2frac (problem 3.3.1)

Average Error: 14.0 → 0.1

Time: 3.3s

Precision: binary64

Cost: 448

Math

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

↓

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

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(Float64(-1.0 / 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[(N[(-1.0 / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.0

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

2. Applied egg-rr 13.5

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

3. Applied egg-rr 32.1

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x + \left(-1 - x\right)}{x + x \cdot x}\right)} - 1} \]

4. Simplified 0.1

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

   [Start] 32.1	\[ e^{\mathsf{log1p}\left(\frac{x + \left(-1 - x\right)}{x + x \cdot x}\right)} - 1 \]
   expm1-def [=>] 31.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + \left(-1 - x\right)}{x + x \cdot x}\right)\right)} \]
   expm1-log1p [=>] 13.5	\[ \color{blue}{\frac{x + \left(-1 - x\right)}{x + x \cdot x}} \]
   distribute-rgt1-in [=>] 13.5	\[ \frac{x + \left(-1 - x\right)}{\color{blue}{\left(x + 1\right) \cdot x}} \]
   associate-/l/ [<=] 13.5	\[ \color{blue}{\frac{\frac{x + \left(-1 - x\right)}{x}}{x + 1}} \]
   +-commutative [=>] 13.5	\[ \frac{\frac{\color{blue}{\left(-1 - x\right) + x}}{x}}{x + 1} \]
   associate--r- [<=] 0.1	\[ \frac{\frac{\color{blue}{-1 - \left(x - x\right)}}{x}}{x + 1} \]
   sub-neg [=>] 0.1	\[ \frac{\frac{\color{blue}{-1 + \left(-\left(x - x\right)\right)}}{x}}{x + 1} \]
   +-inverses [=>] 0.1	\[ \frac{\frac{-1 + \left(-\color{blue}{0}\right)}{x}}{x + 1} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{-1 + \color{blue}{0}}{x}}{x + 1} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\color{blue}{-1}}{x}}{x + 1} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.1
Cost	712

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

Alternative 2
Error	1.3
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 3
Error	1.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	448

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

Alternative 5
Error	30.2
Cost	192

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

Alternative 6
Error	62.0
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023041 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))