2frac (problem 3.3.1)

Percentage Accurate: 77.6% → 99.9%

Time: 6.5s

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 78.6%

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

   1. frac-sub 79.9%

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

   2. *-rgt-identity 79.9%

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

   3. metadata-eval 79.9%

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

   4. div-inv 79.9%

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

   5. associate-/r* 79.9%

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

   6. *-un-lft-identity 79.9%

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

   7. *-rgt-identity 79.9%

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

   8. +-commutative 79.9%

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

   9. div-inv 79.9%

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

   10. metadata-eval 79.9%

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

   11. *-rgt-identity 79.9%

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

   12. +-commutative 79.9%

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

4. Applied egg-rr 79.9%

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

   1. frac-2neg 79.9%

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

   2. div-inv 79.9%

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

   3. sub-neg 79.9%

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

   4. distribute-neg-in 79.9%

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

   5. metadata-eval 79.9%

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

   6. sub-neg 79.9%

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

   7. distribute-neg-in 79.9%

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

   8. metadata-eval 79.9%

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

   9. sub-neg 79.9%

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

6. Applied egg-rr 79.9%

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

   1. associate-+r- 79.9%

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

   2. +-commutative 79.9%

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

   3. associate--l+ 99.9%

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

   4. +-inverses 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. *-lft-identity 99.9%

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

8. Simplified 99.9%

   \[\leadsto \frac{\color{blue}{\frac{1}{-1 - x}}}{x} \]
9. 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 56.8%

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

      1. frac-sub 59.5%

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

      2. div-inv 59.5%

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

      3. *-un-lft-identity 59.5%

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

      4. *-rgt-identity 59.5%

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

      5. +-commutative 59.5%

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

      6. metadata-eval 59.5%

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

      7. frac-times 59.5%

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

      8. clear-num 59.5%

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

      9. associate-*l/ 59.5%

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

      10. *-un-lft-identity 59.5%

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

      11. div-inv 59.5%

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

      12. metadata-eval 59.5%

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

      13. *-rgt-identity 59.5%

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

      14. +-commutative 59.5%

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

   4. Applied egg-rr 59.5%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. add-sqr-sqrt 48.8%

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

      2. sqrt-unprod 52.3%

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

      3. swap-sqr 52.3%

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

      4. metadata-eval 52.3%

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

      5. *-un-lft-identity 52.3%

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

      6. sqrt-unprod 51.9%

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

      7. add-sqr-sqrt 51.9%

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

      8. associate-/l/ 51.9%

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

      9. associate-/r* 51.9%

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

      10. frac-2neg 51.9%

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

      11. metadata-eval 51.9%

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

      12. distribute-neg-in 51.9%

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

      13. metadata-eval 51.9%

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

      14. add-sqr-sqrt 34.0%

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

      15. sqrt-unprod 76.3%

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

      16. sqr-neg 76.3%

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

      17. sqrt-unprod 42.6%

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

      18. add-sqr-sqrt 96.8%

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

   7. Applied egg-rr 96.8%

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

   8. Taylor expanded in x around inf 97.0%

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

   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 99.1%

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

      1. neg-mul-1 99.1%

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

      2. sub-neg 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 98.0%

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

Alternative 3: 98.3% accurate, 0.6× speedup

Math

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

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 57.1%

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

      1. frac-sub 59.9%

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

      2. div-inv 59.9%

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

      3. *-un-lft-identity 59.9%

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

      4. *-rgt-identity 59.9%

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

      5. +-commutative 59.9%

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

      6. metadata-eval 59.9%

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

      7. frac-times 59.9%

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

      8. clear-num 59.9%

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

      9. associate-*l/ 59.8%

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

      10. *-un-lft-identity 59.8%

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

      11. div-inv 59.8%

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

      12. metadata-eval 59.8%

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

      13. *-rgt-identity 59.8%

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

      14. +-commutative 59.8%

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

   4. Applied egg-rr 59.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. add-sqr-sqrt 48.4%

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

      2. sqrt-unprod 51.9%

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

      3. swap-sqr 51.9%

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

      4. metadata-eval 51.9%

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

      5. *-un-lft-identity 51.9%

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

      6. sqrt-unprod 51.5%

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

      7. add-sqr-sqrt 51.5%

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

      8. associate-/l/ 51.6%

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

      9. associate-/r* 51.5%

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

      10. frac-2neg 51.5%

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

      11. metadata-eval 51.5%

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

      12. distribute-neg-in 51.5%

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

      13. metadata-eval 51.5%

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

      14. add-sqr-sqrt 33.8%

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

      15. sqrt-unprod 75.7%

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

      16. sqr-neg 75.7%

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

      17. sqrt-unprod 42.3%

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

      18. add-sqr-sqrt 96.1%

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

   7. Applied egg-rr 96.1%

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

   8. Taylor expanded in x around inf 96.3%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.5%

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

4. Final simplification 97.9%

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

Alternative 4: 99.3% 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 78.6%

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

   1. clear-num 78.6%

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

   2. frac-sub 79.9%

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

   3. *-un-lft-identity 79.9%

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

   4. div-inv 79.9%

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

   5. metadata-eval 79.9%

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

   6. *-rgt-identity 79.9%

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

   7. *-rgt-identity 79.9%

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

   8. +-commutative 79.9%

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

   9. *-commutative 79.9%

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

   10. div-inv 79.9%

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

   11. metadata-eval 79.9%

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

   12. *-rgt-identity 79.9%

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

   13. +-commutative 79.9%

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

4. Applied egg-rr 79.9%

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

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: 52.8% 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 78.6%

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

3. Taylor expanded in x around 0 52.6%

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

Alternative 6: 3.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

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

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in x around 0 52.6%

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

   1. add-sqr-sqrt 25.6%

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

   2. sqrt-unprod 39.5%

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

   3. frac-times 40.0%

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

   4. metadata-eval 40.0%

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

   5. metadata-eval 40.0%

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

   6. frac-times 39.5%

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

   7. sqrt-unprod 1.1%

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

   8. add-sqr-sqrt 3.3%

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

   9. clear-num 3.3%

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

   10. associate-/r/ 3.3%

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

5. Applied egg-rr 3.3%

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

7. Applied egg-rr 2.5%

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

   1. fma-undefine 2.5%

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

   2. distribute-rgt-in 2.5%

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

   3. associate-*r/ 2.5%

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

   4. metadata-eval 2.5%

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

   5. associate-+l+ 2.5%

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

   6. associate-*r/ 2.5%

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

   7. *-commutative 2.5%

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

   8. mul-1-neg 2.5%

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

   9. distribute-frac-neg 2.5%

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

   10. *-inverses 2.5%

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

   11. metadata-eval 2.5%

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

   12. metadata-eval 2.5%

       \[\leadsto \frac{1}{x} + \color{blue}{-2} \]

9. Simplified 2.5%

   \[\leadsto \color{blue}{\frac{1}{x} + -2} \]

10. Taylor expanded in x around inf 3.8%

    \[\leadsto \color{blue}{-2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))