2frac (problem 3.3.1)

Average Error: 14.2 → 0.1

Time: 2.6s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return (-1.0 / (1.0 + 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) / (1.0d0 + 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 / (1.0 + x)) / x;
}

Python

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

↓

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

Derivation

1. Initial program 14.2

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

2. Applied egg-rr 13.5

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

3. Taylor expanded in x around 0 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.0
Cost	712

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

Alternative 2
Error	1.5
Cost	584

\[\begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.5
Cost	584

\[\begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;\frac{-1 + x}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.1
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;\frac{-1 + x}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.4
Cost	448

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

Alternative 6
Error	30.6
Cost	192

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

Alternative 7
Error	62.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022338 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))