?

?

(FPCore (x a) :precision binary64 (/ x (+ (pow x 2.0) (pow a 2.0))))

(FPCore (x a) :precision binary64 (/ x (fma a a (* x x))))

double code(double x, double a) {
	return x / (pow(x, 2.0) + pow(a, 2.0));
}

double code(double x, double a) {
	return x / fma(a, a, (x * x));
}

function code(x, a)
	return Float64(x / Float64((x ^ 2.0) + (a ^ 2.0)))
end

function code(x, a)
	return Float64(x / fma(a, a, Float64(x * x)))
end

code[x_, a_] := N[(x / N[(N[Power[x, 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

code[x_, a_] := N[(x / N[(a * a + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{x}{{x}^{2} + {a}^{2}}

\frac{x}{\mathsf{fma}\left(a, a, x \cdot x\right)}

Error?

Derivation?

1. Initial program 0.3

   \[\frac{x}{{x}^{2} + {a}^{2}} \]

2. Simplified0.3

   \[\leadsto \color{blue}{\frac{x}{x \cdot x + a \cdot a}} \]
   Proof

   [Start]0.3	\[ \frac{x}{{x}^{2} + {a}^{2}} \]
   unpow2 [=>]0.3	\[ \frac{x}{\color{blue}{x \cdot x} + {a}^{2}} \]
   unpow2 [=>]0.3	\[ \frac{x}{x \cdot x + \color{blue}{a \cdot a}} \]

3. Taylor expanded in x around 0 0.3

   \[\leadsto \frac{x}{\color{blue}{{x}^{2} + {a}^{2}}} \]

4. Simplified0.3

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(a, a, x \cdot x\right)}} \]
   Proof

   [Start]0.3	\[ \frac{x}{{x}^{2} + {a}^{2}} \]
   +-commutative [=>]0.3	\[ \frac{x}{\color{blue}{{a}^{2} + {x}^{2}}} \]
   unpow2 [=>]0.3	\[ \frac{x}{\color{blue}{a \cdot a} + {x}^{2}} \]
   unpow2 [=>]0.3	\[ \frac{x}{a \cdot a + \color{blue}{x \cdot x}} \]
   fma-udef [<=]0.3	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(a, a, x \cdot x\right)}} \]

5. Final simplification0.3

   \[\leadsto \frac{x}{\mathsf{fma}\left(a, a, x \cdot x\right)} \]

Alternatives

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x a)
  :name "x/(x^2+a^2)"
  :precision binary64
  :pre (and (and (<= 1.0 x) (<= x 1000.0)) (and (<= 1.0 a) (<= a 1000.0)))
  (/ x (+ (pow x 2.0) (pow a 2.0))))