Average Error: 0.1 → 0.1
Time: 1.0m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1.0 + \frac{1.0}{\sqrt{9.0 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[1.0 \cdot \mathsf{fma}\left(rand \cdot \frac{{\left(a - \frac{1.0}{3.0}\right)}^{\frac{-1}{2}}}{\sqrt{9.0}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1.0 + \frac{1.0}{\sqrt{9.0 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
↓
1.0 \cdot \mathsf{fma}\left(rand \cdot \frac{{\left(a - \frac{1.0}{3.0}\right)}^{\frac{-1}{2}}}{\sqrt{9.0}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r2738033 = a;
        double r2738034 = 1.0;
        double r2738035 = 3.0;
        double r2738036 = r2738034 / r2738035;
        double r2738037 = r2738033 - r2738036;
        double r2738038 = 9.0;
        double r2738039 = r2738038 * r2738037;
        double r2738040 = sqrt(r2738039);
        double r2738041 = r2738034 / r2738040;
        double r2738042 = rand;
        double r2738043 = r2738041 * r2738042;
        double r2738044 = r2738034 + r2738043;
        double r2738045 = r2738037 * r2738044;
        return r2738045;
}
↓
double f(double a, double rand) {
        double r2738046 = 1.0;
        double r2738047 = rand;
        double r2738048 = a;
        double r2738049 = 3.0;
        double r2738050 = r2738046 / r2738049;
        double r2738051 = r2738048 - r2738050;
        double r2738052 = -0.5;
        double r2738053 = pow(r2738051, r2738052);
        double r2738054 = 9.0;
        double r2738055 = sqrt(r2738054);
        double r2738056 = r2738053 / r2738055;
        double r2738057 = r2738047 * r2738056;
        double r2738058 = fma(r2738057, r2738051, r2738051);
        double r2738059 = r2738046 * r2738058;
        return r2738059;
}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.1

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1.0 + \frac{1.0}{\sqrt{9.0 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9.0}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0}\]
  3. Using strategy rm

  4. Applied sqrt-prod0.1

    \[\leadsto \mathsf{fma}\left(\frac{rand}{\color{blue}{\sqrt{a - \frac{1.0}{3.0}} \cdot \sqrt{9.0}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0\]

  5. Applied associate-/r*0.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{rand}{\sqrt{a - \frac{1.0}{3.0}}}}{\sqrt{9.0}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.1

    \[\leadsto \mathsf{fma}\left(\frac{\frac{rand}{\sqrt{a - \frac{1.0}{3.0}}}}{\sqrt{\color{blue}{1 \cdot 9.0}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0\]

  8. Applied sqrt-prod0.1

    \[\leadsto \mathsf{fma}\left(\frac{\frac{rand}{\sqrt{a - \frac{1.0}{3.0}}}}{\color{blue}{\sqrt{1} \cdot \sqrt{9.0}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0\]

  9. Applied div-inv0.1

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{rand \cdot \frac{1}{\sqrt{a - \frac{1.0}{3.0}}}}}{\sqrt{1} \cdot \sqrt{9.0}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0\]

  10. Applied times-frac0.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{rand}{\sqrt{1}} \cdot \frac{\frac{1}{\sqrt{a - \frac{1.0}{3.0}}}}{\sqrt{9.0}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0\]

  11. Simplified0.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{rand} \cdot \frac{\frac{1}{\sqrt{a - \frac{1.0}{3.0}}}}{\sqrt{9.0}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0\]
  12. Using strategy rm

  13. Applied pow1/20.1

    \[\leadsto \mathsf{fma}\left(rand \cdot \frac{\frac{1}{\color{blue}{{\left(a - \frac{1.0}{3.0}\right)}^{\frac{1}{2}}}}}{\sqrt{9.0}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0\]

  14. Applied pow-flip0.1

    \[\leadsto \mathsf{fma}\left(rand \cdot \frac{\color{blue}{{\left(a - \frac{1.0}{3.0}\right)}^{\left(-\frac{1}{2}\right)}}}{\sqrt{9.0}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0\]

  15. Simplified0.1

    \[\leadsto \mathsf{fma}\left(rand \cdot \frac{{\left(a - \frac{1.0}{3.0}\right)}^{\color{blue}{\frac{-1}{2}}}}{\sqrt{9.0}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right) \cdot 1.0\]

  16. Final simplification0.1

    \[\leadsto 1.0 \cdot \mathsf{fma}\left(rand \cdot \frac{{\left(a - \frac{1.0}{3.0}\right)}^{\frac{-1}{2}}}{\sqrt{9.0}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))