2frac (problem 3.3.1)

Percentage Accurate: 77.6% → 99.9%

Time: 4.5s

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: 77.6% 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 82.8%

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

   1. frac-sub 83.5%

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

   2. *-rgt-identity 83.5%

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

   3. metadata-eval 83.5%

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

   4. div-inv 83.5%

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

   5. associate-/r* 83.5%

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

   6. *-un-lft-identity 83.5%

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

   7. *-rgt-identity 83.5%

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

   8. +-commutative 83.5%

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

   9. div-inv 83.5%

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

   10. metadata-eval 83.5%

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

   11. *-rgt-identity 83.5%

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

   12. +-commutative 83.5%

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

3. Applied egg-rr 83.5%

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

   1. frac-2neg 83.5%

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

   2. div-inv 83.5%

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

   3. +-commutative 83.5%

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

   4. +-commutative 83.5%

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

   5. distribute-neg-in 83.5%

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

   6. neg-mul-1 83.5%

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

   7. metadata-eval 83.5%

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

   8. fma-def 83.5%

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

5. Applied egg-rr 83.5%

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

   1. associate-*r/ 83.5%

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

   2. *-rgt-identity 83.5%

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

   3. distribute-neg-frac 83.5%

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

   4. neg-mul-1 83.5%

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

   5. associate--r+ 99.9%

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

   6. +-inverses 99.9%

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

   7. metadata-eval 99.9%

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

   8. associate-*r/ 99.9%

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

   9. metadata-eval 99.9%

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

   10. fma-udef 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%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = 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;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 64.2%

      \[\frac{1}{x + 1} - \frac{1}{x} \]

   2. Taylor expanded in x around inf 94.7%

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

      1. unpow2 94.7%

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

   4. Simplified 94.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]

   2. Taylor expanded in x around 0 99.9%

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

4. Final simplification 97.4%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.76))) (/ -1.0 (* x x)) (/ (+ -1.0 x) x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.76)) {
		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 <= 0.76d0))) 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 <= 0.76)) {
		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 <= 0.76):
		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 <= 0.76))
		tmp = Float64(-1.0 / Float64(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 <= 0.76)))
		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, 0.76]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.76000000000000001 < x

   1. Initial program 64.2%

      \[\frac{1}{x + 1} - \frac{1}{x} \]

   2. Taylor expanded in x around inf 94.7%

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

      1. unpow2 94.7%

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

   4. Simplified 94.7%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]

   2. Taylor expanded in x around 0 100.0%

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

      1. *-inverses 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x}{x}\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.76))) (/ (/ -1.0 x) x) (/ (+ -1.0 x) x)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.76)) {
		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 <= 0.76d0))) 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 <= 0.76)) {
		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 <= 0.76):
		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 <= 0.76))
		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 <= 0.76)))
		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, 0.76]], $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 0.76000000000000001 < x

   1. Initial program 64.2%

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

      1. frac-sub 65.7%

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

      2. *-rgt-identity 65.7%

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

      3. metadata-eval 65.7%

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

      4. div-inv 65.7%

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

      5. associate-/r* 65.7%

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

      6. *-un-lft-identity 65.7%

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

      7. *-rgt-identity 65.7%

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

      8. +-commutative 65.7%

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

      9. div-inv 65.7%

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

      10. metadata-eval 65.7%

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

      11. *-rgt-identity 65.7%

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

      12. +-commutative 65.7%

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

   3. Applied egg-rr 65.7%

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

      1. frac-2neg 65.7%

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

      2. div-inv 65.7%

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

      3. +-commutative 65.7%

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

      4. +-commutative 65.7%

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

      5. distribute-neg-in 65.7%

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

      6. neg-mul-1 65.7%

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

      7. metadata-eval 65.7%

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

      8. fma-def 65.7%

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

   5. Applied egg-rr 65.7%

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

      1. associate-*r/ 65.7%

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

      2. *-rgt-identity 65.7%

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

      3. distribute-neg-frac 65.7%

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

      4. neg-mul-1 65.7%

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

      5. associate--r+ 99.9%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-*r/ 99.9%

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

      9. metadata-eval 99.9%

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

      10. fma-udef 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%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 94.7%

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

      1. unpow2 94.7%

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

      2. associate-/r* 97.5%

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

   10. Simplified 97.5%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]

   2. Taylor expanded in x around 0 100.0%

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

      1. *-inverses 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x}{x}\\ \end{array} \]

Alternative 5: 99.5% 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 82.8%

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

   1. sub-neg 82.8%

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

   2. +-commutative 82.8%

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

   3. distribute-neg-frac 82.8%

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

   4. metadata-eval 82.8%

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

3. Applied egg-rr 82.8%

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

   1. +-commutative 82.8%

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

   2. *-rgt-identity 82.8%

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

   3. associate-*l/ 82.8%

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

   4. associate-*r/ 82.8%

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

   5. neg-mul-1 82.8%

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

   6. neg-sub0 82.8%

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

   7. associate-+l- 82.8%

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

   8. sub0-neg 82.8%

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

   9. sub-neg 82.8%

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

   10. distribute-neg-out 82.8%

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

   11. distribute-neg-frac 82.8%

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

   12. metadata-eval 82.8%

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

   13. remove-double-neg 82.8%

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

   14. remove-double-neg 82.8%

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

   15. +-commutative 82.8%

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

   16. distribute-neg-in 82.8%

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

   17. neg-mul-1 82.8%

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

   18. metadata-eval 82.8%

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

   19. fma-udef 82.8%

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

   20. neg-mul-1 82.8%

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

   21. associate-/r* 82.8%

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

   22. metadata-eval 82.8%

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

   23. metadata-eval 82.8%

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

   24. associate-*r/ 82.8%

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

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 6: 52.7% 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 82.8%

   \[\frac{1}{x + 1} - \frac{1}{x} \]

2. Taylor expanded in x around 0 54.8%

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

3. Final simplification 54.8%

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

Reproduce

herbie shell --seed 2023173 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))