Average Error: 0.2 → 0.2
Time: 22.9s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(\left(a - \frac{1.0}{3.0}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot rand\]
\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)
↓
\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(\left(a - \frac{1.0}{3.0}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot rand
double f(double a, double rand) {
        double r3669940 = a;
        double r3669941 = 1.0;
        double r3669942 = /* ERROR: no posit support in C */;
        double r3669943 = 3.0;
        double r3669944 = /* ERROR: no posit support in C */;
        double r3669945 = r3669942 / r3669944;
        double r3669946 = r3669940 - r3669945;
        double r3669947 = 1.0;
        double r3669948 = /* ERROR: no posit support in C */;
        double r3669949 = 9.0;
        double r3669950 = /* ERROR: no posit support in C */;
        double r3669951 = r3669950 * r3669946;
        double r3669952 = sqrt(r3669951);
        double r3669953 = r3669948 / r3669952;
        double r3669954 = rand;
        double r3669955 = r3669953 * r3669954;
        double r3669956 = r3669948 + r3669955;
        double r3669957 = r3669946 * r3669956;
        return r3669957;
}
↓
double f(double a, double rand) {
        double r3669958 = a;
        double r3669959 = 1.0;
        double r3669960 = 3.0;
        double r3669961 = r3669959 / r3669960;
        double r3669962 = r3669958 - r3669961;
        double r3669963 = 1.0;
        double r3669964 = r3669962 * r3669963;
        double r3669965 = 9.0;
        double r3669966 = r3669965 * r3669962;
        double r3669967 = sqrt(r3669966);
        double r3669968 = r3669963 / r3669967;
        double r3669969 = r3669962 * r3669968;
        double r3669970 = rand;
        double r3669971 = r3669969 * r3669970;
        double r3669972 = r3669964 + r3669971;
        return r3669972;
}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.2

    \[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
  2. Using strategy rm

  3. Applied distribute-lft-in0.2

    \[\leadsto \color{blue}{\frac{\left(\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(1\right)\right)}{\left(\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)\right)}}\]
  4. Using strategy rm

  5. Applied associate-*r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (*.p16 (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0))) (+.p16 (real->posit16 1) (*.p16 (/.p16 (real->posit16 1) (sqrt.p16 (*.p16 (real->posit16 9) (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0)))))) rand))))