2frac (problem 3.3.1)

Percentage Accurate: 77.4% → 99.9%

Time: 6.8s

Alternatives: 9

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: 77.4% 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: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

4. Applied rewrites 99.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + 1} - \frac{1}{x}\\ t_1 := \left(1 - x\right) - \frac{1}{x}\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -100.0) 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 <= -100.0) {
		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 <= (-100.0d0)) 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 <= -100.0) {
		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 <= -100.0:
		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 <= -100.0)
		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 <= -100.0)
		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, -100.0], 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)) < -100 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 98.9

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

   5. Applied rewrites 98.9%

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

   if -100 < (-.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 95.2

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

   5. Applied rewrites 95.2%

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

Alternative 3: 97.6% accurate, 1.0× 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 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 95.2

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

   5. Applied rewrites 95.2%

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

   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

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

   5. Applied rewrites 98.4%

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

Alternative 4: 99.4% 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 74.5%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 98.9%

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

Alternative 5: 51.9% 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 74.5%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 46.8

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

5. Applied rewrites 46.8%

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

Alternative 6: 3.3% 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 74.5%

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

4. Applied rewrites 99.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 99.9

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

6. Applied rewrites 99.9%

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

8. Applied rewrites 16.2%

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

9. Taylor expanded in x around 0

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

    1. lower-/.f64 3.6

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

11. Applied rewrites 3.6%

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

Alternative 7: 3.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-inverses N/A

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

   6. *-rgt-identity N/A

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

   7. +-commutative N/A

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

   8. associate-+r+ N/A

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

   9. neg-sub0 N/A

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

   10. associate-+l- N/A

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

   11. neg-sub0 N/A

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

   12. sub-neg N/A

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

   13. remove-double-neg N/A

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

   14. mul-1-neg N/A

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

   15. distribute-neg-in N/A

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

   16. +-commutative N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. distribute-neg-in N/A

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

   19. +-commutative N/A

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

5. Applied rewrites 45.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 2.1%

   \[\leadsto x \cdot \color{blue}{x} \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 2.7%

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

13. Applied rewrites 3.5%

    \[\leadsto -\mathsf{fma}\left(x, x, x\right) \]
14. Add Preprocessing

Alternative 8: 3.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x))

C

double code(double x) {
	return -x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -x
end function

Java

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

Python

def code(x):
	return -x

Julia

function code(x)
	return Float64(-x)
end

MATLAB

function tmp = code(x)
	tmp = -x;
end

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 74.5%

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

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-inverses N/A

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

   6. *-rgt-identity N/A

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

   7. +-commutative N/A

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

   8. associate-+r+ N/A

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

   9. neg-sub0 N/A

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

   10. associate-+l- N/A

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

   11. neg-sub0 N/A

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

   12. sub-neg N/A

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

   13. remove-double-neg N/A

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

   14. mul-1-neg N/A

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

   15. distribute-neg-in N/A

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

   16. +-commutative N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. distribute-neg-in N/A

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

   19. +-commutative N/A

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

5. Applied rewrites 45.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 2.1%

   \[\leadsto x \cdot \color{blue}{x} \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 2.7%

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

12. Taylor expanded in x around 0

    \[\leadsto -1 \cdot x \]
13. Step-by-step derivation

14. Applied rewrites 3.4%

    \[\leadsto -x \]
15. Add Preprocessing

Alternative 9: 3.0% accurate, 29.0× 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 74.5%

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

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-inverses N/A

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

   6. *-rgt-identity N/A

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

   7. +-commutative N/A

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

   8. associate-+r+ N/A

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

   9. neg-sub0 N/A

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

   10. associate-+l- N/A

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

   11. neg-sub0 N/A

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

   12. sub-neg N/A

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

   13. remove-double-neg N/A

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

   14. mul-1-neg N/A

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

   15. distribute-neg-in N/A

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

   16. +-commutative N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. distribute-neg-in N/A

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

   19. +-commutative N/A

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

5. Applied rewrites 45.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 2.5%

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

9. Taylor expanded in x around 0

   \[\leadsto 1 \]
10. Step-by-step derivation

11. Applied rewrites 3.1%

    \[\leadsto 1 \]
12. 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 2024240 
(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)))