2frac (problem 3.3.1)

Percentage Accurate: 77.0% → 99.4%

Time: 8.0s

Alternatives: 6

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: 77.0% 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: 99.4% 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]

Derivation

1. Initial program 75.7%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
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. associate--r+ N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. associate-+r- N/A

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

   5. neg-sub0 N/A

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

   6. associate--r- N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. div-sub N/A

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

   10. neg-sub0 N/A

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

   11. mul-1-neg N/A

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

   12. /-lowering-/.f64 N/A

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

5. Simplified 48.9%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

   6. /-lowering-/.f64 N/A

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

   7. +-commutative N/A

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

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

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

   9. /-lowering-/.f64 48.9%

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

7. Applied egg-rr 48.9%

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

8. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. mul-1-neg N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. rgt-mult-inverse N/A

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

   12. distribute-lft-out-- N/A

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

   13. *-lowering-*.f64 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. mul-1-neg N/A

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

   18. unsub-neg N/A

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

   19. --lowering--.f64 99.5%

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

10. Simplified 99.5%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (/ -1.0 x) x)))
   (if (<= x -1.0) t_0 (if (<= x 1.0) (+ (- 1.0 x) (/ -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 <= 1.0) {
		tmp = (1.0 - x) + (-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 <= 1.0d0) then
        tmp = (1.0d0 - x) + ((-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 <= 1.0) {
		tmp = (1.0 - x) + (-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 <= 1.0:
		tmp = (1.0 - x) + (-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 <= 1.0)
		tmp = Float64(Float64(1.0 - x) + 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 <= 1.0)
		tmp = (1.0 - x) + (-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, 1.0], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 49.5%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 96.1%

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

   8. Simplified 96.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 98.1%

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

Alternative 3: 98.2% 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 49.5%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 96.1%

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

   8. Simplified 96.1%

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

   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 99.8%

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

4. Final simplification 98.0%

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

Alternative 4: 97.7% 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 49.5%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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 95.5%

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

   5. Simplified 95.5%

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

   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 99.8%

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

4. Final simplification 97.7%

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

Alternative 5: 51.1% 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 75.7%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 54.3%

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

5. Simplified 54.3%

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

Alternative 6: 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 75.7%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
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 53.1%

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

6. Taylor expanded in x around inf

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

8. Simplified 2.7%

   \[\leadsto \color{blue}{1} \]
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 2024192 
(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)))