ENA, Section 1.4, Mentioned, B

Average Error: 7.9 → 0.3

Time: 9.3s

Precision: binary64

Cost: 6720

\[0.999 \leq x \land x \leq 1.001\]

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (/ 10.0 (- 1.0 (* x x))))

↓

(FPCore (x) :precision binary64 (/ -10.0 (fma x x -1.0)))

C

double code(double x) {
	return 10.0 / (1.0 - (x * x));
}

↓

double code(double x) {
	return -10.0 / fma(x, x, -1.0);
}

Julia

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

Wolfram

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]

Derivation

1. Initial program 7.9

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

2. Simplified 0.3

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

   [Start] 7.9	\[ \frac{10}{1 - x \cdot x} \]
   sub-neg [=>] 7.9	\[ \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]
   +-commutative [=>] 7.9	\[ \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]
   neg-sub0 [=>] 7.9	\[ \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]
   associate-+l- [=>] 7.9	\[ \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]
   sub0-neg [=>] 7.9	\[ \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]
   neg-mul-1 [=>] 7.9	\[ \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]
   associate-/r* [=>] 7.9	\[ \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]
   metadata-eval [=>] 7.9	\[ \frac{\color{blue}{-10}}{x \cdot x - 1} \]
   fma-neg [=>] 0.3	\[ \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]
   metadata-eval [=>] 0.3	\[ \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	55.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;10\\ \mathbf{else}:\\ \;\;\;\;\frac{-10}{x \cdot x}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	576

\[\frac{-10}{\left(x + -1\right) \cdot \left(x + 1\right)} \]

Alternative 3
Error	0.4
Cost	576

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

Alternative 4
Error	7.9
Cost	448

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

Alternative 5
Error	52.0
Cost	320

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

Alternative 6
Error	57.9
Cost	64

\[10 \]

Reproduce

herbie shell --seed 2023045 
(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))))