2frac (problem 3.3.1)

Average Error: 14.9 → 0.4

Time: 3.5s

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 14.9

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

2. Applied egg-rr 14.3

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

3. Taylor expanded in x around 0 0.4

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.0
Cost	713

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

Alternative 2
Error	1.4
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
Error	1.1
Cost	585

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

Alternative 4
Error	31.4
Cost	192

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

Reproduce

herbie shell --seed 2022354 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))