Average Error: 0.1 → 0.1
Time: 4.1s
Precision: binary64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
↓
\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(a, 9, -3\right)\right)}^{-0.5}\\ \mathsf{fma}\left(rand \cdot t_0, a, a\right) + -0.3333333333333333 \cdot \mathsf{fma}\left(rand, t_0, 1\right) \end{array} \]
(FPCore (a rand)
 :precision binary64
 (*
  (- a (/ 1.0 3.0))
  (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))
↓
(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (pow (fma a 9.0 -3.0) -0.5)))
   (+ (fma (* rand t_0) a a) (* -0.3333333333333333 (fma rand t_0 1.0)))))
double code(double a, double rand) {
	return (a - (1.0 / 3.0)) * (1.0 + ((1.0 / sqrt((9.0 * (a - (1.0 / 3.0))))) * rand));
}
↓
double code(double a, double rand) {
	double t_0 = pow(fma(a, 9.0, -3.0), -0.5);
	return fma((rand * t_0), a, a) + (-0.3333333333333333 * fma(rand, t_0, 1.0));
}

function code(a, rand)
	return Float64(Float64(a - Float64(1.0 / 3.0)) * Float64(1.0 + Float64(Float64(1.0 / sqrt(Float64(9.0 * Float64(a - Float64(1.0 / 3.0))))) * rand)))
end
↓
function code(a, rand)
	t_0 = fma(a, 9.0, -3.0) ^ -0.5
	return Float64(fma(Float64(rand * t_0), a, a) + Float64(-0.3333333333333333 * fma(rand, t_0, 1.0)))
end

code[a_, rand_] := N[(N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / N[Sqrt[N[(9.0 * N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[a_, rand_] := Block[{t$95$0 = N[Power[N[(a * 9.0 + -3.0), $MachinePrecision], -0.5], $MachinePrecision]}, N[(N[(N[(rand * t$95$0), $MachinePrecision] * a + a), $MachinePrecision] + N[(-0.3333333333333333 * N[(rand * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\begin{array}{l}
t_0 := {\left(\mathsf{fma}\left(a, 9, -3\right)\right)}^{-0.5}\\
\mathsf{fma}\left(rand \cdot t_0, a, a\right) + -0.3333333333333333 \cdot \mathsf{fma}\left(rand, t_0, 1\right)
\end{array}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.1

    \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\left(a - 0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}\right)} \]

  3. Applied egg-rr0.1

    \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(rand, {\left(\mathsf{fma}\left(a, 9, -3\right)\right)}^{-0.5}, 1\right) + -0.3333333333333333 \cdot \mathsf{fma}\left(rand, {\left(\mathsf{fma}\left(a, 9, -3\right)\right)}^{-0.5}, 1\right)} \]

  4. Applied egg-rr0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(rand \cdot {\left(\mathsf{fma}\left(a, 9, -3\right)\right)}^{-0.5}, a, a\right)} + -0.3333333333333333 \cdot \mathsf{fma}\left(rand, {\left(\mathsf{fma}\left(a, 9, -3\right)\right)}^{-0.5}, 1\right) \]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(rand \cdot {\left(\mathsf{fma}\left(a, 9, -3\right)\right)}^{-0.5}, a, a\right) + -0.3333333333333333 \cdot \mathsf{fma}\left(rand, {\left(\mathsf{fma}\left(a, 9, -3\right)\right)}^{-0.5}, 1\right) \]

Reproduce

herbie shell --seed 2022159 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))