2frac (problem 3.3.1)

Average Accuracy: 77.2% → 99.3%

Time: 3.2s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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 + 1.0d0)) - (1.0d0 / x)
end function

↓

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 + 1.0)) - (1.0 / x);
}

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 77.2%

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

2. Applied egg-rr 77.2%

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

   [Start] 77.2	\[ \frac{1}{x + 1} - \frac{1}{x} \]
   sub-neg [=>] 77.2	\[ \color{blue}{\frac{1}{x + 1} + \left(-\frac{1}{x}\right)} \]
   +-commutative [=>] 77.2	\[ \frac{1}{\color{blue}{1 + x}} + \left(-\frac{1}{x}\right) \]
   distribute-neg-frac [=>] 77.2	\[ \frac{1}{1 + x} + \color{blue}{\frac{-1}{x}} \]
   metadata-eval [=>] 77.2	\[ \frac{1}{1 + x} + \frac{\color{blue}{-1}}{x} \]

3. Simplified 99.3%

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

   [Start] 77.2	\[ \frac{1}{1 + x} + \frac{-1}{x} \]
   +-commutative [<=] 77.2	\[ \color{blue}{\frac{-1}{x} + \frac{1}{1 + x}} \]
   *-rgt-identity [<=] 77.2	\[ \color{blue}{\frac{-1}{x} \cdot 1} + \frac{1}{1 + x} \]
   associate-*l/ [=>] 77.2	\[ \color{blue}{\frac{-1 \cdot 1}{x}} + \frac{1}{1 + x} \]
   associate-*r/ [<=] 77.2	\[ \color{blue}{-1 \cdot \frac{1}{x}} + \frac{1}{1 + x} \]
   neg-mul-1 [<=] 77.2	\[ \color{blue}{\left(-\frac{1}{x}\right)} + \frac{1}{1 + x} \]
   neg-sub0 [=>] 77.2	\[ \color{blue}{\left(0 - \frac{1}{x}\right)} + \frac{1}{1 + x} \]
   associate-+l- [=>] 77.2	\[ \color{blue}{0 - \left(\frac{1}{x} - \frac{1}{1 + x}\right)} \]
   sub0-neg [=>] 77.2	\[ \color{blue}{-\left(\frac{1}{x} - \frac{1}{1 + x}\right)} \]
   sub-neg [=>] 77.2	\[ -\color{blue}{\left(\frac{1}{x} + \left(-\frac{1}{1 + x}\right)\right)} \]
   distribute-neg-out [<=] 77.2	\[ \color{blue}{\left(-\frac{1}{x}\right) + \left(-\left(-\frac{1}{1 + x}\right)\right)} \]
   distribute-neg-frac [=>] 77.2	\[ \color{blue}{\frac{-1}{x}} + \left(-\left(-\frac{1}{1 + x}\right)\right) \]
   metadata-eval [=>] 77.2	\[ \frac{\color{blue}{-1}}{x} + \left(-\left(-\frac{1}{1 + x}\right)\right) \]
   remove-double-neg [=>] 77.2	\[ \frac{-1}{x} + \color{blue}{\frac{1}{1 + x}} \]
   remove-double-neg [<=] 77.2	\[ \frac{-1}{x} + \frac{1}{\color{blue}{-\left(-\left(1 + x\right)\right)}} \]
   +-commutative [=>] 77.2	\[ \frac{-1}{x} + \frac{1}{-\left(-\color{blue}{\left(x + 1\right)}\right)} \]
   distribute-neg-in [=>] 77.2	\[ \frac{-1}{x} + \frac{1}{-\color{blue}{\left(\left(-x\right) + \left(-1\right)\right)}} \]
   neg-mul-1 [=>] 77.2	\[ \frac{-1}{x} + \frac{1}{-\left(\color{blue}{-1 \cdot x} + \left(-1\right)\right)} \]
   metadata-eval [=>] 77.2	\[ \frac{-1}{x} + \frac{1}{-\left(-1 \cdot x + \color{blue}{-1}\right)} \]
   fma-udef [<=] 77.2	\[ \frac{-1}{x} + \frac{1}{-\color{blue}{\mathsf{fma}\left(-1, x, -1\right)}} \]
   neg-mul-1 [=>] 77.2	\[ \frac{-1}{x} + \frac{1}{\color{blue}{-1 \cdot \mathsf{fma}\left(-1, x, -1\right)}} \]
   associate-/r* [=>] 77.2	\[ \frac{-1}{x} + \color{blue}{\frac{\frac{1}{-1}}{\mathsf{fma}\left(-1, x, -1\right)}} \]
   metadata-eval [=>] 77.2	\[ \frac{-1}{x} + \frac{\color{blue}{-1}}{\mathsf{fma}\left(-1, x, -1\right)} \]
   metadata-eval [<=] 77.2	\[ \frac{-1}{x} + \frac{\color{blue}{-1 \cdot 1}}{\mathsf{fma}\left(-1, x, -1\right)} \]
   associate-*r/ [<=] 77.2	\[ \frac{-1}{x} + \color{blue}{-1 \cdot \frac{1}{\mathsf{fma}\left(-1, x, -1\right)}} \]

4. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	97.4%
Cost	585

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

Alternative 2
Accuracy	97.9%
Cost	585

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

Alternative 3
Accuracy	98.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;\frac{-1 + x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \end{array} \]

Alternative 4
Accuracy	52.1%
Cost	192

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

Alternative 5
Accuracy	3.0%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))