3frac (problem 3.3.3)

Percentage Accurate: 84.3% → 98.7%

Time: 8.5s

Alternatives: 9

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: 84.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - x \cdot 0.5\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot t_0 - \left(1 + x\right) \cdot \left(1 - x \cdot 0.5\right)}{x \cdot \left(\left(1 + x\right) \cdot t_0\right)}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (* x 0.5)))
        (t_1 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -1e-12)
     (/ (- (* x t_0) (* (+ 1.0 x) (- 1.0 (* x 0.5)))) (* x (* (+ 1.0 x) t_0)))
     (if (<= t_1 0.0) (* 2.0 (pow x -3.0)) (- (* x -2.0) (/ 2.0 x))))))

C

double code(double x) {
	double t_0 = 0.5 - (x * 0.5);
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-12) {
		tmp = ((x * t_0) - ((1.0 + x) * (1.0 - (x * 0.5)))) / (x * ((1.0 + x) * t_0));
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * pow(x, -3.0);
	} else {
		tmp = (x * -2.0) - (2.0 / 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 = 0.5d0 - (x * 0.5d0)
    t_1 = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_1 <= (-1d-12)) then
        tmp = ((x * t_0) - ((1.0d0 + x) * (1.0d0 - (x * 0.5d0)))) / (x * ((1.0d0 + x) * t_0))
    else if (t_1 <= 0.0d0) then
        tmp = 2.0d0 * (x ** (-3.0d0))
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 - (x * 0.5);
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-12) {
		tmp = ((x * t_0) - ((1.0 + x) * (1.0 - (x * 0.5)))) / (x * ((1.0 + x) * t_0));
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * Math.pow(x, -3.0);
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 - (x * 0.5)
	t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_1 <= -1e-12:
		tmp = ((x * t_0) - ((1.0 + x) * (1.0 - (x * 0.5)))) / (x * ((1.0 + x) * t_0))
	elif t_1 <= 0.0:
		tmp = 2.0 * math.pow(x, -3.0)
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(x * 0.5))
	t_1 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -1e-12)
		tmp = Float64(Float64(Float64(x * t_0) - Float64(Float64(1.0 + x) * Float64(1.0 - Float64(x * 0.5)))) / Float64(x * Float64(Float64(1.0 + x) * t_0)));
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 * (x ^ -3.0));
	else
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 - (x * 0.5);
	t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= -1e-12)
		tmp = ((x * t_0) - ((1.0 + x) * (1.0 - (x * 0.5)))) / (x * ((1.0 + x) * t_0));
	elseif (t_1 <= 0.0)
		tmp = 2.0 * (x ^ -3.0);
	else
		tmp = (x * -2.0) - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e-12], N[(N[(N[(x * t$95$0), $MachinePrecision] - N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[(1.0 + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $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))) < -9.9999999999999998e-13

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. +-commutative 99.2%

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

      3. sub-neg 99.2%

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

      4. distribute-neg-frac 99.2%

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

      5. metadata-eval 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. associate-/r* 99.2%

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

      9. metadata-eval 99.2%

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

      10. neg-mul-1 99.2%

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

      11. sub0-neg 99.2%

          \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} + \frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

      12. associate-+l- 99.2%

          \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} + \frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) \]

      13. neg-sub0 99.2%

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

      14. +-commutative 99.2%

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

      15. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Applied egg-rr 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-*l* 97.5%

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

      3. metadata-eval 97.5%

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

      4. +-commutative 97.5%

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

      5. *-commutative 97.5%

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

      6. associate-*l* 97.5%

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

      7. +-commutative 97.5%

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

   7. Simplified 97.5%

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

   9. Applied egg-rr 100.0%

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

       1. neg-sub0 100.0%

          \[\leadsto \frac{x \cdot \color{blue}{\left(0 - \left(-0.5 + x \cdot 0.5\right)\right)} - \left(x + 1\right) \cdot \left(-\left(x \cdot 0.5 + -1\right)\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(-\left(-0.5 + x \cdot 0.5\right)\right)\right)} \]

       2. associate--r+ 100.0%

          \[\leadsto \frac{x \cdot \color{blue}{\left(\left(0 - -0.5\right) - x \cdot 0.5\right)} - \left(x + 1\right) \cdot \left(-\left(x \cdot 0.5 + -1\right)\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(-\left(-0.5 + x \cdot 0.5\right)\right)\right)} \]

       3. metadata-eval 100.0%

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

       4. neg-sub0 100.0%

          \[\leadsto \frac{x \cdot \left(0.5 - x \cdot 0.5\right) - \left(x + 1\right) \cdot \color{blue}{\left(0 - \left(x \cdot 0.5 + -1\right)\right)}}{\left(x + 1\right) \cdot \left(x \cdot \left(-\left(-0.5 + x \cdot 0.5\right)\right)\right)} \]

       5. +-commutative 100.0%

          \[\leadsto \frac{x \cdot \left(0.5 - x \cdot 0.5\right) - \left(x + 1\right) \cdot \left(0 - \color{blue}{\left(-1 + x \cdot 0.5\right)}\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(-\left(-0.5 + x \cdot 0.5\right)\right)\right)} \]

       6. associate--r+ 100.0%

          \[\leadsto \frac{x \cdot \left(0.5 - x \cdot 0.5\right) - \left(x + 1\right) \cdot \color{blue}{\left(\left(0 - -1\right) - x \cdot 0.5\right)}}{\left(x + 1\right) \cdot \left(x \cdot \left(-\left(-0.5 + x \cdot 0.5\right)\right)\right)} \]

       7. metadata-eval 100.0%

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

       8. *-commutative 100.0%

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

       9. associate-*l* 100.0%

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

       10. neg-sub0 100.0%

           \[\leadsto \frac{x \cdot \left(0.5 - x \cdot 0.5\right) - \left(x + 1\right) \cdot \left(1 - x \cdot 0.5\right)}{x \cdot \left(\color{blue}{\left(0 - \left(-0.5 + x \cdot 0.5\right)\right)} \cdot \left(x + 1\right)\right)} \]

       11. associate--r+ 100.0%

           \[\leadsto \frac{x \cdot \left(0.5 - x \cdot 0.5\right) - \left(x + 1\right) \cdot \left(1 - x \cdot 0.5\right)}{x \cdot \left(\color{blue}{\left(\left(0 - -0.5\right) - x \cdot 0.5\right)} \cdot \left(x + 1\right)\right)} \]

       12. metadata-eval 100.0%

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

   11. Simplified 100.0%

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

   if -9.9999999999999998e-13 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around inf 99.2%

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

      1. div-inv 99.2%

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

      2. *-commutative 99.2%

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

      3. pow-flip 100.0%

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

      4. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \left(0.5 - x \cdot 0.5\right) - \left(1 + x\right) \cdot \left(1 - x \cdot 0.5\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(0.5 - x \cdot 0.5\right)\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 2: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + x} + \frac{x \cdot 0.5 + -1}{x} \cdot \frac{-1}{x \cdot 0.5 + -0.5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-+l- 84.3%

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

   2. +-commutative 84.3%

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

   3. sub-neg 84.3%

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

   4. distribute-neg-frac 84.3%

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

   5. metadata-eval 84.3%

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

   6. metadata-eval 84.3%

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

   7. metadata-eval 84.3%

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

   8. associate-/r* 84.3%

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

   9. metadata-eval 84.3%

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

   10. neg-mul-1 84.3%

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

   11. sub0-neg 84.3%

       \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} + \frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) \]

   12. associate-+l- 84.3%

       \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} + \frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) \]

   13. neg-sub0 84.3%

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

   14. +-commutative 84.3%

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

   15. unsub-neg 84.3%

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

3. Simplified 84.3%

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

5. Applied egg-rr 57.0%

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

   1. *-commutative 57.0%

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

   2. associate-*l* 57.0%

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

   3. metadata-eval 57.0%

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

   4. +-commutative 57.0%

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

   5. *-commutative 57.0%

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

   6. associate-*l* 57.0%

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

   7. +-commutative 57.0%

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

7. Simplified 57.0%

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

   1. associate-/r* 83.1%

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

   2. div-inv 83.1%

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

9. Applied egg-rr 84.3%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{x \cdot 0.5 + -1}{x} \cdot \frac{1}{-0.5 + x \cdot 0.5}} \]

10. Final simplification 84.3%

    \[\leadsto \frac{1}{1 + x} + \frac{x \cdot 0.5 + -1}{x} \cdot \frac{-1}{x \cdot 0.5 + -0.5} \]

Alternative 3: 84.3% 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 84.3%

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

2. Final simplification 84.3%

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

Alternative 4: 83.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 1.0) (- (* x -2.0) (/ 2.0 x)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 1.0d0) then
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 1.0:
		tmp = (x * -2.0) - (2.0 / x)
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 1.0], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 68.1%

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

   3. Simplified 68.1%

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

   5. Applied egg-rr 26.3%

      \[\leadsto \frac{-2}{x} + \color{blue}{\left(\left(\frac{1}{1 + x} + \frac{1}{1 + x}\right) + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+l+ 24.1%

         \[\leadsto \frac{-2}{x} + \color{blue}{\left(\frac{1}{1 + x} + \left(\frac{1}{1 + x} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)} \]

      2. expm1-log1p 24.1%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + x}\right)\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

      3. expm1-def 2.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - 1\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

      4. associate-+l- 2.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)}\right) \]

      5. fma-udef 2.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) \cdot {\left(1 + x\right)}^{-0.5} + \frac{1}{1 + x}\right)}\right)\right)\right) \]

      6. distribute-lft-neg-out 2.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\color{blue}{\left(-{\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}\right)} + \frac{1}{1 + x}\right)\right)\right)\right) \]

      7. pow-sqr 5.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{{\left(1 + x\right)}^{\left(2 \cdot -0.5\right)}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

      8. metadata-eval 5.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-{\left(1 + x\right)}^{\color{blue}{-1}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

      9. unpow-1 5.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{\frac{1}{1 + x}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

      10. +-commutative 5.2%

          \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} + \left(-\frac{1}{1 + x}\right)\right)}\right)\right)\right) \]

      11. sub-neg 5.2%

          \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} - \frac{1}{1 + x}\right)}\right)\right)\right) \]

      12. +-inverses 5.2%

          \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{0}\right)\right)\right) \]

      13. metadata-eval 5.2%

          \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \color{blue}{1}\right)\right) \]

   7. Simplified 67.2%

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

      1. clear-num 67.2%

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

      2. frac-2neg 67.2%

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

      3. metadata-eval 67.2%

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

      4. frac-add 67.2%

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

      5. *-un-lft-identity 67.2%

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

      6. neg-sub0 67.2%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(x + 1\right)\right)} + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

      7. +-commutative 67.2%

         \[\leadsto \frac{\left(0 - \color{blue}{\left(1 + x\right)}\right) + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

      8. associate--r+ 67.2%

         \[\leadsto \frac{\color{blue}{\left(\left(0 - 1\right) - x\right)} + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

      9. metadata-eval 67.2%

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

      10. div-inv 67.2%

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

      11. metadata-eval 67.2%

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

      12. div-inv 67.2%

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

      13. metadata-eval 67.2%

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

      14. neg-sub0 67.2%

          \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}} \]

      15. +-commutative 67.2%

          \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)} \]

      16. associate--r+ 67.2%

          \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}} \]

      17. metadata-eval 67.2%

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

   9. Applied egg-rr 67.2%

      \[\leadsto \color{blue}{\frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* 67.2%

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

       2. *-commutative 67.2%

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

       3. associate-+l- 54.5%

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

       4. *-commutative 54.5%

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

       5. associate-*l* 54.5%

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

       6. metadata-eval 54.5%

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

   11. Simplified 54.5%

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

   13. Applied egg-rr 67.0%

       \[\leadsto \color{blue}{0} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 82.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

3. Simplified 84.3%

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

5. Applied egg-rr 63.0%

   \[\leadsto \frac{-2}{x} + \color{blue}{\left(\left(\frac{1}{1 + x} + \frac{1}{1 + x}\right) + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)} \]
6. Step-by-step derivation

   1. associate-+l+ 61.9%

      \[\leadsto \frac{-2}{x} + \color{blue}{\left(\frac{1}{1 + x} + \left(\frac{1}{1 + x} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)} \]

   2. expm1-log1p 61.9%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + x}\right)\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

   3. expm1-def 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - 1\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

   4. associate-+l- 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)}\right) \]

   5. fma-udef 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) \cdot {\left(1 + x\right)}^{-0.5} + \frac{1}{1 + x}\right)}\right)\right)\right) \]

   6. distribute-lft-neg-out 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\color{blue}{\left(-{\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}\right)} + \frac{1}{1 + x}\right)\right)\right)\right) \]

   7. pow-sqr 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{{\left(1 + x\right)}^{\left(2 \cdot -0.5\right)}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   8. metadata-eval 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-{\left(1 + x\right)}^{\color{blue}{-1}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   9. unpow-1 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{\frac{1}{1 + x}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   10. +-commutative 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} + \left(-\frac{1}{1 + x}\right)\right)}\right)\right)\right) \]

   11. sub-neg 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} - \frac{1}{1 + x}\right)}\right)\right)\right) \]

   12. +-inverses 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{0}\right)\right)\right) \]

   13. metadata-eval 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \color{blue}{1}\right)\right) \]

7. Simplified 83.1%

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

8. Final simplification 83.1%

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

Alternative 6: 82.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 1.0) (/ -2.0 x) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 1.0d0) then
        tmp = (-2.0d0) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 1.0:
		tmp = -2.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 1.0], N[(-2.0 / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 68.1%

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

   3. Simplified 68.1%

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

   5. Applied egg-rr 26.3%

      \[\leadsto \frac{-2}{x} + \color{blue}{\left(\left(\frac{1}{1 + x} + \frac{1}{1 + x}\right) + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+l+ 24.1%

         \[\leadsto \frac{-2}{x} + \color{blue}{\left(\frac{1}{1 + x} + \left(\frac{1}{1 + x} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)} \]

      2. expm1-log1p 24.1%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + x}\right)\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

      3. expm1-def 2.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - 1\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

      4. associate-+l- 2.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)}\right) \]

      5. fma-udef 2.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) \cdot {\left(1 + x\right)}^{-0.5} + \frac{1}{1 + x}\right)}\right)\right)\right) \]

      6. distribute-lft-neg-out 2.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\color{blue}{\left(-{\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}\right)} + \frac{1}{1 + x}\right)\right)\right)\right) \]

      7. pow-sqr 5.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{{\left(1 + x\right)}^{\left(2 \cdot -0.5\right)}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

      8. metadata-eval 5.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-{\left(1 + x\right)}^{\color{blue}{-1}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

      9. unpow-1 5.2%

         \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{\frac{1}{1 + x}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

      10. +-commutative 5.2%

          \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} + \left(-\frac{1}{1 + x}\right)\right)}\right)\right)\right) \]

      11. sub-neg 5.2%

          \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} - \frac{1}{1 + x}\right)}\right)\right)\right) \]

      12. +-inverses 5.2%

          \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{0}\right)\right)\right) \]

      13. metadata-eval 5.2%

          \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \color{blue}{1}\right)\right) \]

   7. Simplified 67.2%

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

      1. clear-num 67.2%

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

      2. frac-2neg 67.2%

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

      3. metadata-eval 67.2%

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

      4. frac-add 67.2%

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

      5. *-un-lft-identity 67.2%

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

      6. neg-sub0 67.2%

         \[\leadsto \frac{\color{blue}{\left(0 - \left(x + 1\right)\right)} + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

      7. +-commutative 67.2%

         \[\leadsto \frac{\left(0 - \color{blue}{\left(1 + x\right)}\right) + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

      8. associate--r+ 67.2%

         \[\leadsto \frac{\color{blue}{\left(\left(0 - 1\right) - x\right)} + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

      9. metadata-eval 67.2%

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

      10. div-inv 67.2%

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

      11. metadata-eval 67.2%

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

      12. div-inv 67.2%

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

      13. metadata-eval 67.2%

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

      14. neg-sub0 67.2%

          \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}} \]

      15. +-commutative 67.2%

          \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)} \]

      16. associate--r+ 67.2%

          \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}} \]

      17. metadata-eval 67.2%

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

   9. Applied egg-rr 67.2%

      \[\leadsto \color{blue}{\frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* 67.2%

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

       2. *-commutative 67.2%

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

       3. associate-+l- 54.5%

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

       4. *-commutative 54.5%

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

       5. associate-*l* 54.5%

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

       6. metadata-eval 54.5%

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

   11. Simplified 54.5%

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

   13. Applied egg-rr 67.0%

       \[\leadsto \color{blue}{0} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 3.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -8 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -8.0)

C

double code(double x) {
	return -8.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -8.0d0
end function

Java

public static double code(double x) {
	return -8.0;
}

Python

def code(x):
	return -8.0

Julia

function code(x)
	return -8.0
end

MATLAB

function tmp = code(x)
	tmp = -8.0;
end

Wolfram

code[x_] := -8.0

Derivation

1. Initial program 84.3%

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

3. Simplified 84.3%

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

5. Applied egg-rr 63.0%

   \[\leadsto \frac{-2}{x} + \color{blue}{\left(\left(\frac{1}{1 + x} + \frac{1}{1 + x}\right) + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)} \]
6. Step-by-step derivation

   1. associate-+l+ 61.9%

      \[\leadsto \frac{-2}{x} + \color{blue}{\left(\frac{1}{1 + x} + \left(\frac{1}{1 + x} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)} \]

   2. expm1-log1p 61.9%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + x}\right)\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

   3. expm1-def 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - 1\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

   4. associate-+l- 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)}\right) \]

   5. fma-udef 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) \cdot {\left(1 + x\right)}^{-0.5} + \frac{1}{1 + x}\right)}\right)\right)\right) \]

   6. distribute-lft-neg-out 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\color{blue}{\left(-{\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}\right)} + \frac{1}{1 + x}\right)\right)\right)\right) \]

   7. pow-sqr 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{{\left(1 + x\right)}^{\left(2 \cdot -0.5\right)}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   8. metadata-eval 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-{\left(1 + x\right)}^{\color{blue}{-1}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   9. unpow-1 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{\frac{1}{1 + x}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   10. +-commutative 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} + \left(-\frac{1}{1 + x}\right)\right)}\right)\right)\right) \]

   11. sub-neg 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} - \frac{1}{1 + x}\right)}\right)\right)\right) \]

   12. +-inverses 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{0}\right)\right)\right) \]

   13. metadata-eval 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \color{blue}{1}\right)\right) \]

7. Simplified 83.1%

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

   1. clear-num 83.1%

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

   2. frac-2neg 83.1%

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

   3. metadata-eval 83.1%

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

   4. frac-add 83.1%

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

   5. *-un-lft-identity 83.1%

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

   6. neg-sub0 83.1%

      \[\leadsto \frac{\color{blue}{\left(0 - \left(x + 1\right)\right)} + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

   7. +-commutative 83.1%

      \[\leadsto \frac{\left(0 - \color{blue}{\left(1 + x\right)}\right) + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

   8. associate--r+ 83.1%

      \[\leadsto \frac{\color{blue}{\left(\left(0 - 1\right) - x\right)} + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

   9. metadata-eval 83.1%

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

   10. div-inv 83.1%

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

   11. metadata-eval 83.1%

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

   12. div-inv 83.1%

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

   13. metadata-eval 83.1%

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

   14. neg-sub0 83.1%

       \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}} \]

   15. +-commutative 83.1%

       \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)} \]

   16. associate--r+ 83.1%

       \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}} \]

   17. metadata-eval 83.1%

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

9. Applied egg-rr 83.1%

   \[\leadsto \color{blue}{\frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)}} \]
10. Step-by-step derivation

    1. associate-/r* 83.1%

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

    2. *-commutative 83.1%

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

    3. associate-+l- 76.9%

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

    4. *-commutative 76.9%

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

    5. associate-*l* 76.9%

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

    6. metadata-eval 76.9%

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

11. Simplified 76.9%

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

13. Applied egg-rr 3.2%

    \[\leadsto \color{blue}{-8} \]

14. Final simplification 3.2%

    \[\leadsto -8 \]

Alternative 8: 3.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -0.5)

C

double code(double x) {
	return -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.5d0
end function

Java

public static double code(double x) {
	return -0.5;
}

Python

def code(x):
	return -0.5

Julia

function code(x)
	return -0.5
end

MATLAB

function tmp = code(x)
	tmp = -0.5;
end

Wolfram

code[x_] := -0.5

Derivation

1. Initial program 84.3%

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

3. Simplified 84.3%

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

5. Applied egg-rr 63.0%

   \[\leadsto \frac{-2}{x} + \color{blue}{\left(\left(\frac{1}{1 + x} + \frac{1}{1 + x}\right) + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)} \]
6. Step-by-step derivation

   1. associate-+l+ 61.9%

      \[\leadsto \frac{-2}{x} + \color{blue}{\left(\frac{1}{1 + x} + \left(\frac{1}{1 + x} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)} \]

   2. expm1-log1p 61.9%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + x}\right)\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

   3. expm1-def 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - 1\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

   4. associate-+l- 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)}\right) \]

   5. fma-udef 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) \cdot {\left(1 + x\right)}^{-0.5} + \frac{1}{1 + x}\right)}\right)\right)\right) \]

   6. distribute-lft-neg-out 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\color{blue}{\left(-{\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}\right)} + \frac{1}{1 + x}\right)\right)\right)\right) \]

   7. pow-sqr 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{{\left(1 + x\right)}^{\left(2 \cdot -0.5\right)}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   8. metadata-eval 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-{\left(1 + x\right)}^{\color{blue}{-1}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   9. unpow-1 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{\frac{1}{1 + x}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   10. +-commutative 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} + \left(-\frac{1}{1 + x}\right)\right)}\right)\right)\right) \]

   11. sub-neg 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} - \frac{1}{1 + x}\right)}\right)\right)\right) \]

   12. +-inverses 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{0}\right)\right)\right) \]

   13. metadata-eval 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \color{blue}{1}\right)\right) \]

7. Simplified 83.1%

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

   1. clear-num 83.1%

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

   2. frac-2neg 83.1%

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

   3. metadata-eval 83.1%

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

   4. frac-add 83.1%

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

   5. *-un-lft-identity 83.1%

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

   6. neg-sub0 83.1%

      \[\leadsto \frac{\color{blue}{\left(0 - \left(x + 1\right)\right)} + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

   7. +-commutative 83.1%

      \[\leadsto \frac{\left(0 - \color{blue}{\left(1 + x\right)}\right) + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

   8. associate--r+ 83.1%

      \[\leadsto \frac{\color{blue}{\left(\left(0 - 1\right) - x\right)} + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

   9. metadata-eval 83.1%

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

   10. div-inv 83.1%

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

   11. metadata-eval 83.1%

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

   12. div-inv 83.1%

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

   13. metadata-eval 83.1%

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

   14. neg-sub0 83.1%

       \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}} \]

   15. +-commutative 83.1%

       \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)} \]

   16. associate--r+ 83.1%

       \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}} \]

   17. metadata-eval 83.1%

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

9. Applied egg-rr 83.1%

   \[\leadsto \color{blue}{\frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)}} \]
10. Step-by-step derivation

    1. associate-/r* 83.1%

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

    2. *-commutative 83.1%

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

    3. associate-+l- 76.9%

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

    4. *-commutative 76.9%

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

    5. associate-*l* 76.9%

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

    6. metadata-eval 76.9%

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

11. Simplified 76.9%

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

13. Applied egg-rr 3.2%

    \[\leadsto \color{blue}{-0.5} \]

14. Final simplification 3.2%

    \[\leadsto -0.5 \]

Alternative 9: 34.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 84.3%

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

3. Simplified 84.3%

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

5. Applied egg-rr 63.0%

   \[\leadsto \frac{-2}{x} + \color{blue}{\left(\left(\frac{1}{1 + x} + \frac{1}{1 + x}\right) + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)} \]
6. Step-by-step derivation

   1. associate-+l+ 61.9%

      \[\leadsto \frac{-2}{x} + \color{blue}{\left(\frac{1}{1 + x} + \left(\frac{1}{1 + x} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)} \]

   2. expm1-log1p 61.9%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + x}\right)\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

   3. expm1-def 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - 1\right)} + \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right) \]

   4. associate-+l- 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \mathsf{fma}\left(-{\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}, \frac{1}{1 + x}\right)\right)\right)}\right) \]

   5. fma-udef 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) \cdot {\left(1 + x\right)}^{-0.5} + \frac{1}{1 + x}\right)}\right)\right)\right) \]

   6. distribute-lft-neg-out 51.1%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\color{blue}{\left(-{\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}\right)} + \frac{1}{1 + x}\right)\right)\right)\right) \]

   7. pow-sqr 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{{\left(1 + x\right)}^{\left(2 \cdot -0.5\right)}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   8. metadata-eval 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-{\left(1 + x\right)}^{\color{blue}{-1}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   9. unpow-1 52.6%

      \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \left(\left(-\color{blue}{\frac{1}{1 + x}}\right) + \frac{1}{1 + x}\right)\right)\right)\right) \]

   10. +-commutative 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} + \left(-\frac{1}{1 + x}\right)\right)}\right)\right)\right) \]

   11. sub-neg 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{\left(\frac{1}{1 + x} - \frac{1}{1 + x}\right)}\right)\right)\right) \]

   12. +-inverses 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \left(1 - \color{blue}{0}\right)\right)\right) \]

   13. metadata-eval 52.6%

       \[\leadsto \frac{-2}{x} + \left(\frac{1}{1 + x} + \left(e^{\mathsf{log1p}\left(\frac{1}{1 + x}\right)} - \color{blue}{1}\right)\right) \]

7. Simplified 83.1%

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

   1. clear-num 83.1%

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

   2. frac-2neg 83.1%

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

   3. metadata-eval 83.1%

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

   4. frac-add 83.1%

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

   5. *-un-lft-identity 83.1%

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

   6. neg-sub0 83.1%

      \[\leadsto \frac{\color{blue}{\left(0 - \left(x + 1\right)\right)} + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

   7. +-commutative 83.1%

      \[\leadsto \frac{\left(0 - \color{blue}{\left(1 + x\right)}\right) + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

   8. associate--r+ 83.1%

      \[\leadsto \frac{\color{blue}{\left(\left(0 - 1\right) - x\right)} + \frac{x}{-2} \cdot -2}{\frac{x}{-2} \cdot \left(-\left(x + 1\right)\right)} \]

   9. metadata-eval 83.1%

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

   10. div-inv 83.1%

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

   11. metadata-eval 83.1%

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

   12. div-inv 83.1%

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

   13. metadata-eval 83.1%

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

   14. neg-sub0 83.1%

       \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}} \]

   15. +-commutative 83.1%

       \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)} \]

   16. associate--r+ 83.1%

       \[\leadsto \frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}} \]

   17. metadata-eval 83.1%

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

9. Applied egg-rr 83.1%

   \[\leadsto \color{blue}{\frac{\left(-1 - x\right) + \left(x \cdot -0.5\right) \cdot -2}{\left(x \cdot -0.5\right) \cdot \left(-1 - x\right)}} \]
10. Step-by-step derivation

    1. associate-/r* 83.1%

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

    2. *-commutative 83.1%

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

    3. associate-+l- 76.9%

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

    4. *-commutative 76.9%

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

    5. associate-*l* 76.9%

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

    6. metadata-eval 76.9%

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

11. Simplified 76.9%

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

13. Applied egg-rr 34.1%

    \[\leadsto \color{blue}{0} \]

14. Final simplification 34.1%

    \[\leadsto 0 \]

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 2023305 
(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))))