2frac (problem 3.3.1)

Average Error: 14.3 → 0.1

Time: 2.9s

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 14.3

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

2. Applied egg-rr 13.7

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

3. Applied egg-rr 13.7

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

4. Applied egg-rr 13.7

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

5. Simplified 0.1

   \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{-1}}{x + 1} \]
   Proof
   -1: 0 points increase in error, 0 points decrease in error
   (Rewrite<= metadata-eval (-.f64 0 1)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= +-inverses_binary64 (-.f64 x x)) 1): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate--l-_binary64 (-.f64 x (+.f64 x 1))): 23 points increase in error, 98 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 x (neg.f64 (+.f64 x 1)))): 0 points increase in error, 0 points decrease in error

6. Applied egg-rr 0.1

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

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.4
Cost	448

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

Reproduce

herbie shell --seed 2022334 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))