Average Error: 0.5 → 0.4
Time: 14.9s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot d2 + d1 \cdot \left(d4 - \left(d3 + d1\right)\right)\]
\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 d2 + d1 \cdot \left(d4 - \left(d3 + d1\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r3376914 = d1;
        double r3376915 = d2;
        double r3376916 = r3376914 * r3376915;
        double r3376917 = d3;
        double r3376918 = r3376914 * r3376917;
        double r3376919 = r3376916 - r3376918;
        double r3376920 = d4;
        double r3376921 = r3376920 * r3376914;
        double r3376922 = r3376919 + r3376921;
        double r3376923 = r3376914 * r3376914;
        double r3376924 = r3376922 - r3376923;
        return r3376924;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r3376925 = d1;
        double r3376926 = d2;
        double r3376927 = r3376925 * r3376926;
        double r3376928 = d4;
        double r3376929 = d3;
        double r3376930 = r3376929 + r3376925;
        double r3376931 = r3376928 - r3376930;
        double r3376932 = r3376925 * r3376931;
        double r3376933 = r3376927 + r3376932;
        return r3376933;
}

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. Simplified0.4

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

  4. Applied associate--l+0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 +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)))