2frac (problem 3.3.1)

Percentage Accurate: 77.8% → 99.8%

Time: 3.7s

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: 77.8% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. clear-num 75.8%

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

   2. frac-sub 76.3%

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

   3. *-un-lft-identity 76.3%

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

   4. div-inv 76.3%

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

   5. metadata-eval 76.3%

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

   6. *-rgt-identity 76.3%

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

   7. *-rgt-identity 76.3%

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

   8. +-commutative 76.3%

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

   9. *-commutative 76.3%

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

   10. div-inv 76.3%

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

   11. metadata-eval 76.3%

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

   12. *-rgt-identity 76.3%

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

   13. +-commutative 76.3%

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

4. Applied egg-rr 76.3%

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

5. Taylor expanded in x around 0 99.3%

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

   1. clear-num 99.3%

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

   2. inv-pow 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-/l* 99.3%

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

   5. unpow-prod-down 99.8%

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

   6. inv-pow 99.8%

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

   7. +-commutative 99.8%

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

   8. inv-pow 99.8%

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

   9. clear-num 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + x} \cdot \frac{-1}{x} \]
9. Add Preprocessing

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (* (/ -1.0 x) (/ 1.0 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) * (1.0 / 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) * (1.0d0 / 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) * (1.0 / 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) * (1.0 / 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(Float64(-1.0 / x) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(-1.0 / 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) * (1.0 / 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[(N[(-1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 50.5%

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

      1. clear-num 50.5%

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

      2. frac-sub 51.5%

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

      3. *-un-lft-identity 51.5%

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

      4. div-inv 51.5%

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

      5. metadata-eval 51.5%

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

      6. *-rgt-identity 51.5%

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

      7. *-rgt-identity 51.5%

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

      8. +-commutative 51.5%

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

      9. *-commutative 51.5%

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

      10. div-inv 51.5%

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

      11. metadata-eval 51.5%

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

      12. *-rgt-identity 51.5%

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

      13. +-commutative 51.5%

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

   4. Applied egg-rr 51.5%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-/l* 98.6%

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

      5. unpow-prod-down 99.6%

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

      6. inv-pow 99.6%

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

      7. +-commutative 99.6%

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

      8. inv-pow 99.6%

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

      9. clear-num 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around inf 99.4%

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

   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 99.8%

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

      1. neg-mul-1 99.8%

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

      2. unsub-neg 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.6%

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

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{-1}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.75)))
   (* (/ -1.0 x) (/ 1.0 x))
   (/ (+ x -1.0) x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.75)) {
		tmp = (-1.0 / x) * (1.0 / 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 <= 0.75d0))) then
        tmp = ((-1.0d0) / x) * (1.0d0 / 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 <= 0.75)) {
		tmp = (-1.0 / x) * (1.0 / x);
	} else {
		tmp = (x + -1.0) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.75):
		tmp = (-1.0 / x) * (1.0 / x)
	else:
		tmp = (x + -1.0) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.75))
		tmp = Float64(Float64(-1.0 / x) * Float64(1.0 / 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 <= 0.75)))
		tmp = (-1.0 / x) * (1.0 / 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, 0.75]], $MachinePrecision]], N[(N[(-1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.75 < x

   1. Initial program 50.5%

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

      1. clear-num 50.5%

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

      2. frac-sub 51.5%

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

      3. *-un-lft-identity 51.5%

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

      4. div-inv 51.5%

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

      5. metadata-eval 51.5%

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

      6. *-rgt-identity 51.5%

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

      7. *-rgt-identity 51.5%

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

      8. +-commutative 51.5%

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

      9. *-commutative 51.5%

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

      10. div-inv 51.5%

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

      11. metadata-eval 51.5%

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

      12. *-rgt-identity 51.5%

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

      13. +-commutative 51.5%

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

   4. Applied egg-rr 51.5%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-/l* 98.6%

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

      5. unpow-prod-down 99.6%

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

      6. inv-pow 99.6%

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

      7. +-commutative 99.6%

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

      8. inv-pow 99.6%

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

      9. clear-num 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around inf 99.4%

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

   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 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{-1}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.3× 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]

Derivation

1. Initial program 75.8%

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

   1. clear-num 75.8%

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

   2. frac-sub 76.3%

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

   3. *-un-lft-identity 76.3%

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

   4. div-inv 76.3%

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

   5. metadata-eval 76.3%

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

   6. *-rgt-identity 76.3%

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

   7. *-rgt-identity 76.3%

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

   8. +-commutative 76.3%

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

   9. *-commutative 76.3%

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

   10. div-inv 76.3%

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

   11. metadata-eval 76.3%

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

   12. *-rgt-identity 76.3%

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

   13. +-commutative 76.3%

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

4. Applied egg-rr 76.3%

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

5. Taylor expanded in x around 0 99.3%

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

Alternative 5: 51.8% 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 75.8%

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

3. Taylor expanded in x around 0 52.2%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
4. 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 75.8%

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

3. Taylor expanded in x around 0 52.2%

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

4. Taylor expanded in x around inf 3.1%

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

Reproduce

herbie shell --seed 2024105 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))