2frac (problem 3.3.1)

Percentage Accurate: 77.6% → 98.7%

Time: 8.3s

Alternatives: 9

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -1e-11) t_0 (if (<= t_0 0.0) (/ (/ -1.0 x) x) (/ -1.0 x)))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -1e-11) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    if (t_0 <= (-1d-11)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = ((-1.0d0) / x) / x
    else
        tmp = (-1.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -1e-11) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	tmp = 0
	if t_0 <= -1e-11:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (-1.0 / x) / x
	else:
		tmp = -1.0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -1e-11)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 / x) / x);
	else
		tmp = Float64(-1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -1e-11)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (-1.0 / x) / x;
	else
		tmp = -1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-11], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(-1.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -9.99999999999999939e-12

   1. Initial program 98.9%

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

   if -9.99999999999999939e-12 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 43.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 99.1%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -2.0)
     (* (+ x -1.0) (/ (fma x x 1.0) x))
     (if (<= t_0 0.0) (/ (/ -1.0 x) x) (/ -1.0 x)))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = (x + -1.0) * (fma(x, x, 1.0) / x);
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = Float64(Float64(x + -1.0) * Float64(fma(x, x, 1.0) / x));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 / x) / x);
	else
		tmp = Float64(-1.0 / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], N[(N[(x + -1.0), $MachinePrecision] * N[(N[(x * x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(-1.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. *-inverses N/A

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

      9. div-sub N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. sub-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower-+.f64 N/A

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

      24. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 98.2%

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

   if -2 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   7. Applied rewrites 97.7%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -2:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -2.0)
     (* (+ x -1.0) (+ x (/ 1.0 x)))
     (if (<= t_0 0.0) (/ (/ -1.0 x) x) (/ -1.0 x)))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = (x + -1.0) * (x + (1.0 / x));
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    if (t_0 <= (-2.0d0)) then
        tmp = (x + (-1.0d0)) * (x + (1.0d0 / x))
    else if (t_0 <= 0.0d0) then
        tmp = ((-1.0d0) / x) / x
    else
        tmp = (-1.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = (x + -1.0) * (x + (1.0 / x));
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	tmp = 0
	if t_0 <= -2.0:
		tmp = (x + -1.0) * (x + (1.0 / x))
	elif t_0 <= 0.0:
		tmp = (-1.0 / x) / x
	else:
		tmp = -1.0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = Float64(Float64(x + -1.0) * Float64(x + Float64(1.0 / x)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 / x) / x);
	else
		tmp = Float64(-1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = (x + -1.0) * (x + (1.0 / x));
	elseif (t_0 <= 0.0)
		tmp = (-1.0 / x) / x;
	else
		tmp = -1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], N[(N[(x + -1.0), $MachinePrecision] * N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(-1.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. *-inverses N/A

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

      9. div-sub N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. sub-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower-+.f64 N/A

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

      24. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -2 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   7. Applied rewrites 97.7%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -2:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -2.0)
     (* (+ x -1.0) (+ x (/ 1.0 x)))
     (if (<= t_0 0.0) (/ -1.0 (* x x)) (/ -1.0 x)))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = (x + -1.0) * (x + (1.0 / x));
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    if (t_0 <= (-2.0d0)) then
        tmp = (x + (-1.0d0)) * (x + (1.0d0 / x))
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = (-1.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = (x + -1.0) * (x + (1.0 / x));
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	tmp = 0
	if t_0 <= -2.0:
		tmp = (x + -1.0) * (x + (1.0 / x))
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = -1.0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = Float64(Float64(x + -1.0) * Float64(x + Float64(1.0 / x)));
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(-1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = (x + -1.0) * (x + (1.0 / x));
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = -1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], N[(N[(x + -1.0), $MachinePrecision] * N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. *-inverses N/A

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

      9. div-sub N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. remove-double-neg N/A

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

      15. sub-neg N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. sub-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower-+.f64 N/A

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

      24. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -2 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -2:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -2.0)
     (+ (- 1.0 x) (/ -1.0 x))
     (if (<= t_0 0.0) (/ -1.0 (* x x)) (/ -1.0 x)))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = (1.0 - x) + (-1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    if (t_0 <= (-2.0d0)) then
        tmp = (1.0d0 - x) + ((-1.0d0) / x)
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = (-1.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = (1.0 - x) + (-1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	tmp = 0
	if t_0 <= -2.0:
		tmp = (1.0 - x) + (-1.0 / x)
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = -1.0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / x));
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(-1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = (1.0 - x) + (-1.0 / x);
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = -1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -2 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -2:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -2.0)
     (+ 1.0 (/ -1.0 x))
     (if (<= t_0 0.0) (/ -1.0 (* x x)) (/ -1.0 x)))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    if (t_0 <= (-2.0d0)) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = (-1.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	tmp = 0
	if t_0 <= -2.0:
		tmp = 1.0 + (-1.0 / x)
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = -1.0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(-1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = -1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.3%

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

   if -2 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 44.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.7%

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

Alternative 7: 51.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 56.4

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

5. Applied rewrites 56.4%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
6. Add Preprocessing

Alternative 8: 3.8% accurate, 29.0× 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 74.6%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. associate-+l- N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. *-inverses N/A

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

   9. associate-/r* N/A

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

   10. unpow2 N/A

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

   11. div-sub N/A

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

   12. lower-/.f64 N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 46.0

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

5. Applied rewrites 46.0%

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

6. Taylor expanded in x around -inf

   \[\leadsto -1 \]
7. Step-by-step derivation

8. Applied rewrites 3.5%

   \[\leadsto -1 \]
9. Add Preprocessing

Alternative 9: 3.0% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 74.6%

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

3. Taylor expanded in x around -inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

      \[\leadsto x + \color{blue}{2} \]

   6. lower-+.f64 2.8

      \[\leadsto \color{blue}{x + 2} \]

5. Applied rewrites 2.8%

   \[\leadsto \color{blue}{x + 2} \]

6. Taylor expanded in x around 0

   \[\leadsto 2 \]
7. Step-by-step derivation

8. Applied rewrites 3.0%

   \[\leadsto 2 \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 99.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024223 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ -1 x) (+ x 1)))

  :alt
  (! :herbie-platform default (/ 1 (* x (- -1 x))))

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