2frac (problem 3.3.1)

Percentage Accurate: 78.2% → 99.4%

Time: 5.7s

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: 78.2% 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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. frac-2neg 76.7%

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

   2. frac-2neg 76.7%

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

   3. frac-sub 77.1%

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

   4. distribute-rgt-neg-in 77.1%

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

   5. distribute-lft-neg-in 77.1%

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

   6. metadata-eval 77.1%

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

   7. metadata-eval 77.1%

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

   8. *-un-lft-identity 77.1%

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

   9. cancel-sign-sub 77.1%

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

   10. distribute-rgt-neg-in 77.1%

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

   11. *-rgt-identity 77.1%

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

   12. neg-sub0 77.1%

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

   13. +-commutative 77.1%

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

   14. associate--r+ 77.1%

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

   15. metadata-eval 77.1%

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

   16. *-commutative 77.1%

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

   17. neg-sub0 77.1%

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

   18. +-commutative 77.1%

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

   19. associate--r+ 77.1%

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

   20. metadata-eval 77.1%

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

3. Applied egg-rr 77.1%

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

   1. associate-+r- 77.1%

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

   2. +-commutative 77.1%

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

   3. associate--l+ 99.9%

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

   4. distribute-lft-neg-out 99.9%

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

   5. distribute-rgt-neg-in 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.8 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{1}{x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 3.8e+61)))
   (+ (/ 1.0 x) (/ -1.0 x))
   (+ (- 1.0 x) (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 3.8e+61)) {
		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 <= 3.8d+61))) 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 <= 3.8e+61)) {
		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 <= 3.8e+61):
		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 <= 3.8e+61))
		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 <= 3.8e+61)))
		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, 3.8e+61]], $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 3.79999999999999995e61 < x

   1. Initial program 56.6%

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

   2. Taylor expanded in x around inf 55.0%

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

   if -1 < x < 3.79999999999999995e61

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0 90.0%

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

      1. mul-1-neg 90.0%

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

      2. sub-neg 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.8 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{1}{x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.55)))
   (/ -1.0 (* x (+ -1.0 x)))
   (+ (- 1.0 x) (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.55)) {
		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 <= 1.55d0))) 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 <= 1.55)) {
		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 <= 1.55):
		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 <= 1.55))
		tmp = Float64(-1.0 / Float64(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 <= 1.55)))
		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, 1.55]], $MachinePrecision]], N[(-1.0 / N[(x * N[(-1.0 + x), $MachinePrecision]), $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 1.55000000000000004 < x

   1. Initial program 52.0%

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

      1. frac-sub 52.7%

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

      2. *-commutative 52.7%

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

      3. associate-/r* 52.7%

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

      4. *-un-lft-identity 52.7%

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

      5. *-rgt-identity 52.7%

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

      6. sub-neg 52.7%

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

      7. neg-sub0 52.7%

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

      8. +-commutative 52.7%

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

      9. associate--r+ 52.7%

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

      10. metadata-eval 52.7%

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

      11. +-commutative 52.7%

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

   3. Applied egg-rr 52.7%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. div-inv 99.5%

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

      2. add-sqr-sqrt 48.8%

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

      3. associate-*l* 48.9%

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

      4. add-sqr-sqrt 48.9%

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

      5. sqrt-unprod 48.9%

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

      6. frac-times 48.9%

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

      7. metadata-eval 48.9%

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

      8. associate-/l/ 48.9%

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

      9. un-div-inv 48.9%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. inv-pow 0.0%

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

      13. sqrt-pow1 0.0%

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

      14. metadata-eval 0.0%

          \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \left(\sqrt{\frac{-1}{x}} \cdot \frac{1}{1 + x}\right) \]

      15. *-commutative 0.0%

          \[\leadsto {x}^{-0.5} \cdot \color{blue}{\left(\frac{1}{1 + x} \cdot \sqrt{\frac{-1}{x}}\right)} \]

   6. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{{x}^{-0.5} \cdot \left(\frac{-1}{-1 + x} \cdot {x}^{-0.5}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 49.2%

         \[\leadsto \color{blue}{\left(\frac{-1}{-1 + x} \cdot {x}^{-0.5}\right) \cdot {x}^{-0.5}} \]

      2. associate-*l* 49.3%

         \[\leadsto \color{blue}{\frac{-1}{-1 + x} \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]

      3. associate-*l/ 49.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)}{-1 + x}} \]

      4. pow-sqr 97.3%

         \[\leadsto \frac{-1 \cdot \color{blue}{{x}^{\left(2 \cdot -0.5\right)}}}{-1 + x} \]

      5. metadata-eval 97.3%

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

      6. unpow-1 97.3%

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

      7. neg-mul-1 97.3%

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

      8. distribute-neg-frac 97.3%

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

      9. metadata-eval 97.3%

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

      10. associate-/l/ 97.4%

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

      11. *-commutative 97.4%

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

   8. Simplified 97.4%

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

   if -1 < x < 1.55000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. sub-neg 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. frac-sub 77.1%

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

   2. *-commutative 77.1%

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

   3. associate-/r* 77.1%

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

   4. *-un-lft-identity 77.1%

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

   5. *-rgt-identity 77.1%

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

   6. sub-neg 77.1%

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

   7. neg-sub0 77.1%

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

   8. +-commutative 77.1%

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

   9. associate--r+ 77.1%

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

   10. metadata-eval 77.1%

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

   11. +-commutative 77.1%

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

3. Applied egg-rr 77.1%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 5: 52.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

2. Taylor expanded in x around 0 53.5%

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

3. Final simplification 53.5%

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

Alternative 6: 3.7% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 76.7%

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

   1. frac-2neg 76.7%

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

   2. frac-2neg 76.7%

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

   3. frac-sub 77.1%

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

   4. distribute-rgt-neg-in 77.1%

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

   5. distribute-lft-neg-in 77.1%

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

   6. metadata-eval 77.1%

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

   7. metadata-eval 77.1%

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

   8. *-un-lft-identity 77.1%

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

   9. cancel-sign-sub 77.1%

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

   10. distribute-rgt-neg-in 77.1%

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

   11. *-rgt-identity 77.1%

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

   12. neg-sub0 77.1%

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

   13. +-commutative 77.1%

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

   14. associate--r+ 77.1%

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

   15. metadata-eval 77.1%

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

   16. *-commutative 77.1%

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

   17. neg-sub0 77.1%

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

   18. +-commutative 77.1%

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

   19. associate--r+ 77.1%

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

   20. metadata-eval 77.1%

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

3. Applied egg-rr 77.1%

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

   1. associate-+r- 77.1%

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

   2. +-commutative 77.1%

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

   3. associate--l+ 99.9%

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

   4. distribute-lft-neg-out 99.9%

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

   5. distribute-rgt-neg-in 99.9%

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

5. Simplified 99.9%

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

7. Applied egg-rr 1.1%

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

   1. associate-*l/ 1.1%

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

   2. associate-*r/ 1.1%

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

   3. associate-/l/ 1.1%

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

   4. mul-1-neg 1.1%

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

   5. rem-square-sqrt 2.9%

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

   6. +-commutative 2.9%

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

9. Simplified 2.9%

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

10. Taylor expanded in x around inf 3.9%

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

11. Final simplification 3.9%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))