FastMath test2

Average Error: 0.3 → 0.3

Time: 9.0s

Precision: 64

Math

\[\frac{\left(\frac{\left(d1 \cdot \left(10\right)\right)}{\left(d1 \cdot d2\right)}\right)}{\left(d1 \cdot \left(20\right)\right)}\]

↓

\[\left(\left(20 + 10\right) + d2\right) \cdot d1\]

C

double f(double d1, double d2) {
        double r1023212 = d1;
        double r1023213 = 10.0;
        double r1023214 = /* ERROR: no posit support in C */;
        double r1023215 = r1023212 * r1023214;
        double r1023216 = d2;
        double r1023217 = r1023212 * r1023216;
        double r1023218 = r1023215 + r1023217;
        double r1023219 = 20.0;
        double r1023220 = /* ERROR: no posit support in C */;
        double r1023221 = r1023212 * r1023220;
        double r1023222 = r1023218 + r1023221;
        return r1023222;
}

↓

double f(double d1, double d2) {
        double r1023223 = 20.0;
        double r1023224 = 10.0;
        double r1023225 = r1023223 + r1023224;
        double r1023226 = d2;
        double r1023227 = r1023225 + r1023226;
        double r1023228 = d1;
        double r1023229 = r1023227 * r1023228;
        return r1023229;
}

Error

Bits error versus d1

Bits error versus d2

Derivation

1. Initial program 0.3

   \[\frac{\left(\frac{\left(d1 \cdot \left(10\right)\right)}{\left(d1 \cdot d2\right)}\right)}{\left(d1 \cdot \left(20\right)\right)}\]

2. Simplified 0.3

   \[\leadsto \color{blue}{\left(\frac{\left(20\right)}{\left(\frac{\left(10\right)}{d2}\right)}\right) \cdot d1}\]
3. Using strategy rm

4. Applied associate-+r+ 0.3

   \[\leadsto \color{blue}{\left(\frac{\left(\frac{\left(20\right)}{\left(10\right)}\right)}{d2}\right)} \cdot d1\]

5. Final simplification 0.3

   \[\leadsto \left(\left(20 + 10\right) + d2\right) \cdot d1\]

Reproduce

herbie shell --seed 2019107 
(FPCore (d1 d2)
  :name "FastMath test2"
  (+.p16 (+.p16 (*.p16 d1 (real->posit16 10)) (*.p16 d1 d2)) (*.p16 d1 (real->posit16 20))))