2frac (problem 3.3.1)

Average Error: 14.6 → 0.1

Time: 2.6s

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 14.6

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

2. Simplified 14.6

   \[\leadsto \color{blue}{\frac{-1}{x} - \frac{-1}{1 + x}} \]
   Proof
   (-.f64 (/.f64 -1 x) (/.f64 -1 (+.f64 1 x))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) x) (/.f64 -1 (+.f64 1 x))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 x))) (/.f64 -1 (+.f64 1 x))): 0 points increase in error, 0 points decrease in error
   (-.f64 (neg.f64 (/.f64 1 x)) (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) (+.f64 1 x))): 0 points increase in error, 0 points decrease in error
   (-.f64 (neg.f64 (/.f64 1 x)) (/.f64 (neg.f64 1) (Rewrite<= +-commutative_binary64 (+.f64 x 1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (neg.f64 (/.f64 1 x)) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (/.f64 1 x)) (neg.f64 (neg.f64 (/.f64 1 (+.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (neg.f64 (/.f64 1 x)) (Rewrite=> remove-double-neg_binary64 (/.f64 1 (+.f64 x 1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 (+.f64 x 1)) (neg.f64 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 1 x))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 14.0

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

4. Taylor expanded in x around 0 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.9
Cost	584

\[\begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -51841659923.4166:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.06660068974399568:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.6
Cost	584

\[\begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -51841659923.4166:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.06660068974399568:\\ \;\;\;\;\frac{-1}{x} + 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.3
Cost	448

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

Alternative 4
Error	31.1
Cost	192

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

Reproduce

herbie shell --seed 2022300 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))