2frac (problem 3.3.1)

Percentage Accurate: 77.3% → 99.9%

Time: 4.3s

Alternatives: 5

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: 77.3% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \frac{\frac{1 - \left(x - x\right)}{-1 - x}}{x} \]
6. Add Preprocessing

Alternative 2: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} - \frac{-1}{1 + x}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 x) (/ -1.0 (+ 1.0 x)))))
   (if (<= t_0 -2000000.0)
     (- (- 1.0 x) (/ 1.0 x))
     (if (<= t_0 0.0) (/ -1.0 (* x x)) (- 1.0 (/ 1.0 x))))))

C

double code(double x) {
	double t_0 = (-1.0 / x) - (-1.0 / (1.0 + x));
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = (1.0 - x) - (1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (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) / x) - ((-1.0d0) / (1.0d0 + x))
    if (t_0 <= (-2000000.0d0)) then
        tmp = (1.0d0 - x) - (1.0d0 / x)
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = 1.0d0 - (1.0d0 / x)
    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 tmp;
	if (t_0 <= -2000000.0) {
		tmp = (1.0 - x) - (1.0 / x);
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (-1.0 / x) - (-1.0 / (1.0 + x))
	tmp = 0
	if t_0 <= -2000000.0:
		tmp = (1.0 - x) - (1.0 / x)
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = 1.0 - (1.0 / x)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(-1.0 / x) - Float64(-1.0 / Float64(1.0 + x)))
	tmp = 0.0
	if (t_0 <= -2000000.0)
		tmp = Float64(Float64(1.0 - x) - Float64(1.0 / x));
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (-1.0 / x) - (-1.0 / (1.0 + x));
	tmp = 0.0;
	if (t_0 <= -2000000.0)
		tmp = (1.0 - x) - (1.0 / x);
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = 1.0 - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] - N[(-1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000.0], N[(N[(1.0 - x), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $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)) < -2e6

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -2e6 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 60.1%

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

   3. Taylor expanded in x around inf

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

      1. 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. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 98.5%

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

   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

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1}{x} - \frac{-1}{1 + x} \leq -2000000:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x}\\ \mathbf{elif}\;\frac{-1}{x} - \frac{-1}{1 + x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} - \frac{-1}{1 + x}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;\frac{x - 1}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 x) (/ -1.0 (+ 1.0 x)))))
   (if (<= t_0 -2000000.0)
     (/ (- x 1.0) x)
     (if (<= t_0 0.0) (/ -1.0 (* x x)) (- 1.0 (/ 1.0 x))))))

C

double code(double x) {
	double t_0 = (-1.0 / x) - (-1.0 / (1.0 + x));
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = (x - 1.0) / x;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (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) / x) - ((-1.0d0) / (1.0d0 + x))
    if (t_0 <= (-2000000.0d0)) then
        tmp = (x - 1.0d0) / x
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = 1.0d0 - (1.0d0 / x)
    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 tmp;
	if (t_0 <= -2000000.0) {
		tmp = (x - 1.0) / x;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (-1.0 / x) - (-1.0 / (1.0 + x))
	tmp = 0
	if t_0 <= -2000000.0:
		tmp = (x - 1.0) / x
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = 1.0 - (1.0 / x)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(-1.0 / x) - Float64(-1.0 / Float64(1.0 + x)))
	tmp = 0.0
	if (t_0 <= -2000000.0)
		tmp = Float64(Float64(x - 1.0) / x);
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (-1.0 / x) - (-1.0 / (1.0 + x));
	tmp = 0.0;
	if (t_0 <= -2000000.0)
		tmp = (x - 1.0) / x;
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = 1.0 - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] - N[(-1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000.0], N[(N[(x - 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $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)) < -2e6

   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 99.8

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

   5. Applied rewrites 99.8%

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

   if -2e6 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 60.1%

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

   3. Taylor expanded in x around inf

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

      1. 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. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 98.5%

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

   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

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1}{x} - \frac{-1}{1 + x} \leq -2000000:\\ \;\;\;\;\frac{x - 1}{x}\\ \mathbf{elif}\;\frac{-1}{x} - \frac{-1}{1 + x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]
5. 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 81.9%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
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-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. neg-mul-1 N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift--.f64 N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. +-commutative N/A

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

   14. metadata-eval N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. frac-2neg N/A

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

   18. lift-/.f64 N/A

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

6. Applied rewrites 99.7%

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

Alternative 5: 52.4% 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 81.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 56.9

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

5. Applied rewrites 56.9%

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

Developer Target 1: 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 2024308 
(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)))