2frac (problem 3.3.1)

Percentage Accurate: 77.6% → 99.9%

Time: 6.1s

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

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

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 78.3%

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

   1. frac-sub 79.0%

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

   2. *-commutative 79.0%

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

   3. associate-/r* 79.0%

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

   4. sub-neg 79.0%

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

   5. *-rgt-identity 79.0%

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

   6. *-lft-identity 79.0%

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

   7. +-commutative 79.0%

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

   8. distribute-neg-in 79.0%

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

   9. unsub-neg 79.0%

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

   10. metadata-eval 79.0%

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

   11. +-commutative 79.0%

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

3. Applied egg-rr 79.0%

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

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 2: 98.5% accurate, 0.8× 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 56.6%

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

      1. frac-2neg 56.6%

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

      2. frac-sub 58.1%

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

      3. div-inv 58.1%

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

      4. *-lft-identity 58.1%

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

      5. distribute-rgt-neg-in 58.1%

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

      6. *-rgt-identity 58.1%

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

      7. neg-mul-1 58.1%

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

      8. metadata-eval 58.1%

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

      9. *-rgt-identity 58.1%

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

      10. fma-neg 58.1%

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

      11. metadata-eval 58.1%

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

      12. *-rgt-identity 58.1%

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

      13. remove-double-neg 58.1%

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

      14. +-commutative 58.1%

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

      15. distribute-rgt-neg-in 58.1%

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

      16. distribute-lft-neg-in 58.1%

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

      17. *-commutative 58.1%

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

      18. +-commutative 58.1%

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

      19. distribute-neg-in 58.1%

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

      20. unsub-neg 58.1%

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

      21. metadata-eval 58.1%

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

   3. Applied egg-rr 58.1%

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

      1. associate-*r/ 58.1%

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

      2. *-rgt-identity 58.1%

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

      3. fma-udef 58.1%

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

      4. neg-mul-1 58.1%

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

      5. +-commutative 58.1%

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

      6. unsub-neg 58.1%

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

      7. +-commutative 58.1%

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

   5. Simplified 58.1%

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

   6. Taylor expanded in x around 0 98.4%

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

      1. sub-neg 98.4%

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

      2. +-commutative 98.4%

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

      3. distribute-rgt-in 98.3%

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

      4. neg-mul-1 98.3%

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

      5. distribute-lft-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

      7. distribute-rgt-neg-in 98.3%

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

   8. Applied egg-rr 98.3%

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

   9. Taylor expanded in x around inf 95.6%

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

       1. unpow2 95.6%

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

       2. associate-/r* 97.1%

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

   11. Simplified 97.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. 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. neg-mul-1 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 98.2%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.77\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.77))) (/ -1.0 (* x x)) (+ 1.0 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.77)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.77d0))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.77)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.77):
		tmp = -1.0 / (x * x)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.77))
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.77)))
		tmp = -1.0 / (x * x);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.77]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.77000000000000002 < x

   1. Initial program 56.6%

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

   2. Taylor expanded in x around inf 95.6%

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

      1. unpow2 95.6%

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

   4. Simplified 95.6%

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

   if -1 < x < 0.77000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.77\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.77\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.77))) (/ (/ -1.0 x) x) (+ 1.0 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.77)) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.77d0))) then
        tmp = ((-1.0d0) / x) / x
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.77)) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.77):
		tmp = (-1.0 / x) / x
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.77))
		tmp = Float64(Float64(-1.0 / x) / x);
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.77)))
		tmp = (-1.0 / x) / x;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.77]], $MachinePrecision]], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.77000000000000002 < x

   1. Initial program 56.6%

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

      1. frac-2neg 56.6%

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

      2. frac-sub 58.1%

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

      3. div-inv 58.1%

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

      4. *-lft-identity 58.1%

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

      5. distribute-rgt-neg-in 58.1%

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

      6. *-rgt-identity 58.1%

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

      7. neg-mul-1 58.1%

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

      8. metadata-eval 58.1%

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

      9. *-rgt-identity 58.1%

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

      10. fma-neg 58.1%

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

      11. metadata-eval 58.1%

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

      12. *-rgt-identity 58.1%

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

      13. remove-double-neg 58.1%

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

      14. +-commutative 58.1%

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

      15. distribute-rgt-neg-in 58.1%

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

      16. distribute-lft-neg-in 58.1%

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

      17. *-commutative 58.1%

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

      18. +-commutative 58.1%

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

      19. distribute-neg-in 58.1%

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

      20. unsub-neg 58.1%

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

      21. metadata-eval 58.1%

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

   3. Applied egg-rr 58.1%

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

      1. associate-*r/ 58.1%

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

      2. *-rgt-identity 58.1%

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

      3. fma-udef 58.1%

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

      4. neg-mul-1 58.1%

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

      5. +-commutative 58.1%

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

      6. unsub-neg 58.1%

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

      7. +-commutative 58.1%

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

   5. Simplified 58.1%

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

   6. Taylor expanded in x around 0 98.4%

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

      1. sub-neg 98.4%

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

      2. +-commutative 98.4%

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

      3. distribute-rgt-in 98.3%

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

      4. neg-mul-1 98.3%

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

      5. distribute-lft-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

      7. distribute-rgt-neg-in 98.3%

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

   8. Applied egg-rr 98.3%

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

   9. Taylor expanded in x around inf 95.6%

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

       1. unpow2 95.6%

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

       2. associate-/r* 97.1%

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

   11. Simplified 97.1%

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

   if -1 < x < 0.77000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.77\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

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 78.3%

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

   1. frac-2neg 78.3%

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

   2. frac-sub 79.0%

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

   3. div-inv 79.0%

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

   4. *-lft-identity 79.0%

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

   5. distribute-rgt-neg-in 79.0%

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

   6. *-rgt-identity 79.0%

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

   7. neg-mul-1 79.0%

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

   8. metadata-eval 79.0%

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

   9. *-rgt-identity 79.0%

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

   10. fma-neg 79.0%

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

   11. metadata-eval 79.0%

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

   12. *-rgt-identity 79.0%

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

   13. remove-double-neg 79.0%

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

   14. +-commutative 79.0%

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

   15. distribute-rgt-neg-in 79.0%

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

   16. distribute-lft-neg-in 79.0%

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

   17. *-commutative 79.0%

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

   18. +-commutative 79.0%

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

   19. distribute-neg-in 79.0%

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

   20. unsub-neg 79.0%

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

   21. metadata-eval 79.0%

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

3. Applied egg-rr 79.0%

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

   1. associate-*r/ 79.0%

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

   2. *-rgt-identity 79.0%

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

   3. fma-udef 79.0%

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

   4. neg-mul-1 79.0%

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

   5. +-commutative 79.0%

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

   6. unsub-neg 79.0%

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

   7. +-commutative 79.0%

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

5. Simplified 79.0%

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

6. Taylor expanded in x around 0 99.1%

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

7. Final simplification 99.1%

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

Alternative 6: 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 78.3%

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

2. Taylor expanded in x around 0 51.6%

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

3. Final simplification 51.6%

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))