2frac (problem 3.3.1)

Percentage Accurate: 77.5% → 99.6%

Time: 4.8s

Alternatives: 4

Speedup: 1.6×

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: 77.5% 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: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(x, x, x\right)} \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	return Float64(-1.0 / fma(x, x, x))
end

Wolfram

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

Derivation

1. Initial program 74.8%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower--.f64 75.6

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

4. Applied rewrites 75.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--r+ N/A

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

   7. +-inverses N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. distribute-lft-in N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 99.8

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{fma}\left(x, x, x\right)}} \]
7. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(x - \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (/ -1.0 (* x x))
   (- 1.0 (- x (/ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (x - (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = 1.0d0 - (x - ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = -1.0 / (x * x)
	else:
		tmp = 1.0 - (x - (-1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(1.0 - Float64(x - Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -1.0 / (x * x);
	else
		tmp = 1.0 - (x - (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 50.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.7

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

   5. Applied rewrites 97.7%

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

   7. Applied rewrites 97.7%

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

   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

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

   5. Applied rewrites 98.9%

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

4. Final simplification 98.3%

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

Alternative 3: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.76))) (/ -1.0 (* x x)) (/ (- x 1.0) x)))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.76d0))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = (x - 1.0d0) / x
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.76):
		tmp = -1.0 / (x * x)
	else:
		tmp = (x - 1.0) / x
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.76)))
		tmp = -1.0 / (x * x);
	else
		tmp = (x - 1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.76]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.76000000000000001 < x

   1. Initial program 50.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.7

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

   5. Applied rewrites 97.7%

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

   7. Applied rewrites 97.7%

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

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.8% accurate, 2.4× 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 74.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 50.7

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

5. Applied rewrites 50.7%

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

Developer Target 1: 99.6% 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 2024340 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (* x (- -1 x))))

  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))