FastMath dist4

Average Error: 0.5 → 0.4

Time: 10.5s

Precision: 64

Math

\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]

↓

\[d1 \cdot \left(\left(d2 + d4\right) - \left(d3 + d1\right)\right)\]

C

double f(double d1, double d2, double d3, double d4) {
        double r911522 = d1;
        double r911523 = d2;
        double r911524 = r911522 * r911523;
        double r911525 = d3;
        double r911526 = r911522 * r911525;
        double r911527 = r911524 - r911526;
        double r911528 = d4;
        double r911529 = r911528 * r911522;
        double r911530 = r911527 + r911529;
        double r911531 = r911522 * r911522;
        double r911532 = r911530 - r911531;
        return r911532;
}

↓

double f(double d1, double d2, double d3, double d4) {
        double r911533 = d1;
        double r911534 = d2;
        double r911535 = d4;
        double r911536 = r911534 + r911535;
        double r911537 = d3;
        double r911538 = r911537 + r911533;
        double r911539 = r911536 - r911538;
        double r911540 = r911533 * r911539;
        return r911540;
}

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Bits error versus d4

Derivation

1. Initial program 0.5

   \[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]

2. Simplified 0.4

   \[\leadsto \color{blue}{d1 \cdot \left(\frac{\left(d2 - d3\right)}{\left(d4 - d1\right)}\right)}\]
3. Using strategy rm

4. Applied associate-+r- 0.4

   \[\leadsto d1 \cdot \color{blue}{\left(\left(\frac{\left(d2 - d3\right)}{d4}\right) - d1\right)}\]
5. Using strategy rm

6. Applied sub-neg 0.4

   \[\leadsto d1 \cdot \color{blue}{\left(\frac{\left(\frac{\left(d2 - d3\right)}{d4}\right)}{\left(-d1\right)}\right)}\]

7. Applied distribute-rgt-in 0.5

   \[\leadsto \color{blue}{\frac{\left(\left(\frac{\left(d2 - d3\right)}{d4}\right) \cdot d1\right)}{\left(\left(-d1\right) \cdot d1\right)}}\]
8. Using strategy rm

9. Applied distribute-rgt-out 0.4

   \[\leadsto \color{blue}{d1 \cdot \left(\frac{\left(\frac{\left(d2 - d3\right)}{d4}\right)}{\left(-d1\right)}\right)}\]

10. Simplified 0.4

    \[\leadsto d1 \cdot \color{blue}{\left(\left(\frac{d2}{d4}\right) - \left(\frac{d3}{d1}\right)\right)}\]

11. Final simplification 0.4

    \[\leadsto d1 \cdot \left(\left(d2 + d4\right) - \left(d3 + d1\right)\right)\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))