2frac (problem 3.3.1)

Percentage Accurate: 77.4% → 99.9%

Time: 7.8s

Alternatives: 5

Speedup: 1.6×

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: 77.4% 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: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. div-sub N/A

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

   6. sub-neg N/A

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

   7. *-lft-identity N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 76.9%

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

   1. lift-fma.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. un-div-inv N/A

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

   11. lift-/.f64 N/A

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

   12. frac-2neg N/A

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

   13. frac-2neg N/A

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

   14. sub-div N/A

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

   15. lower-/.f64 N/A

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

6. Applied rewrites 77.4%

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

   1. lift-neg.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. lift-neg.f64 N/A

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

   5. distribute-neg-in N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-in N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. sub-neg N/A

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

   13. lift-+.f64 N/A

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

   14. associate--r+ N/A

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

   15. +-inverses N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. sub-neg N/A

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

   19. metadata-eval N/A

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

   20. distribute-neg-in N/A

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

   21. +-commutative N/A

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

   22. lift-+.f64 N/A

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

   23. distribute-lft-neg-in N/A

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

   24. lift-+.f64 N/A

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

   25. distribute-lft1-in N/A

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

   26. lift-fma.f64 N/A

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

8. Applied rewrites 99.9%

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

Alternative 2: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ t_1 := \left(1 - x\right) + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))) (t_1 (+ (- 1.0 x) (/ -1.0 x))))
   (if (<= t_0 -2.0) t_1 (if (<= t_0 0.0) (/ -1.0 (* x x)) t_1))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = (1.0 - x) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    t_1 = (1.0d0 - x) + ((-1.0d0) / x)
    if (t_0 <= (-2.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = t_1
    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 t_1 = (1.0 - x) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	t_1 = (1.0 - x) + (-1.0 / x)
	tmp = 0
	if t_0 <= -2.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = t_1
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	t_1 = (1.0 - x) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

      \[\leadsto \color{blue}{\left(1 + -1 \cdot x\right)} - \frac{1}{x} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   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 56.3%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 97.2%

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

Alternative 3: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ t_1 := 1 + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))) (t_1 (+ 1.0 (/ -1.0 x))))
   (if (<= t_0 -2.0) t_1 (if (<= t_0 0.0) (/ -1.0 (* x x)) t_1))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = 1.0 + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    t_1 = 1.0d0 + ((-1.0d0) / x)
    if (t_0 <= (-2.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = t_1
    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 t_1 = 1.0 + (-1.0 / x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	t_1 = 1.0 + (-1.0 / x)
	tmp = 0
	if t_0 <= -2.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = t_1
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	t_1 = 1.0 + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

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

   5. Applied rewrites 97.2%

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

   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 56.3%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 97.0%

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

Alternative 4: 99.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.1%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. div-sub N/A

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

   6. sub-neg N/A

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

   7. *-lft-identity N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 76.9%

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

   1. lift-fma.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. unsub-neg N/A

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

   9. lift-/.f64 N/A

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

   10. un-div-inv N/A

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

   11. lift-/.f64 N/A

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

   12. sub-div N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 77.4

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 77.4

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

6. Applied rewrites 77.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{-1}}{\mathsf{fma}\left(x, x, x\right)} \]
10. Add Preprocessing

Alternative 5: 52.0% 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 77.1%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 48.8

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

5. Applied rewrites 48.8%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
6. 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 2024212 
(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)))