2frac (problem 3.3.1)

Percentage Accurate: 78.1% → 99.9%

Time: 36.1s

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: 78.1% 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 77.7%

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

   1. sub-neg 77.7%

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

   2. add-exp-log 50.4%

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

   3. log-rec 50.4%

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

   4. +-commutative 50.4%

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

   5. log1p-define 50.4%

      \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(x\right)}} + \left(-\frac{1}{x}\right) \]

   6. distribute-neg-frac 50.4%

      \[\leadsto e^{-\mathsf{log1p}\left(x\right)} + \color{blue}{\frac{-1}{x}} \]

   7. metadata-eval 50.4%

      \[\leadsto e^{-\mathsf{log1p}\left(x\right)} + \frac{\color{blue}{-1}}{x} \]

4. Applied egg-rr 50.4%

   \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(x\right)} + \frac{-1}{x}} \]
5. Step-by-step derivation

   1. *-rgt-identity 50.4%

      \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(x\right)} \cdot 1} + \frac{-1}{x} \]

   2. fma-undefine 50.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{-\mathsf{log1p}\left(x\right)}, 1, \frac{-1}{x}\right)} \]

   3. *-inverses 50.4%

      \[\leadsto \mathsf{fma}\left(e^{-\mathsf{log1p}\left(x\right)}, \color{blue}{\frac{-x}{-x}}, \frac{-1}{x}\right) \]

   4. metadata-eval 50.4%

      \[\leadsto \mathsf{fma}\left(e^{-\mathsf{log1p}\left(x\right)}, \frac{-x}{-x}, \frac{\color{blue}{-1}}{x}\right) \]

   5. distribute-neg-frac 50.4%

      \[\leadsto \mathsf{fma}\left(e^{-\mathsf{log1p}\left(x\right)}, \frac{-x}{-x}, \color{blue}{-\frac{1}{x}}\right) \]

   6. fma-neg 50.4%

      \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(x\right)} \cdot \frac{-x}{-x} - \frac{1}{x}} \]

   7. *-inverses 50.4%

      \[\leadsto e^{-\mathsf{log1p}\left(x\right)} \cdot \color{blue}{1} - \frac{1}{x} \]

   8. *-rgt-identity 50.4%

      \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(x\right)}} - \frac{1}{x} \]

   9. exp-neg 50.4%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right)}}} - \frac{1}{x} \]

   10. log1p-undefine 50.4%

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

   11. rem-exp-log 77.7%

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

   12. *-inverses 77.7%

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

   13. associate-/l/ 53.3%

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

   14. associate-/r* 77.7%

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

   15. div-sub 77.6%

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

   16. *-inverses 77.6%

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

   17. div-sub 78.6%

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

   18. associate-/l/ 78.6%

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

   19. +-commutative 78.6%

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

   20. associate--r+ 99.1%

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

6. Simplified 99.9%

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

Alternative 2: 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 56.4%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. unpow2 96.6%

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

      2. associate-/r* 98.1%

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

   5. Applied egg-rr 98.1%

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

   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 99.4%

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

4. Final simplification 98.7%

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

Alternative 3: 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 77.7%

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

   1. sub-neg 77.7%

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

   2. add-exp-log 50.4%

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

   3. log-rec 50.4%

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

   4. +-commutative 50.4%

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

   5. log1p-define 50.4%

      \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(x\right)}} + \left(-\frac{1}{x}\right) \]

   6. distribute-neg-frac 50.4%

      \[\leadsto e^{-\mathsf{log1p}\left(x\right)} + \color{blue}{\frac{-1}{x}} \]

   7. metadata-eval 50.4%

      \[\leadsto e^{-\mathsf{log1p}\left(x\right)} + \frac{\color{blue}{-1}}{x} \]

4. Applied egg-rr 50.4%

   \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(x\right)} + \frac{-1}{x}} \]
5. Step-by-step derivation

   1. *-rgt-identity 50.4%

      \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(x\right)} \cdot 1} + \frac{-1}{x} \]

   2. fma-undefine 50.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{-\mathsf{log1p}\left(x\right)}, 1, \frac{-1}{x}\right)} \]

   3. *-inverses 50.4%

      \[\leadsto \mathsf{fma}\left(e^{-\mathsf{log1p}\left(x\right)}, \color{blue}{\frac{-x}{-x}}, \frac{-1}{x}\right) \]

   4. metadata-eval 50.4%

      \[\leadsto \mathsf{fma}\left(e^{-\mathsf{log1p}\left(x\right)}, \frac{-x}{-x}, \frac{\color{blue}{-1}}{x}\right) \]

   5. distribute-neg-frac 50.4%

      \[\leadsto \mathsf{fma}\left(e^{-\mathsf{log1p}\left(x\right)}, \frac{-x}{-x}, \color{blue}{-\frac{1}{x}}\right) \]

   6. fma-neg 50.4%

      \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(x\right)} \cdot \frac{-x}{-x} - \frac{1}{x}} \]

   7. *-inverses 50.4%

      \[\leadsto e^{-\mathsf{log1p}\left(x\right)} \cdot \color{blue}{1} - \frac{1}{x} \]

   8. *-rgt-identity 50.4%

      \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(x\right)}} - \frac{1}{x} \]

   9. exp-neg 50.4%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right)}}} - \frac{1}{x} \]

   10. log1p-undefine 50.4%

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

   11. rem-exp-log 77.7%

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

   12. *-inverses 77.7%

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

   13. associate-/l/ 53.3%

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

   14. associate-/r* 77.7%

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

   15. div-sub 77.6%

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

   16. *-inverses 77.6%

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

   17. div-sub 78.6%

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

   18. associate-/l/ 78.6%

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

   19. +-commutative 78.6%

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

   20. associate--r+ 99.1%

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

6. Simplified 99.9%

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

   1. associate-/l/ 99.1%

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

   2. *-commutative 99.1%

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

8. Applied egg-rr 99.1%

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

Alternative 4: 52.4% 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 77.7%

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

3. Taylor expanded in x around 0 51.2%

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

Alternative 5: 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 77.7%

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

3. Taylor expanded in x around inf 51.1%

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

   1. unpow2 51.1%

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

   2. associate-/r* 51.9%

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

5. Applied egg-rr 51.9%

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

   1. div-inv 51.8%

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

   2. frac-2neg 51.8%

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

   3. metadata-eval 51.8%

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

   4. frac-times 51.1%

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

   5. metadata-eval 51.1%

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

7. Applied egg-rr 51.1%

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

9. Applied egg-rr 3.7%

   \[\leadsto \color{blue}{\frac{x}{\left(0 + \left(x + 0 \cdot x\right)\right) \cdot -1}} \]
10. Step-by-step derivation

    1. +-lft-identity 3.7%

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

    2. mul0-lft 3.7%

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

    3. +-rgt-identity 3.7%

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

    4. associate-/r* 3.7%

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

    5. *-inverses 3.7%

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

    6. metadata-eval 3.7%

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

11. Simplified 3.7%

    \[\leadsto \color{blue}{-1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))