2frac (problem 3.3.1)

Percentage Accurate: 77.0% → 99.9%

Time: 6.9s

Alternatives: 6

Speedup: 1.3×

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: 77.0% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(-1 - x\right)}^{-2}}{x \cdot \frac{1}{-1 - x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (pow (- -1.0 x) -2.0) (* x (/ 1.0 (- -1.0 x)))))

C

double code(double x) {
	return pow((-1.0 - x), -2.0) / (x * (1.0 / (-1.0 - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-1.0d0) - x) ** (-2.0d0)) / (x * (1.0d0 / ((-1.0d0) - x)))
end function

Java

public static double code(double x) {
	return Math.pow((-1.0 - x), -2.0) / (x * (1.0 / (-1.0 - x)));
}

Python

def code(x):
	return math.pow((-1.0 - x), -2.0) / (x * (1.0 / (-1.0 - x)))

Julia

function code(x)
	return Float64((Float64(-1.0 - x) ^ -2.0) / Float64(x * Float64(1.0 / Float64(-1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = ((-1.0 - x) ^ -2.0) / (x * (1.0 / (-1.0 - x)));
end

Wolfram

code[x_] := N[(N[Power[N[(-1.0 - x), $MachinePrecision], -2.0], $MachinePrecision] / N[(x * N[(1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

4. Applied egg-rr 99.8%

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

   1. frac-times N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot 1}{\left(1 + x\right) \cdot \left(1 + x\right)}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, x\right)\right)}, x\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(1 + x\right) \cdot \left(1 + x\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(-1, x\right)\right), x\right)\right) \]

   3. sqr-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\mathsf{\_.f64}\left(-1, x\right)}\right), x\right)\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\color{blue}{-1}, x\right)\right), x\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, x\right)\right), x\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(-1 - x\right) \cdot \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\color{blue}{-1}, x\right)\right), x\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(-1 - x\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, \color{blue}{x}\right)\right), x\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(-1 - x\right) \cdot \left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, x\right)\right), x\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(-1 - x\right) \cdot \left(-1 - x\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, \color{blue}{x}\right)\right), x\right)\right) \]

   10. pow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{\left(-1 - x\right)}^{2}}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\mathsf{\_.f64}\left(-1, x\right)}\right), x\right)\right) \]

   11. pow-flip N/A

       \[\leadsto \mathsf{/.f64}\left(\left({\left(-1 - x\right)}^{\left(\mathsf{neg}\left(2\right)\right)}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, x\right)\right)}, x\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\left({\left(-1 - x\right)}^{-2}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\mathsf{\_.f64}\left(-1, x\right)}\right), x\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\left({\left(-1 - x\right)}^{\left(-1 \cdot 2\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\mathsf{\_.f64}\left(-1, x\right)}\right), x\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(-1 - x\right), \left(-1 \cdot 2\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, x\right)\right)}, x\right)\right) \]

   15. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(-1, x\right), \left(-1 \cdot 2\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(-1, x\right)\right), x\right)\right) \]

   16. metadata-eval 99.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(-1, x\right), -2\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\mathsf{\_.f64}\left(-1, x\right)}\right), x\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \frac{{\left(-1 - x\right)}^{-2}}{x \cdot \frac{1}{-1 - x}} \]
8. Add Preprocessing

Alternative 2: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (/ -1.0 x) x)))
   (if (<= x -1.0) t_0 (if (<= x 0.75) (- 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 <= 0.75) {
		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 <= 0.75d0) 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 <= 0.75) {
		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 <= 0.75:
		tmp = 1.0 - (1.0 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(-1.0 / x) / x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.75)
		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 <= 0.75)
		tmp = 1.0 - (1.0 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.75], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.75 < x

   1. Initial program 53.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 96.8%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 96.8%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{x}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, x\right), x\right) \]

   7. Applied egg-rr 98.7%

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

   if -1 < x < 0.75

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, x\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.3%

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

Alternative 3: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (* x x))))
   (if (<= x -1.0) t_0 (if (<= x 0.75) (- 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 <= 0.75) {
		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 <= 0.75d0) 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 <= 0.75) {
		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 <= 0.75:
		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 <= 0.75)
		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 <= 0.75)
		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, 0.75], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.75 < x

   1. Initial program 53.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 96.8%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 96.8%

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

   if -1 < x < 0.75

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, x\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.3%

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

Alternative 4: 99.9% accurate, 1.3× speedup

Math

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

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(Float64(1.0 / x) / Float64(-1.0 - x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-neg-frac2 N/A

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

   4. clear-num N/A

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

   5. frac-add N/A

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

   6. *-lft-identity N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-rgt-identity N/A

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

   10. +-commutative N/A

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

   11. *-rgt-identity N/A

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

   12. associate-+l+ N/A

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

   13. sub-neg N/A

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

   14. +-inverses N/A

       \[\leadsto \frac{1 + 0}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x + 1}{1}} \]

   15. metadata-eval N/A

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

   16. distribute-lft-neg-in N/A

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

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

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. *-rgt-identity N/A

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

   21. associate-/r* N/A

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

   22. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)}\right) \]

   23. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\mathsf{neg}\left(\color{blue}{\left(x + 1\right)}\right)\right)\right) \]

4. Applied egg-rr 99.9%

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

Alternative 5: 51.2% accurate, 3.0× 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 76.1%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 51.1%

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{x}\right) \]

5. Simplified 51.1%

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

Alternative 6: 2.9% accurate, 9.0× 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 76.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(1, x\right)\right) \]
4. Step-by-step derivation

5. Simplified 49.8%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 3.1%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))