Average Error: 0.1 → 0.1
Time: 3.0s
Precision: binary64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
↓
\[\left(a + -0.3333333333333333\right) \cdot \left(1 + {\left(\mathsf{fma}\left(9, a, -3\right)\right)}^{-0.5} \cdot rand\right) \]
(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
 (* (+ a -0.3333333333333333) (+ 1.0 (* (pow (fma 9.0 a -3.0) -0.5) rand))))
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) {
	return (a + -0.3333333333333333) * (1.0 + (pow(fma(9.0, a, -3.0), -0.5) * rand));
}

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)
	return Float64(Float64(a + -0.3333333333333333) * Float64(1.0 + Float64((fma(9.0, a, -3.0) ^ -0.5) * rand)))
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_] := N[(N[(a + -0.3333333333333333), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(9.0 * a + -3.0), $MachinePrecision], -0.5], $MachinePrecision] * rand), $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)
↓
\left(a + -0.3333333333333333\right) \cdot \left(1 + {\left(\mathsf{fma}\left(9, a, -3\right)\right)}^{-0.5} \cdot rand\right)

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. Applied egg-rr0.1

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

  3. Applied egg-rr0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2022153 
(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))))