2frac (problem 3.3.1)

Average Error: 14.5 → 0.1

Time: 5.2s

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 14.5

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

2. Applied egg-rr 13.8

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

3. Simplified 0.1

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

   [Start] 13.8	\[ \frac{\left(-x\right) + \left(1 + x\right)}{x \cdot \left(-1 - x\right)} \]
   associate-/l/ [<=] 13.8	\[ \color{blue}{\frac{\frac{\left(-x\right) + \left(1 + x\right)}{-1 - x}}{x}} \]
   +-commutative [=>] 13.8	\[ \frac{\frac{\color{blue}{\left(1 + x\right) + \left(-x\right)}}{-1 - x}}{x} \]
   unsub-neg [=>] 13.8	\[ \frac{\frac{\color{blue}{\left(1 + x\right) - x}}{-1 - x}}{x} \]
   associate--l+ [=>] 0.1	\[ \frac{\frac{\color{blue}{1 + \left(x - x\right)}}{-1 - x}}{x} \]

4. Applied egg-rr 32.2

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

5. Simplified 0.1

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

   [Start] 32.2	\[ e^{\mathsf{log1p}\left(\frac{1}{x \cdot \left(-1 - x\right)}\right)} - 1 \]
   expm1-def [=>] 17.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x \cdot \left(-1 - x\right)}\right)\right)} \]
   expm1-log1p [=>] 0.4	\[ \color{blue}{\frac{1}{x \cdot \left(-1 - x\right)}} \]
   associate-/r* [=>] 0.1	\[ \color{blue}{\frac{\frac{1}{x}}{-1 - x}} \]

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.4
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
Error	1.0
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
Error	0.4
Cost	448

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

Alternative 4
Error	30.6
Cost	192

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

Reproduce

herbie shell --seed 2023060 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))