2frac (problem 3.3.1)

Percentage Accurate: 77.2% → 99.8%

Time: 5.1s

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: 77.2% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. sub-neg 78.4%

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

   2. +-commutative 78.4%

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

   3. distribute-neg-frac 78.4%

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

   4. metadata-eval 78.4%

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

3. Applied egg-rr 78.4%

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

5. Simplified 99.0%

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

   1. metadata-eval 99.0%

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

   2. +-commutative 99.0%

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

   3. *-commutative 99.0%

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

   4. frac-times 99.8%

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

   5. +-commutative 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.75)) {
		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 <= 0.75d0))) 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 <= 0.75)) {
		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 <= 0.75):
		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 <= 0.75))
		tmp = Float64(Float64(1.0 / x) / Float64(-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 <= 0.75)))
		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, 0.75]], $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 0.75 < x

   1. Initial program 57.6%

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

   2. Taylor expanded in x around inf 96.0%

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

      1. metadata-eval 96.0%

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

      2. distribute-neg-frac 96.0%

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

      3. unpow2 96.0%

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

      4. associate-/r* 97.7%

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

      5. metadata-eval 97.7%

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

      6. associate-*l/ 97.7%

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

      7. associate-*r/ 97.5%

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

      8. distribute-rgt-neg-in 97.5%

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

      9. metadata-eval 97.5%

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

      10. distribute-neg-frac 97.5%

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

      11. distribute-neg-frac 97.5%

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

      12. metadata-eval 97.5%

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

      13. distribute-lft-neg-in 97.5%

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

      14. unpow-1 97.5%

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

      15. unpow-1 97.5%

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

      16. pow-sqr 97.8%

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

      17. metadata-eval 97.8%

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

   4. Simplified 97.8%

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

      1. metadata-eval 97.8%

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

      2. pow-flip 96.0%

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

      3. pow2 96.0%

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

      4. metadata-eval 96.0%

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

      5. frac-times 97.5%

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

      6. frac-2neg 97.5%

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

      7. metadata-eval 97.5%

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

      8. un-div-inv 97.7%

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

   6. Applied egg-rr 97.7%

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

   if -1 < x < 0.75

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.3%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.75)) {
		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 <= 0.75d0))) 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 <= 0.75)) {
		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 <= 0.75):
		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 <= 0.75))
		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 <= 0.75)))
		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, 0.75]], $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 0.75 < x

   1. Initial program 57.6%

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

   2. Taylor expanded in x around inf 96.0%

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

      1. unpow2 96.0%

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

   4. Simplified 96.0%

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

   if -1 < x < 0.75

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.9%

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

4. Final simplification 97.4%

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

Alternative 4: 99.4% 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 78.4%

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

   1. sub-neg 78.4%

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

   2. +-commutative 78.4%

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

   3. distribute-neg-frac 78.4%

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

   4. metadata-eval 78.4%

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

3. Applied egg-rr 78.4%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 5: 51.8% 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 78.4%

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

2. Taylor expanded in x around 0 51.1%

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

3. Final simplification 51.1%

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

Alternative 6: 2.9% 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 78.4%

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

2. Taylor expanded in x around 0 50.1%

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

3. Taylor expanded in x around inf 2.8%

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

4. Final simplification 2.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))