2frac (problem 3.3.1)

Percentage Accurate: 77.4% → 99.9%

Time: 10.4s

Alternatives: 5

Speedup: 1.3×

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: 77.4% 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, 1.3× speedup

Math

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

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(Float64(1.0 / x) / Float64(-1.0 - x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. inv-pow N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left({x}^{\color{blue}{-1}}\right)\right) \]

   2. sqr-pow N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left({x}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)\right) \]

   3. pow-prod-down N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left({\left(x \cdot x\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right) \]

   4. sqr-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left({\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)}\right)\right) \]

   5. pow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left({\left({\left(\mathsf{neg}\left(x\right)\right)}^{2}\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)}\right)\right) \]

   6. pow-pow N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left({\left(\mathsf{neg}\left(x\right)\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{2}\right)}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left({\left(\mathsf{neg}\left(x\right)\right)}^{\left(2 \cdot \frac{-1}{2}\right)}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left({\left(\mathsf{neg}\left(x\right)\right)}^{-1}\right)\right) \]

   9. inv-pow N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{1}{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right) \]

   10. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(0 - \color{blue}{\frac{1}{x}}\right)\right) \]

   12. flip-- N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{0 \cdot 0 - \frac{1}{x} \cdot \frac{1}{x}}{\color{blue}{0 + \frac{1}{x}}}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{0 - \frac{1}{x} \cdot \frac{1}{x}}{0 + \frac{1}{x}}\right)\right) \]

   14. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{x} \cdot \frac{1}{x}\right)}{\color{blue}{0} + \frac{1}{x}}\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \frac{1}{x}}{\color{blue}{0} + \frac{1}{x}}\right)\right) \]

   16. inv-pow N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot {x}^{-1}}{0 + \frac{1}{x}}\right)\right) \]

   17. sqr-pow N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \left({x}^{\left(\frac{-1}{2}\right)} \cdot {x}^{\left(\frac{-1}{2}\right)}\right)}{0 + \frac{1}{x}}\right)\right) \]

   18. pow-prod-down N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot {\left(x \cdot x\right)}^{\left(\frac{-1}{2}\right)}}{0 + \frac{1}{x}}\right)\right) \]

   19. sqr-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{-1}{2}\right)}}{0 + \frac{1}{x}}\right)\right) \]

   20. pow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot {\left({\left(\mathsf{neg}\left(x\right)\right)}^{2}\right)}^{\left(\frac{-1}{2}\right)}}{0 + \frac{1}{x}}\right)\right) \]

   21. pow-pow N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(2 \cdot \frac{-1}{2}\right)}}{0 + \frac{1}{x}}\right)\right) \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(2 \cdot \frac{-1}{2}\right)}}{0 + \frac{1}{x}}\right)\right) \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{-1}}{0 + \frac{1}{x}}\right)\right) \]

   24. inv-pow N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(x\right)}}{0 + \frac{1}{x}}\right)\right) \]

   25. div-inv N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}}{\color{blue}{0} + \frac{1}{x}}\right)\right) \]

   26. frac-2neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\frac{\frac{1}{x}}{x}}{\color{blue}{0} + \frac{1}{x}}\right)\right) \]

   27. un-div-inv N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\frac{1}{x} \cdot \frac{1}{x}}{\color{blue}{0} + \frac{1}{x}}\right)\right) \]

   28. inv-pow N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right), \left(\frac{\frac{1}{x} \cdot \frac{1}{x}}{0 + {x}^{\color{blue}{-1}}}\right)\right) \]

4. Applied egg-rr 25.5%

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

   1. +-commutative N/A

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

   2. *-inverses N/A

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

   3. associate-/r* N/A

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

   4. associate-/l/ N/A

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

   5. inv-pow N/A

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

   6. pow2 N/A

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

   7. pow-prod-up N/A

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

   8. metadata-eval N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. inv-pow N/A

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

6. Applied egg-rr 77.3%

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

   1. associate--r+ N/A

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

   2. div-sub N/A

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

   3. +-inverses N/A

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

   4. div0 N/A

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

   5. neg-sub0 N/A

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

   6. associate-/r* N/A

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

   7. distribute-frac-neg2 N/A

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

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)}\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x + 1\right)}\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)\right) \]

   11. distribute-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(-1 + \left(\mathsf{neg}\left(\color{blue}{x}\right)\right)\right)\right) \]

   13. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(-1 - \color{blue}{x}\right)\right) \]

   14. --lowering--.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{\_.f64}\left(-1, \color{blue}{x}\right)\right) \]

8. Applied egg-rr 99.9%

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

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (/ -1.0 x) x)))
   (if (<= x -1.0) t_0 (if (<= x 0.76) (+ 1.0 (/ -1.0 x)) t_0))))

C

double code(double x) {
	double t_0 = (-1.0 / x) / x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.76) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / x) / x
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 0.76d0) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (-1.0 / x) / x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.76) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (-1.0 / x) / x
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 0.76:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(-1.0 / x) / x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.76)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (-1.0 / x) / x;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.76)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.76], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.76000000000000001 < x

   1. Initial program 53.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{x} - 1}{x}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{x} - 1\right), x\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{x} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{x} + -1\right), x\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 + \frac{1}{x}\right), x\right), x\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)\right)\right), x\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 - \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right), x\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), x\right), x\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), x\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \left(\frac{-1}{x}\right)\right), x\right), x\right) \]

      13. /-lowering-/.f64 98.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(-1, x\right)\right), x\right), x\right) \]

   5. Simplified 98.6%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(1\right)}{x}}{x} \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{x} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right), \color{blue}{x}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{x}\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{x}\right), x\right) \]

      8. /-lowering-/.f64 97.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), x\right) \]

   8. Simplified 97.1%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, x\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (* x x))))
   (if (<= x -1.0) t_0 (if (<= x 0.76) (+ 1.0 (/ -1.0 x)) t_0))))

C

double code(double x) {
	double t_0 = -1.0 / (x * x);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.76) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) / (x * x)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 0.76d0) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -1.0 / (x * x);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.76) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -1.0 / (x * x)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 0.76:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(-1.0 / Float64(x * x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.76)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -1.0 / (x * x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.76)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.76], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.76000000000000001 < x

   1. Initial program 53.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 96.1%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 96.1%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, x\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.1% accurate, 3.0× 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 76.6%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 52.1%

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{x}\right) \]

5. Simplified 52.1%

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

Alternative 5: 2.9% accurate, 9.0× 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 76.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, x\right)\right) \]
4. Step-by-step derivation

5. Simplified 50.6%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 2.9%

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

Developer Target 1: 99.9% accurate, 1.3× 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: 99.4% accurate, 1.3× 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 2024191 
(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)))