2frac (problem 3.3.1)

Percentage Accurate: 77.6% → 98.7%

Time: 8.9s

Alternatives: 9

Speedup: 0.8×

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

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

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 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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. sub-neg N/A

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

      9. +-commutative N/A

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

      10. neg-sub0 N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. unsub-neg N/A

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

      14. *-inverses N/A

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

      15. div-sub N/A

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

      16. unsub-neg N/A

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

      17. mul-1-neg N/A

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

      18. *-rgt-identity N/A

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

      19. associate-/l* N/A

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

   5. Applied rewrites 99.5%

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

   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 55.5%

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

   3. Taylor expanded in x around inf

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

      1. 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 96.4

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

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 97.6%

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

   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 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 98.1%

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

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)))