3frac (problem 3.3.3)

Percentage Accurate: 85.0% → 99.9%

Time: 7.3s

Alternatives: 10

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. +-commutative 83.7%

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

   2. sub-neg 83.7%

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

   3. metadata-eval 83.7%

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

   4. frac-sub 55.3%

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

   5. frac-add 55.8%

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

   6. *-un-lft-identity 55.8%

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

   7. *-commutative 55.8%

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

   8. +-commutative 55.8%

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

   9. /-rgt-identity 55.8%

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

   10. *-un-lft-identity 55.8%

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

   11. /-rgt-identity 55.8%

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

   12. +-commutative 55.8%

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

   13. *-commutative 55.8%

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

   14. +-commutative 55.8%

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

3. Applied egg-rr 55.8%

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

4. Taylor expanded in x around 0 99.7%

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

   1. expm1-log1p-u 71.5%

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

   2. expm1-udef 55.2%

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

   3. +-commutative 55.2%

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

   4. distribute-lft-in 55.2%

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

   5. *-rgt-identity 55.2%

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

   6. fma-def 55.2%

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

6. Applied egg-rr 55.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(x + -1\right) \cdot \mathsf{fma}\left(x, x, x\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 71.5%

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

   2. expm1-log1p 99.7%

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

   3. associate-/r* 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{2}{\left(x + -1\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.85)))
   (/ 2.0 (* (+ x -1.0) (* x x)))
   (- (* x -2.0) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.85)) {
		tmp = 2.0 / ((x + -1.0) * (x * x));
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.85d0))) then
        tmp = 2.0d0 / ((x + (-1.0d0)) * (x * x))
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.85)) {
		tmp = 2.0 / ((x + -1.0) * (x * x));
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.85):
		tmp = 2.0 / ((x + -1.0) * (x * x))
	else:
		tmp = (x * -2.0) - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.85))
		tmp = Float64(2.0 / Float64(Float64(x + -1.0) * Float64(x * x)));
	else
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.85)))
		tmp = 2.0 / ((x + -1.0) * (x * x));
	else
		tmp = (x * -2.0) - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.85]], $MachinePrecision]], N[(2.0 / N[(N[(x + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.849999999999999978 < x

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

      2. sub-neg 69.9%

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

      3. metadata-eval 69.9%

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

      4. frac-sub 17.7%

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

      5. frac-add 18.6%

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

      6. *-un-lft-identity 18.6%

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

      7. *-commutative 18.6%

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

      8. +-commutative 18.6%

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

      9. /-rgt-identity 18.6%

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

      10. *-un-lft-identity 18.6%

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

      11. /-rgt-identity 18.6%

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

      12. +-commutative 18.6%

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

      13. *-commutative 18.6%

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

      14. +-commutative 18.6%

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

   3. Applied egg-rr 18.6%

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

   4. Taylor expanded in x around 0 99.5%

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

   5. Taylor expanded in x around inf 96.6%

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

      1. unpow2 96.6%

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

   7. Simplified 96.6%

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

   if -1 < x < 0.849999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

   4. Simplified 98.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{2}{\left(x + -1\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 3: 84.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2}{x} - \frac{2}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ 2.0 x) (/ 2.0 x))
   (if (<= x 0.65)
     (- (* x -2.0) (/ 2.0 x))
     (+ (/ -1.0 x) (/ 1.0 (+ x -1.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / x) - (2.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (2.0d0 / x) - (2.0d0 / x)
    else if (x <= 0.65d0) then
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    else
        tmp = ((-1.0d0) / x) + (1.0d0 / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / x) - (2.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (2.0 / x) - (2.0 / x)
	elif x <= 0.65:
		tmp = (x * -2.0) - (2.0 / x)
	else:
		tmp = (-1.0 / x) + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(2.0 / x) - Float64(2.0 / x));
	elseif (x <= 0.65)
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	else
		tmp = Float64(Float64(-1.0 / x) + Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (2.0 / x) - (2.0 / x);
	elseif (x <= 0.65)
		tmp = (x * -2.0) - (2.0 / x);
	else
		tmp = (-1.0 / x) + (1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(2.0 / x), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.65], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. sub-neg 69.6%

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

      3. metadata-eval 69.6%

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

      4. associate-+r- 69.7%

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

      5. frac-2neg 69.7%

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

      6. metadata-eval 69.7%

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

      7. +-commutative 69.7%

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

      8. distribute-neg-in 69.7%

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

      9. metadata-eval 69.7%

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

      10. sub-neg 69.7%

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

      11. +-commutative 69.7%

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

   3. Applied egg-rr 69.7%

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

   4. Taylor expanded in x around inf 67.5%

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

   if -1 < x < 0.650000000000000022

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

   4. Simplified 98.0%

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

   if 0.650000000000000022 < x

   1. Initial program 70.3%

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

   2. Taylor expanded in x around inf 66.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2}{x} - \frac{2}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + -1}\\ \end{array} \]

Alternative 4: 83.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(2.0 / x) - Float64(2.0 / x));
	else
		tmp = Float64(Float64(x * -2.0) - 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 = (2.0 / x) - (2.0 / x);
	else
		tmp = (x * -2.0) - (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[(N[(2.0 / x), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

      2. sub-neg 69.9%

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

      3. metadata-eval 69.9%

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

      4. associate-+r- 69.9%

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

      5. frac-2neg 69.9%

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

      6. metadata-eval 69.9%

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

      7. +-commutative 69.9%

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

      8. distribute-neg-in 69.9%

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

      9. metadata-eval 69.9%

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

      10. sub-neg 69.9%

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

      11. +-commutative 69.9%

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

   3. Applied egg-rr 69.9%

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

   4. Taylor expanded in x around inf 66.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

   4. Simplified 98.0%

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

4. Final simplification 81.1%

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

Alternative 5: 77.1% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ -1.0 (* x x))
   (if (<= x 1.0) (- (* x -2.0) (/ 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) - (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)) - (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) - (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) - (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 * -2.0) - 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) - (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[(N[(x * -2.0), $MachinePrecision] - 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 69.6%

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

      1. +-commutative 69.6%

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

      2. sub-neg 69.6%

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

      3. metadata-eval 69.6%

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

      4. associate-+r- 69.7%

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

      5. frac-2neg 69.7%

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

      6. metadata-eval 69.7%

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

      7. +-commutative 69.7%

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

      8. distribute-neg-in 69.7%

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

      9. metadata-eval 69.7%

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

      10. sub-neg 69.7%

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

      11. +-commutative 69.7%

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

   3. Applied egg-rr 69.7%

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

   4. Taylor expanded in x around inf 67.9%

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

   5. Taylor expanded in x around inf 56.3%

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

      1. unpow2 56.3%

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

   7. Simplified 56.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

   4. Simplified 98.0%

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

   if 1 < x

   1. Initial program 70.3%

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

   2. Taylor expanded in x around inf 66.9%

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

   3. Taylor expanded in x around inf 56.3%

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

      1. unpow2 56.3%

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

   5. Simplified 56.3%

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

4. Final simplification 75.4%

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

Alternative 6: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. associate-+l- 83.7%

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

   2. sub-neg 83.7%

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

   3. metadata-eval 83.7%

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

   4. frac-sub 55.3%

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

   5. frac-sub 55.8%

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

   6. *-un-lft-identity 55.8%

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

   7. distribute-rgt-in 55.7%

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

   8. neg-mul-1 55.7%

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

   9. sub-neg 55.7%

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

   10. +-commutative 55.7%

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

   11. *-rgt-identity 55.7%

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

   12. +-commutative 55.7%

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

   13. distribute-lft-in 55.7%

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

   14. metadata-eval 55.7%

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

   15. +-commutative 55.7%

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

   16. distribute-rgt-in 55.7%

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

3. Applied egg-rr 55.7%

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

4. Taylor expanded in x around 0 99.7%

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

5. Final simplification 99.7%

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

Alternative 7: 76.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.38 < x

   1. Initial program 70.2%

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

      1. +-commutative 70.2%

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

      2. sub-neg 70.2%

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

      3. metadata-eval 70.2%

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

      4. associate-+r- 70.1%

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

      5. frac-2neg 70.1%

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

      6. metadata-eval 70.1%

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

      7. +-commutative 70.1%

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

      8. distribute-neg-in 70.1%

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

      9. metadata-eval 70.1%

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

      10. sub-neg 70.1%

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

      11. +-commutative 70.1%

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

   3. Applied egg-rr 70.1%

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

   4. Taylor expanded in x around inf 66.8%

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

   5. Taylor expanded in x around inf 55.3%

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

      1. unpow2 55.3%

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

   7. Simplified 55.3%

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

   if -1 < x < 0.38

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.1%

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

4. Final simplification 74.7%

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

Alternative 8: 76.8% 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 69.6%

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

      1. +-commutative 69.6%

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

      2. sub-neg 69.6%

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

      3. metadata-eval 69.6%

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

      4. associate-+r- 69.7%

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

      5. frac-2neg 69.7%

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

      6. metadata-eval 69.7%

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

      7. +-commutative 69.7%

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

      8. distribute-neg-in 69.7%

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

      9. metadata-eval 69.7%

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

      10. sub-neg 69.7%

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

      11. +-commutative 69.7%

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

   3. Applied egg-rr 69.7%

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

   4. Taylor expanded in x around inf 67.9%

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

   5. Taylor expanded in x around inf 56.3%

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

      1. unpow2 56.3%

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

   7. Simplified 56.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.4%

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

   if 1 < x

   1. Initial program 70.3%

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

   2. Taylor expanded in x around inf 66.9%

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

   3. Taylor expanded in x around inf 56.3%

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

      1. unpow2 56.3%

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

   5. Simplified 56.3%

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

4. Final simplification 75.1%

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

Alternative 9: 52.2% 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 83.7%

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

2. Taylor expanded in x around 0 47.3%

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

3. Final simplification 47.3%

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

Alternative 10: 3.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 83.7%

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

2. Taylor expanded in x around 0 46.4%

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

3. Taylor expanded in x around inf 10.4%

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

4. Taylor expanded in x around inf 3.4%

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

5. Final simplification 3.4%

   \[\leadsto -1 \]

Developer target: 99.5% 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 2023213 
(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))))