2frac (problem 3.3.1)

Average Error: 14.3 → 0.1

Time: 6.0s

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 14.3

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

2. Applied egg-rr 14.3

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

3. Simplified 0.1

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

   [Start] 14.3	\[ \frac{1}{1 + x} + \left(-\frac{1}{x}\right) \]
   rational.json-simplify-1 [<=] 14.3	\[ \color{blue}{\left(-\frac{1}{x}\right) + \frac{1}{1 + x}} \]
   rational.json-simplify-10 [=>] 14.3	\[ \color{blue}{\frac{\frac{1}{x}}{-1}} + \frac{1}{1 + x} \]
   rational.json-simplify-47 [=>] 14.3	\[ \color{blue}{\frac{1}{x \cdot -1}} + \frac{1}{1 + x} \]
   rational.json-simplify-2 [<=] 14.3	\[ \frac{1}{\color{blue}{-1 \cdot x}} + \frac{1}{1 + x} \]
   rational.json-simplify-29 [=>] 13.7	\[ \color{blue}{\frac{-1 \cdot x + \left(1 + x\right)}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)}} \]
   rational.json-simplify-1 [=>] 13.7	\[ \frac{-1 \cdot x + \color{blue}{\left(x + 1\right)}}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   rational.json-simplify-41 [=>] 13.7	\[ \frac{\color{blue}{x + \left(1 + -1 \cdot x\right)}}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   rational.json-simplify-5 [<=] 13.7	\[ \frac{\color{blue}{\left(x - 0\right)} + \left(1 + -1 \cdot x\right)}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   rational.json-simplify-17 [=>] 13.7	\[ \frac{\left(x - 0\right) + \color{blue}{\left(-1 \cdot x - -1\right)}}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   rational.json-simplify-2 [=>] 13.7	\[ \frac{\left(x - 0\right) + \left(\color{blue}{x \cdot -1} - -1\right)}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   rational.json-simplify-9 [=>] 13.7	\[ \frac{\left(x - 0\right) + \left(\color{blue}{\left(-x\right)} - -1\right)}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   rational.json-simplify-12 [=>] 13.7	\[ \frac{\left(x - 0\right) + \left(\color{blue}{\left(0 - x\right)} - -1\right)}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   rational.json-simplify-42 [=>] 13.7	\[ \frac{\left(x - 0\right) + \color{blue}{\left(\left(0 - -1\right) - x\right)}}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   metadata-eval [=>] 13.7	\[ \frac{\left(x - 0\right) + \left(\color{blue}{1} - x\right)}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   rational.json-simplify-27 [=>] 0.4	\[ \frac{\color{blue}{1 - 0}}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{\color{blue}{1}}{\left(-1 \cdot x\right) \cdot \left(1 + x\right)} \]
   rational.json-simplify-46 [=>] 0.1	\[ \color{blue}{\frac{\frac{1}{-1 \cdot x}}{1 + x}} \]
   rational.json-simplify-2 [=>] 0.1	\[ \frac{\frac{1}{\color{blue}{x \cdot -1}}}{1 + x} \]
   rational.json-simplify-47 [<=] 0.1	\[ \frac{\color{blue}{\frac{\frac{1}{x}}{-1}}}{1 + x} \]
   rational.json-simplify-10 [<=] 0.1	\[ \frac{\color{blue}{-\frac{1}{x}}}{1 + x} \]
   rational.json-simplify-17 [=>] 0.1	\[ \frac{-\frac{1}{x}}{\color{blue}{x - -1}} \]
   rational.json-simplify-50 [<=] 0.1	\[ \color{blue}{\frac{\frac{1}{x}}{-1 - x}} \]

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	15.7
Cost	648

\[\begin{array}{l} t_0 := \frac{x}{x \cdot \left(-x\right)}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.4
Cost	448

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

Alternative 3
Error	30.4
Cost	192

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

Reproduce

herbie shell --seed 2023075 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))