neg-in 18.4%

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

      10. metadata-eval 18.4%

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

      11. sub-neg 18.4%

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

   5. Applied egg-rr 20.4%

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

   6. Taylor expanded in x around inf 65.8%

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

   7. Taylor expanded in x around inf 54.1%

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

      1. unpow2 54.1%

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

   9. Simplified 54.1%

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

4. Final simplification 78.3%

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

Alternative 4: 76.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		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 <= 1.0d0))) 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 <= 1.0)) {
		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 <= 1.0):
		tmp = -1.0 / (x * x)
	else:
		tmp = -2.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		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 <= 1.0)))
		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, 1.0]], $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 1 < x

   1. Initial program 71.5%

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

      1. associate-+l- 71.5%

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

      2. sub-neg 71.5%

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

      3. neg-mul-1 71.5%

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

      4. metadata-eval 71.5%

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

      5. cancel-sign-sub-inv 71.5%

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

      6. +-commutative 71.5%

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

      7. *-lft-identity 71.5%

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

      8. sub-neg 71.5%

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

      9. metadata-eval 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in x around inf 70.8%

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

      1. flip-+ 21.6%

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

      2. sub-neg 21.6%

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

      3. metadata-eval 21.6%

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

      4. distribute-neg-in 21.6%

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

      5. +-commutative 21.6%

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

      6. associate-/r/ 18.5%

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

      7. metadata-eval 18.5%

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

      8. +-commutative 18.5%

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

      9. distribute-neg-in 18.5%

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

      10. metadata-eval 18.5%

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

      11. sub-neg 18.5%

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

   6. Applied egg-rr 18.5%

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

      1. associate-*l/ 16.5%

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

      2. *-lft-identity 16.5%

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

   8. Simplified 16.5%

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

   9. Taylor expanded in x around inf 55.6%

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

       1. unpow2 55.6%

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

   11. Simplified 55.6%

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

   if -1 < x < 1

   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 99.2%

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

4. Final simplification 77.9%

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

Alternative 5: 76.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0 / (x * x);
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 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 * x)
    else if (x <= 1.0d0) then
        tmp = (-2.0d0) / x
    else
        tmp = 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 * x);
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 1.0 / (x * x);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 75.2%

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

      1. associate-+l- 75.2%

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

      2. sub-neg 75.2%

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

      3. neg-mul-1 75.2%

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

      4. metadata-eval 75.2%

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

      5. cancel-sign-sub-inv 75.2%

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

      6. +-commutative 75.2%

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

      7. *-lft-identity 75.2%

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

      8. sub-neg 75.2%

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

      9. metadata-eval 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around inf 75.3%

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

      1. flip-+ 24.1%

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

      2. sub-neg 24.1%

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

      3. metadata-eval 24.1%

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

      4. distribute-neg-in 24.1%

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

      5. +-commutative 24.1%

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

      6. associate-/r/ 18.6%

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

      7. metadata-eval 18.6%

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

      8. +-commutative 18.6%

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

      9. distribute-neg-in 18.6%

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

      10. metadata-eval 18.6%

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

      11. sub-neg 18.6%

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

   6. Applied egg-rr 18.6%

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

      1. associate-*l/ 19.1%

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

      2. *-lft-identity 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in x around inf 58.3%

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

       1. unpow2 58.3%

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

   11. Simplified 58.3%

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

   if -1 < x < 1

   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 99.2%

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

   if 1 < x

   1. Initial program 66.9%

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

      1. associate-+l- 66.9%

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

      2. sub-neg 66.9%

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

      3. neg-mul-1 66.9%

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

      4. metadata-eval 66.9%

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

      5. cancel-sign-sub-inv 66.9%

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

      6. +-commutative 66.9%

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

      7. *-lft-identity 66.9%

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

      8. sub-neg 66.9%

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

      9. metadata-eval 66.9%

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

   3. Simplified 66.9%

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

      1. flip-+ 18.4%

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

      2. sub-neg 18.4%

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

      3. metadata-eval 18.4%

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

      4. distribute-neg-in 18.4%

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

      5. +-commutative 18.4%

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

      6. associate-/r/ 18.4%

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

      7. metadata-eval 18.4%

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

      8. +-commutative 18.4%

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

      9. distribute-neg-in 18.4%

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

      10. metadata-eval 18.4%

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

      11. sub-neg 18.4%

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

   5. Applied egg-rr 20.4%

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

   6. Taylor expanded in x around inf 65.8%

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

   7. Taylor expanded in x around inf 54.1%

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

      1. unpow2 54.1%

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

   9. Simplified 54.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x}\\ \end{array} \]

Alternative 6: 83.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

   1. associate-+l- 86.1%

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

   2. sub-neg 86.1%

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

   3. neg-mul-1 86.1%

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

   4. metadata-eval 86.1%

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

   5. cancel-sign-sub-inv 86.1%

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

   6. +-commutative 86.1%

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

   7. *-lft-identity 86.1%

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

   8. sub-neg 86.1%

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

   9. metadata-eval 86.1%

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

3. Simplified 86.1%

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

4. Taylor expanded in x around 0 52.3%

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

5. Taylor expanded in x around 0 85.3%

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

6. Final simplification 85.3%

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

Alternative 7: 51.8% 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 86.1%

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

   1. associate-+l- 86.1%

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

   2. sub-neg 86.1%

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

   3. neg-mul-1 86.1%

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

   4. metadata-eval 86.1%

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

   5. cancel-sign-sub-inv 86.1%

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

   6. +-commutative 86.1%

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

   7. *-lft-identity 86.1%

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

   8. sub-neg 86.1%

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

   9. metadata-eval 86.1%

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

3. Simplified 86.1%

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

4. Taylor expanded in x around 0 53.3%

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

5. Final simplification 53.3%

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

Alternative 8: 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 86.1%

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

   1. associate-+l- 86.1%

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

   2. sub-neg 86.1%

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

   3. neg-mul-1 86.1%

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

   4. metadata-eval 86.1%

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

   5. cancel-sign-sub-inv 86.1%

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

   6. +-commutative 86.1%

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

   7. *-lft-identity 86.1%

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

   8. sub-neg 86.1%

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

   9. metadata-eval 86.1%

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

3. Simplified 86.1%

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

4. Taylor expanded in x around 0 52.3%

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

5. Taylor expanded in x around inf 3.4%

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

6. Final simplification 3.4%

   \[\leadsto -1 \]

Developer target: 99.6% 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 2023200 
(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))))