Average Error: 0.1 → 0.1
Time: 24.1s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{\sqrt{a - \frac{1.0}{3.0}}}{3}, rand, a - \frac{1.0}{3.0}\right)\]
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
↓
\mathsf{fma}\left(\frac{\sqrt{a - \frac{1.0}{3.0}}}{3}, rand, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r1118050 = a;
        double r1118051 = 1.0;
        double r1118052 = 3.0;
        double r1118053 = r1118051 / r1118052;
        double r1118054 = r1118050 - r1118053;
        double r1118055 = 1.0;
        double r1118056 = 9.0;
        double r1118057 = r1118056 * r1118054;
        double r1118058 = sqrt(r1118057);
        double r1118059 = r1118055 / r1118058;
        double r1118060 = rand;
        double r1118061 = r1118059 * r1118060;
        double r1118062 = r1118055 + r1118061;
        double r1118063 = r1118054 * r1118062;
        return r1118063;
}
↓
double f(double a, double rand) {
        double r1118064 = a;
        double r1118065 = 1.0;
        double r1118066 = 3.0;
        double r1118067 = r1118065 / r1118066;
        double r1118068 = r1118064 - r1118067;
        double r1118069 = sqrt(r1118068);
        double r1118070 = 3.0;
        double r1118071 = r1118069 / r1118070;
        double r1118072 = rand;
        double r1118073 = fma(r1118071, r1118072, r1118068);
        return r1118073;
}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.1

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a - \frac{1.0}{3.0}}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}, rand, a - \frac{1.0}{3.0}\right)}\]
  3. Using strategy rm

  4. Applied sqrt-prod0.1

    \[\leadsto \mathsf{fma}\left(\frac{a - \frac{1.0}{3.0}}{\color{blue}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}}}, rand, a - \frac{1.0}{3.0}\right)\]

  5. Applied add-sqr-sqrt0.2

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sqrt{a - \frac{1.0}{3.0}} \cdot \sqrt{a - \frac{1.0}{3.0}}}}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}}, rand, a - \frac{1.0}{3.0}\right)\]

  6. Applied times-frac0.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{9}} \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}}}, rand, a - \frac{1.0}{3.0}\right)\]

  7. Simplified0.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt{a - \frac{1.0}{3.0}}}{3}} \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}}, rand, a - \frac{1.0}{3.0}\right)\]

  8. Simplified0.1

    \[\leadsto \mathsf{fma}\left(\frac{\sqrt{a - \frac{1.0}{3.0}}}{3} \cdot \color{blue}{1}, rand, a - \frac{1.0}{3.0}\right)\]

  9. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\frac{\sqrt{a - \frac{1.0}{3.0}}}{3}, rand, a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))