2frac (problem 3.3.1)

Average Accuracy: 77.7% → 99.9%

Time: 5.2s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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)) - (1.0d0 / x)
end function

↓

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)) - (1.0 / x);
}

↓

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

Python

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

↓

def code(x):
	return (1.0 / x) / (-1.0 - 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 / x) / Float64(-1.0 - x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (1.0 / x) / (-1.0 - 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 / x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.7%

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

2. Applied egg-rr 78.6%

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

   [Start] 77.7	\[ \frac{1}{x + 1} - \frac{1}{x} \]
   frac-sub [=>] 78.6	\[ \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}} \]
   frac-2neg [=>] 78.6	\[ \color{blue}{\frac{-\left(1 \cdot x - \left(x + 1\right) \cdot 1\right)}{-\left(x + 1\right) \cdot x}} \]
   *-un-lft-identity [<=] 78.6	\[ \frac{-\left(\color{blue}{x} - \left(x + 1\right) \cdot 1\right)}{-\left(x + 1\right) \cdot x} \]
   cancel-sign-sub-inv [=>] 78.6	\[ \frac{-\color{blue}{\left(x + \left(-\left(x + 1\right)\right) \cdot 1\right)}}{-\left(x + 1\right) \cdot x} \]
   *-commutative [<=] 78.6	\[ \frac{-\left(x + \color{blue}{1 \cdot \left(-\left(x + 1\right)\right)}\right)}{-\left(x + 1\right) \cdot x} \]
   *-un-lft-identity [<=] 78.6	\[ \frac{-\left(x + \color{blue}{\left(-\left(x + 1\right)\right)}\right)}{-\left(x + 1\right) \cdot x} \]
   distribute-neg-in [=>] 78.6	\[ \frac{\color{blue}{\left(-x\right) + \left(-\left(-\left(x + 1\right)\right)\right)}}{-\left(x + 1\right) \cdot x} \]
   *-un-lft-identity [=>] 78.6	\[ \frac{\left(-x\right) + \left(-\color{blue}{1 \cdot \left(-\left(x + 1\right)\right)}\right)}{-\left(x + 1\right) \cdot x} \]
   distribute-lft-neg-in [=>] 78.6	\[ \frac{\left(-x\right) + \color{blue}{\left(-1\right) \cdot \left(-\left(x + 1\right)\right)}}{-\left(x + 1\right) \cdot x} \]
   *-commutative [<=] 78.6	\[ \frac{\left(-x\right) + \color{blue}{\left(-\left(x + 1\right)\right) \cdot \left(-1\right)}}{-\left(x + 1\right) \cdot x} \]
   cancel-sign-sub-inv [<=] 78.6	\[ \frac{\color{blue}{\left(-x\right) - \left(x + 1\right) \cdot \left(-1\right)}}{-\left(x + 1\right) \cdot x} \]
   distribute-rgt-neg-in [<=] 78.6	\[ \frac{\left(-x\right) - \color{blue}{\left(-\left(x + 1\right) \cdot 1\right)}}{-\left(x + 1\right) \cdot x} \]
   distribute-lft-neg-in [=>] 78.6	\[ \frac{\left(-x\right) - \color{blue}{\left(-\left(x + 1\right)\right) \cdot 1}}{-\left(x + 1\right) \cdot x} \]
   cancel-sign-sub [=>] 78.6	\[ \frac{\color{blue}{\left(-x\right) + \left(x + 1\right) \cdot 1}}{-\left(x + 1\right) \cdot x} \]
   *-rgt-identity [=>] 78.6	\[ \frac{\left(-x\right) + \color{blue}{\left(x + 1\right)}}{-\left(x + 1\right) \cdot x} \]
   +-commutative [=>] 78.6	\[ \frac{\left(-x\right) + \color{blue}{\left(1 + x\right)}}{-\left(x + 1\right) \cdot x} \]
   distribute-lft-neg-in [=>] 78.6	\[ \frac{\left(-x\right) + \left(1 + x\right)}{\color{blue}{\left(-\left(x + 1\right)\right) \cdot x}} \]
   *-commutative [=>] 78.6	\[ \frac{\left(-x\right) + \left(1 + x\right)}{\color{blue}{x \cdot \left(-\left(x + 1\right)\right)}} \]
   neg-sub0 [=>] 78.6	\[ \frac{\left(-x\right) + \left(1 + x\right)}{x \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}} \]
   metadata-eval [<=] 78.6	\[ \frac{\left(-x\right) + \left(1 + x\right)}{x \cdot \left(\color{blue}{\log 1} - \left(x + 1\right)\right)} \]
   +-commutative [=>] 78.6	\[ \frac{\left(-x\right) + \left(1 + x\right)}{x \cdot \left(\log 1 - \color{blue}{\left(1 + x\right)}\right)} \]
   associate--r+ [=>] 78.6	\[ \frac{\left(-x\right) + \left(1 + x\right)}{x \cdot \color{blue}{\left(\left(\log 1 - 1\right) - x\right)}} \]

3. Simplified 78.6%

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

   [Start] 78.6	\[ \frac{\left(-x\right) + \left(1 + x\right)}{x \cdot \left(-1 - x\right)} \]
   associate-+r+ [=>] 78.6	\[ \frac{\color{blue}{\left(\left(-x\right) + 1\right) + x}}{x \cdot \left(-1 - x\right)} \]
   +-commutative [<=] 78.6	\[ \frac{\color{blue}{\left(1 + \left(-x\right)\right)} + x}{x \cdot \left(-1 - x\right)} \]
   sub-neg [<=] 78.6	\[ \frac{\color{blue}{\left(1 - x\right)} + x}{x \cdot \left(-1 - x\right)} \]

4. Applied egg-rr 99.3%

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

   [Start] 78.6	\[ \frac{\left(1 - x\right) + x}{x \cdot \left(-1 - x\right)} \]
   associate-+l- [=>] 99.3	\[ \frac{\color{blue}{1 - \left(x - x\right)}}{x \cdot \left(-1 - x\right)} \]
   +-inverses [=>] 99.3	\[ \frac{1 - \color{blue}{0}}{x \cdot \left(-1 - x\right)} \]
   metadata-eval [=>] 99.3	\[ \frac{\color{blue}{1}}{x \cdot \left(-1 - x\right)} \]
   metadata-eval [<=] 99.3	\[ \frac{\color{blue}{1 - 0}}{x \cdot \left(-1 - x\right)} \]
   metadata-eval [<=] 99.3	\[ \frac{1 - \color{blue}{\log 1}}{x \cdot \left(-1 - x\right)} \]
   div-sub [=>] 99.3	\[ \color{blue}{\frac{1}{x \cdot \left(-1 - x\right)} - \frac{\log 1}{x \cdot \left(-1 - x\right)}} \]
   metadata-eval [=>] 99.3	\[ \frac{1}{x \cdot \left(-1 - x\right)} - \frac{\color{blue}{0}}{x \cdot \left(-1 - x\right)} \]

5. Simplified 99.9%

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

   [Start] 99.3	\[ \frac{1}{x \cdot \left(-1 - x\right)} - \frac{0}{x \cdot \left(-1 - x\right)} \]
   div0 [=>] 99.3	\[ \frac{1}{x \cdot \left(-1 - x\right)} - \color{blue}{0} \]
   --rgt-identity [=>] 99.3	\[ \color{blue}{\frac{1}{x \cdot \left(-1 - x\right)}} \]
   associate-/r* [=>] 99.9	\[ \color{blue}{\frac{\frac{1}{x}}{-1 - x}} \]

6. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	97.9%
Cost	713

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

Alternative 2
Accuracy	97.7%
Cost	585

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

Alternative 3
Accuracy	99.3%
Cost	448

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

Alternative 4
Accuracy	99.3%
Cost	448

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

Alternative 5
Accuracy	51.9%
Cost	192

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

Alternative 6
Accuracy	3.2%
Cost	128

\[-x \]

Alternative 7
Accuracy	3.0%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))