Average Error: 7.8 → 0.2
Time: 1.5s
Precision: binary64
\[0.999 \leq x \land x \leq 1.001\]
\[\frac{10}{1 - x \cdot x} \]
↓
\[\frac{10}{-\mathsf{fma}\left(x, x, -1\right)} \]
(FPCore (x) :precision binary64 (/ 10.0 (- 1.0 (* x x))))
↓
(FPCore (x) :precision binary64 (/ 10.0 (- (fma x x -1.0))))
double code(double x) {
	return 10.0 / (1.0 - (x * x));
}
↓
double code(double x) {
	return 10.0 / -fma(x, x, -1.0);
}

function code(x)
	return Float64(10.0 / Float64(1.0 - Float64(x * x)))
end
↓
function code(x)
	return Float64(10.0 / Float64(-fma(x, x, -1.0)))
end

code[x_] := N[(10.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(10.0 / (-N[(x * x + -1.0), $MachinePrecision])), $MachinePrecision]

\frac{10}{1 - x \cdot x}
↓
\frac{10}{-\mathsf{fma}\left(x, x, -1\right)}

Error

Bits error versus x

Derivation

  1. Initial program 7.8

    \[\frac{10}{1 - x \cdot x} \]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]

  3. Applied egg-rr0.6

    \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}\right)}^{3}}} \]

  4. Applied egg-rr0.2

    \[\leadsto \color{blue}{\frac{10}{-\mathsf{fma}\left(x, x, -1\right)}} \]

  5. Final simplification0.2

    \[\leadsto \frac{10}{-\mathsf{fma}\left(x, x, -1\right)} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, B"
  :precision binary64
  :pre (and (<= 0.999 x) (<= x 1.001))
  (/ 10.0 (- 1.0 (* x x))))