Average Error: 0.2 → 0.2
Time: 56.1s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
\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 \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
double f(double a, double rand) {
        double r5461089 = a;
        double r5461090 = 1.0;
        double r5461091 = /* ERROR: no posit support in C */;
        double r5461092 = 3.0;
        double r5461093 = /* ERROR: no posit support in C */;
        double r5461094 = r5461091 / r5461093;
        double r5461095 = r5461089 - r5461094;
        double r5461096 = 1.0;
        double r5461097 = /* ERROR: no posit support in C */;
        double r5461098 = 9.0;
        double r5461099 = /* ERROR: no posit support in C */;
        double r5461100 = r5461099 * r5461095;
        double r5461101 = sqrt(r5461100);
        double r5461102 = r5461097 / r5461101;
        double r5461103 = rand;
        double r5461104 = r5461102 * r5461103;
        double r5461105 = r5461097 + r5461104;
        double r5461106 = r5461095 * r5461105;
        return r5461106;
}
↓
double f(double a, double rand) {
        double r5461107 = a;
        double r5461108 = 1.0;
        double r5461109 = 3.0;
        double r5461110 = r5461108 / r5461109;
        double r5461111 = r5461107 - r5461110;
        double r5461112 = 1.0;
        double r5461113 = 9.0;
        double r5461114 = r5461113 * r5461111;
        double r5461115 = sqrt(r5461114);
        double r5461116 = r5461112 / r5461115;
        double r5461117 = rand;
        double r5461118 = r5461116 * r5461117;
        double r5461119 = r5461112 + r5461118;
        double r5461120 = r5461111 * r5461119;
        return r5461120;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(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))))