2frac (problem 3.3.1)

Percentage Accurate: 76.3% → 99.3%

Time: 2.9s

Alternatives: 5

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.3× speedup

Math

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

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(-1.0 / Float64(x * Float64(x + 1.0)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. sub-neg 76.6%

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

   2. +-commutative 76.6%

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

   3. distribute-neg-frac 76.6%

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

   4. metadata-eval 76.6%

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

3. Applied egg-rr 76.6%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 2: 97.2% 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 53.9%

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

   2. Taylor expanded in x around inf 96.4%

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

      1. unpow2 96.4%

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

   4. Simplified 96.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.6%

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

4. Final simplification 98.0%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.75d0))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = ((-1.0d0) + x) / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.75))
		tmp = Float64(-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 <= 0.75)))
		tmp = -1.0 / (x * x);
	else
		tmp = (-1.0 + x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.75]], $MachinePrecision]], N[(-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 0.75 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 96.4%

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

      1. unpow2 96.4%

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

   4. Simplified 96.4%

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

   if -1 < x < 0.75

   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 98.2%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.75d0))) then
        tmp = ((-1.0d0) / x) / x
    else
        tmp = ((-1.0d0) + x) / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.75))
		tmp = Float64(Float64(-1.0 / x) / x);
	else
		tmp = Float64(Float64(-1.0 + x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.75)))
		tmp = (-1.0 / x) / x;
	else
		tmp = (-1.0 + x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.75]], $MachinePrecision]], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(-1.0 + x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.75 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 96.4%

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

      1. metadata-eval 96.4%

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

      2. distribute-neg-frac 96.4%

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

      3. unpow2 96.4%

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

      4. associate-/r* 97.9%

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

      5. metadata-eval 97.9%

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

      6. associate-*l/ 97.9%

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

      7. associate-*r/ 97.7%

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

      8. distribute-rgt-neg-in 97.7%

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

      9. metadata-eval 97.7%

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

      10. distribute-neg-frac 97.7%

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

      11. distribute-neg-frac 97.7%

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

      12. metadata-eval 97.7%

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

      13. distribute-lft-neg-in 97.7%

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

      14. unpow-1 97.7%

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

      15. unpow-1 97.7%

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

      16. pow-sqr 98.0%

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

      17. metadata-eval 98.0%

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

   4. Simplified 98.0%

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

      1. metadata-eval 98.0%

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

      2. pow-div 97.9%

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

      3. unpow-1 97.9%

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

      4. pow1 97.9%

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

   6. Applied egg-rr 97.9%

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

   if -1 < x < 0.75

   1. Initial program 100.0%

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

   2. 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 98.9%

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

Alternative 5: 50.9% 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 76.6%

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

2. Taylor expanded in x around 0 52.0%

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

3. Final simplification 52.0%

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

Reproduce

herbie shell --seed 2023222 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))