?

Average Error: 7.9 → 0.2
Time: 2.0min
Precision: binary64
Cost: 6720

?

\[0.999 \leq x \land x \leq 1.001\]
\[\frac{10}{1 - x \cdot x} \]
↓
\[\frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \]
(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 / 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?

Derivation?

  1. Initial program 7.9

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

  2. Simplified0.2

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

Alternatives

Alternative 1
Error0.3
Cost704
\[\frac{10}{\left(1 - x\right) - \left(-1 + x\right) \cdot x} \]
Alternative 2
Error0.4
Cost576
\[\frac{10}{\left(1 - x\right) \cdot \left(1 + x\right)} \]
Alternative 3
Error0.4
Cost576
\[\frac{\frac{10}{1 - x}}{1 + x} \]
Alternative 4
Error7.9
Cost448
\[\frac{10}{1 - x \cdot x} \]
Alternative 5
Error58.0
Cost64
\[10 \]

Error

Reproduce?

herbie shell --seed 2023033 
(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))))