2frac (problem 3.3.1)

Percentage Accurate: 76.8% → 99.1%

Time: 7.6s

Alternatives: 7

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: 76.8% 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: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -160000000:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 305000000:\\ \;\;\;\;\frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -160000000.0)
   (/ (/ -1.0 x) x)
   (if (<= x 305000000.0) (+ (/ 1.0 (+ x 1.0)) (/ -1.0 x)) (/ -1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -160000000.0) {
		tmp = (-1.0 / x) / x;
	} else if (x <= 305000000.0) {
		tmp = (1.0 / (x + 1.0)) + (-1.0 / x);
	} else {
		tmp = -1.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-160000000.0d0)) then
        tmp = ((-1.0d0) / x) / x
    else if (x <= 305000000.0d0) then
        tmp = (1.0d0 / (x + 1.0d0)) + ((-1.0d0) / x)
    else
        tmp = (-1.0d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -160000000.0) {
		tmp = (-1.0 / x) / x;
	} else if (x <= 305000000.0) {
		tmp = (1.0 / (x + 1.0)) + (-1.0 / x);
	} else {
		tmp = -1.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -160000000.0:
		tmp = (-1.0 / x) / x
	elif x <= 305000000.0:
		tmp = (1.0 / (x + 1.0)) + (-1.0 / x)
	else:
		tmp = -1.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -160000000.0)
		tmp = Float64(Float64(-1.0 / x) / x);
	elseif (x <= 305000000.0)
		tmp = Float64(Float64(1.0 / Float64(x + 1.0)) + Float64(-1.0 / x));
	else
		tmp = Float64(-1.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -160000000.0)
		tmp = (-1.0 / x) / x;
	elseif (x <= 305000000.0)
		tmp = (1.0 / (x + 1.0)) + (-1.0 / x);
	else
		tmp = -1.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -160000000.0], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 305000000.0], N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6e8

   1. Initial program 49.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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 97.0

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

   5. Simplified 97.0%

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

      1. associate-/r* N/A

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

      2. lift-/.f64 N/A

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

      3. lower-/.f64 99.6

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

   7. Applied egg-rr 99.6%

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

   if -1.6e8 < x < 3.05e8

   1. Initial program 99.9%

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

   if 3.05e8 < x

   1. Initial program 56.9%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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 99.8

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

   5. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -160000000:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 305000000:\\ \;\;\;\;\frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + 1} + \frac{-1}{x}\\ t_1 := \left(1 - x\right) + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ x 1.0)) (/ -1.0 x))) (t_1 (+ (- 1.0 x) (/ -1.0 x))))
   (if (<= t_0 -50.0) t_1 (if (<= t_0 0.0) (/ -1.0 (* x x)) t_1))))

C

double code(double x) {
	double t_0 = (1.0 / (x + 1.0)) + (-1.0 / x);
	double t_1 = (1.0 - x) + (-1.0 / x);
	double tmp;
	if (t_0 <= -50.0) {
		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 / (x + 1.0d0)) + ((-1.0d0) / x)
    t_1 = (1.0d0 - x) + ((-1.0d0) / x)
    if (t_0 <= (-50.0d0)) 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 / (x + 1.0)) + (-1.0 / x);
	double t_1 = (1.0 - x) + (-1.0 / x);
	double tmp;
	if (t_0 <= -50.0) {
		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 / (x + 1.0)) + (-1.0 / x)
	t_1 = (1.0 - x) + (-1.0 / x)
	tmp = 0
	if t_0 <= -50.0:
		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(x + 1.0)) + Float64(-1.0 / x))
	t_1 = Float64(Float64(1.0 - x) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -50.0)
		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 / (x + 1.0)) + (-1.0 / x);
	t_1 = (1.0 - x) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -50.0)
		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[(x + 1.0), $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, -50.0], 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)) < -50 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 99.5

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

   5. Simplified 99.5%

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

   if -50 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 54.1%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   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 96.3

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

   5. Simplified 96.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x + 1} + \frac{-1}{x} \leq -50:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{x + 1} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \]
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 2024218 
(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)))