2frac (problem 3.3.1)

Average Accuracy: 77.8% → 99.4%

Time: 3.2s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

def code(x):
	return -1.0 / (x + (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(-1.0 / Float64(x + Float64(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 / (x + (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[(-1.0 / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

2. Applied egg-rr 79.2%

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

   [Start] 79.2	\[ \frac{1}{x + 1} - \frac{1}{x} \]
   sub-neg [=>] 79.2	\[ \color{blue}{\frac{1}{x + 1} + \left(-\frac{1}{x}\right)} \]
   +-commutative [=>] 79.2	\[ \frac{1}{\color{blue}{1 + x}} + \left(-\frac{1}{x}\right) \]
   distribute-neg-frac [=>] 79.2	\[ \frac{1}{1 + x} + \color{blue}{\frac{-1}{x}} \]
   metadata-eval [=>] 79.2	\[ \frac{1}{1 + x} + \frac{\color{blue}{-1}}{x} \]

3. Simplified 99.3%

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

   [Start] 79.2	\[ \frac{1}{1 + x} + \frac{-1}{x} \]
   *-rgt-identity [<=] 79.2	\[ \frac{1}{1 + x} + \color{blue}{\frac{-1}{x} \cdot 1} \]
   cancel-sign-sub [<=] 79.2	\[ \color{blue}{\frac{1}{1 + x} - \left(-\frac{-1}{x}\right) \cdot 1} \]
   distribute-neg-frac [=>] 79.2	\[ \frac{1}{1 + x} - \color{blue}{\frac{--1}{x}} \cdot 1 \]
   metadata-eval [=>] 79.2	\[ \frac{1}{1 + x} - \frac{\color{blue}{1}}{x} \cdot 1 \]
   *-rgt-identity [<=] 79.2	\[ \color{blue}{\frac{1}{1 + x} \cdot 1} - \frac{1}{x} \cdot 1 \]
   *-inverses [<=] 79.2	\[ \frac{1}{1 + x} \cdot \color{blue}{\frac{x}{x}} - \frac{1}{x} \cdot 1 \]
   associate-*r/ [=>] 79.2	\[ \color{blue}{\frac{\frac{1}{1 + x} \cdot x}{x}} - \frac{1}{x} \cdot 1 \]
   associate-*l/ [<=] 58.0	\[ \color{blue}{\frac{\frac{1}{1 + x}}{x} \cdot x} - \frac{1}{x} \cdot 1 \]
   associate-/r* [<=] 58.0	\[ \color{blue}{\frac{1}{\left(1 + x\right) \cdot x}} \cdot x - \frac{1}{x} \cdot 1 \]
   associate-/l/ [<=] 58.0	\[ \color{blue}{\frac{\frac{1}{x}}{1 + x}} \cdot x - \frac{1}{x} \cdot 1 \]
   associate-*l/ [=>] 79.2	\[ \color{blue}{\frac{\frac{1}{x} \cdot x}{1 + x}} - \frac{1}{x} \cdot 1 \]
   associate-*r/ [<=] 79.2	\[ \color{blue}{\frac{1}{x} \cdot \frac{x}{1 + x}} - \frac{1}{x} \cdot 1 \]
   distribute-lft-out-- [=>] 79.2	\[ \color{blue}{\frac{1}{x} \cdot \left(\frac{x}{1 + x} - 1\right)} \]
   *-inverses [<=] 79.2	\[ \frac{1}{x} \cdot \left(\frac{x}{1 + x} - \color{blue}{\frac{1 + x}{1 + x}}\right) \]
   div-sub [<=] 79.5	\[ \frac{1}{x} \cdot \color{blue}{\frac{x - \left(1 + x\right)}{1 + x}} \]
   associate-*l/ [=>] 79.5	\[ \color{blue}{\frac{1 \cdot \frac{x - \left(1 + x\right)}{1 + x}}{x}} \]
   associate-*r/ [<=] 79.5	\[ \color{blue}{1 \cdot \frac{\frac{x - \left(1 + x\right)}{1 + x}}{x}} \]
   *-lft-identity [=>] 79.5	\[ \color{blue}{\frac{\frac{x - \left(1 + x\right)}{1 + x}}{x}} \]
   associate-/l/ [=>] 79.5	\[ \color{blue}{\frac{x - \left(1 + x\right)}{x \cdot \left(1 + x\right)}} \]

4. Applied egg-rr 99.4%

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

   [Start] 99.3	\[ \frac{-1}{x \cdot \left(x + 1\right)} \]
   distribute-rgt-in [=>] 99.4	\[ \frac{-1}{\color{blue}{x \cdot x + 1 \cdot x}} \]
   *-un-lft-identity [<=] 99.4	\[ \frac{-1}{x \cdot x + \color{blue}{x}} \]

5. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	97.9%
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 2
Accuracy	98.4%
Cost	585

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

Alternative 3
Accuracy	99.4%
Cost	448

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

Alternative 4
Accuracy	52.0%
Cost	192

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

Reproduce

herbie shell --seed 2023159 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))