2frac (problem 3.3.1)

Percentage Accurate: 76.9% → 99.9%

Time: 3.4s

Precision: binary64

Cost: 448

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 76.1%

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

2. Applied egg-rr 77.0%

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

   [Start] 76.1%	\[ \frac{1}{x + 1} - \frac{1}{x} \]
   frac-sub [=>] 77.0%	\[ \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}} \]
   *-rgt-identity [<=] 77.0%	\[ \frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\color{blue}{\left(\left(x + 1\right) \cdot 1\right)} \cdot x} \]
   metadata-eval [<=] 77.0%	\[ \frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(\left(x + 1\right) \cdot \color{blue}{\frac{1}{1}}\right) \cdot x} \]
   div-inv [<=] 77.0%	\[ \frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\color{blue}{\frac{x + 1}{1}} \cdot x} \]
   associate-/r* [=>] 77.0%	\[ \color{blue}{\frac{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\frac{x + 1}{1}}}{x}} \]
   *-un-lft-identity [<=] 77.0%	\[ \frac{\frac{\color{blue}{x} - \left(x + 1\right) \cdot 1}{\frac{x + 1}{1}}}{x} \]
   *-rgt-identity [=>] 77.0%	\[ \frac{\frac{x - \color{blue}{\left(x + 1\right)}}{\frac{x + 1}{1}}}{x} \]
   +-commutative [=>] 77.0%	\[ \frac{\frac{x - \color{blue}{\left(1 + x\right)}}{\frac{x + 1}{1}}}{x} \]
   div-inv [=>] 77.0%	\[ \frac{\frac{x - \left(1 + x\right)}{\color{blue}{\left(x + 1\right) \cdot \frac{1}{1}}}}{x} \]
   metadata-eval [=>] 77.0%	\[ \frac{\frac{x - \left(1 + x\right)}{\left(x + 1\right) \cdot \color{blue}{1}}}{x} \]
   *-rgt-identity [=>] 77.0%	\[ \frac{\frac{x - \left(1 + x\right)}{\color{blue}{x + 1}}}{x} \]
   +-commutative [=>] 77.0%	\[ \frac{\frac{x - \left(1 + x\right)}{\color{blue}{1 + x}}}{x} \]

3. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	448

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

Alternative 2
Accuracy	98.1%
Cost	585

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

Alternative 3
Accuracy	99.4%
Cost	448

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

Alternative 4
Accuracy	99.9%
Cost	448

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

Alternative 5
Accuracy	52.5%
Cost	192

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

Reproduce

herbie shell --seed 2023178 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))