2frac (problem 3.3.1)

Percentage Accurate: 77.3% → 99.9%

Time: 5.7s

Alternatives: 5

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: 77.3% 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, 1.3× 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 79.8%

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

   1. frac-sub 80.3%

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

   2. *-rgt-identity 80.3%

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

   3. metadata-eval 80.3%

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

   4. div-inv 80.3%

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

   5. associate-/r* 80.3%

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

   6. *-un-lft-identity 80.3%

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

   7. *-rgt-identity 80.3%

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

   8. +-commutative 80.3%

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

   9. div-inv 80.3%

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

   10. metadata-eval 80.3%

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

   11. *-rgt-identity 80.3%

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

   12. +-commutative 80.3%

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

4. Applied egg-rr 80.3%

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

   1. frac-2neg 80.3%

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

   2. div-inv 80.3%

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

   3. +-commutative 80.3%

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

   4. +-commutative 80.3%

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

   5. distribute-neg-in 80.3%

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

   6. neg-mul-1 80.3%

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

   7. metadata-eval 80.3%

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

   8. fma-define 80.3%

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

6. Applied egg-rr 80.3%

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

   1. distribute-lft-neg-out 80.3%

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

   2. distribute-rgt-neg-in 80.3%

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

   3. associate--r+ 99.9%

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

   4. +-inverses 99.9%

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

   5. metadata-eval 99.9%

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

   6. neg-mul-1 99.9%

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

   7. associate-*l* 99.9%

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

   8. metadata-eval 99.9%

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

   9. *-lft-identity 99.9%

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

   10. fma-undefine 99.9%

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

   11. neg-mul-1 99.9%

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

   12. +-commutative 99.9%

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

   13. unsub-neg 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \frac{\frac{1}{-1 - x}}{x} \]
10. Add Preprocessing

Alternative 2: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(-1.0 / x) / 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) / 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] / x), $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 52.9%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. expm1-log1p-u 97.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{{x}^{2}}\right)\right)} \]

      2. expm1-undefine 51.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{{x}^{2}}\right)} - 1} \]

      3. div-inv 51.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-1 \cdot \frac{1}{{x}^{2}}}\right)} - 1 \]

      4. mul-1-neg 51.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-\frac{1}{{x}^{2}}}\right)} - 1 \]

      5. pow-flip 51.9%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{{x}^{\left(-2\right)}}\right)} - 1 \]

      6. metadata-eval 51.9%

         \[\leadsto e^{\mathsf{log1p}\left(-{x}^{\color{blue}{-2}}\right)} - 1 \]

   5. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{x}^{-2}\right)} - 1} \]
   6. Step-by-step derivation

      1. sub-neg 51.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{x}^{-2}\right)} + \left(-1\right)} \]

      2. metadata-eval 51.9%

         \[\leadsto e^{\mathsf{log1p}\left(-{x}^{-2}\right)} + \color{blue}{-1} \]

      3. +-commutative 51.9%

         \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(-{x}^{-2}\right)}} \]

      4. log1p-undefine 51.9%

         \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \left(-{x}^{-2}\right)\right)}} \]

      5. rem-exp-log 51.9%

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

      6. associate-+r+ 99.1%

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

      7. metadata-eval 99.1%

         \[\leadsto \color{blue}{0} + \left(-{x}^{-2}\right) \]

      8. +-lft-identity 99.1%

         \[\leadsto \color{blue}{-{x}^{-2}} \]

   7. Simplified 99.1%

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

      1. sqr-pow 98.7%

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

      2. metadata-eval 98.7%

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

      3. unpow-1 98.7%

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

      4. un-div-inv 98.9%

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

      5. metadata-eval 98.9%

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

      6. unpow-1 98.9%

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

   9. Applied egg-rr 98.9%

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

   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 99.5%

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

      1. neg-mul-1 99.5%

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

      2. unsub-neg 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.3%

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

Alternative 3: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.75 < x

   1. Initial program 52.9%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. expm1-log1p-u 97.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{{x}^{2}}\right)\right)} \]

      2. expm1-undefine 51.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{{x}^{2}}\right)} - 1} \]

      3. div-inv 51.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-1 \cdot \frac{1}{{x}^{2}}}\right)} - 1 \]

      4. mul-1-neg 51.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-\frac{1}{{x}^{2}}}\right)} - 1 \]

      5. pow-flip 51.9%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{{x}^{\left(-2\right)}}\right)} - 1 \]

      6. metadata-eval 51.9%

         \[\leadsto e^{\mathsf{log1p}\left(-{x}^{\color{blue}{-2}}\right)} - 1 \]

   5. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{x}^{-2}\right)} - 1} \]
   6. Step-by-step derivation

      1. sub-neg 51.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-{x}^{-2}\right)} + \left(-1\right)} \]

      2. metadata-eval 51.9%

         \[\leadsto e^{\mathsf{log1p}\left(-{x}^{-2}\right)} + \color{blue}{-1} \]

      3. +-commutative 51.9%

         \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(-{x}^{-2}\right)}} \]

      4. log1p-undefine 51.9%

         \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \left(-{x}^{-2}\right)\right)}} \]

      5. rem-exp-log 51.9%

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

      6. associate-+r+ 99.1%

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

      7. metadata-eval 99.1%

         \[\leadsto \color{blue}{0} + \left(-{x}^{-2}\right) \]

      8. +-lft-identity 99.1%

         \[\leadsto \color{blue}{-{x}^{-2}} \]

   7. Simplified 99.1%

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

      1. sqr-pow 98.7%

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

      2. metadata-eval 98.7%

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

      3. unpow-1 98.7%

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

      4. un-div-inv 98.9%

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

      5. metadata-eval 98.9%

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

      6. unpow-1 98.9%

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

   9. Applied egg-rr 98.9%

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

   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 99.3%

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

4. Final simplification 99.1%

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

Alternative 4: 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 79.8%

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

3. Taylor expanded in x around 0 58.9%

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

4. Final simplification 58.9%

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

Alternative 5: 3.2% accurate, 4.5× 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 79.8%

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

3. Taylor expanded in x around 0 57.8%

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

   1. neg-mul-1 57.8%

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

   2. unsub-neg 57.8%

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

5. Simplified 57.8%

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

6. Taylor expanded in x around inf 3.2%

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

   1. neg-mul-1 3.2%

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

8. Simplified 3.2%

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

9. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2024084 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))