2frac (problem 3.3.1)

Percentage Accurate: 78.0% → 99.0%

Time: 7.9s

Alternatives: 10

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.0% 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: 99.0% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -2.0)
     (+ (/ (fma x x 1.0) (* (+ 1.0 x) (fma x x 1.0))) (/ -1.0 x))
     (if (<= t_0 0.0)
       (* (/ 1.0 x) (/ (+ -1.0 (/ (+ x -1.0) (* x 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 = (fma(x, x, 1.0) / ((1.0 + x) * fma(x, x, 1.0))) + (-1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = (1.0 / x) * ((-1.0 + ((x + -1.0) / (x * 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(fma(x, x, 1.0) / Float64(Float64(1.0 + x) * fma(x, x, 1.0))) + Float64(-1.0 / x));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(-1.0 + Float64(Float64(x + -1.0) / Float64(x * x))) / x));
	else
		tmp = Float64(-1.0 / x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   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 58.6%

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

   3. Taylor expanded in x around inf

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

      1. associate--r+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r- N/A

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

      5. neg-sub0 N/A

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

      6. associate--r- N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. div-sub N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 99.7%

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

   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 99.8%

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

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, 1\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)} + \frac{-1}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -2e-10)
     (+ (/ (fma x x 1.0) (* (+ 1.0 x) (fma x x 1.0))) (/ -1.0 x))
     (if (<= t_0 0.0) (/ (/ -1.0 x) x) (/ -1.0 x)))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -2e-10) {
		tmp = (fma(x, x, 1.0) / ((1.0 + x) * fma(x, x, 1.0))) + (-1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -2e-10)
		tmp = Float64(Float64(fma(x, x, 1.0) / Float64(Float64(1.0 + x) * fma(x, x, 1.0))) + 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

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, -2e-10], N[(N[(N[(x * x + 1.0), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(-1.0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -2.00000000000000007e-10

   1. Initial program 99.7%

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

   4. Applied rewrites 99.6%

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

   if -2.00000000000000007e-10 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 55.9%

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

   3. Taylor expanded in x around inf

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

      1. 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 97.9

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

   5. Applied rewrites 97.9%

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

   7. Applied rewrites 98.8%

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

   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 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, 1\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)} + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

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: 99.4% 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 2024230 
(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)))