Average Error: 0.5 → 0.4
Time: 10.2s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[\left(\left(\left(d2 + d4\right) - d1\right) - d3\right) \cdot d1\]
\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)
↓
\left(\left(\left(d2 + d4\right) - d1\right) - d3\right) \cdot d1
double f(double d1, double d2, double d3, double d4) {
        double r3527317 = d1;
        double r3527318 = d2;
        double r3527319 = r3527317 * r3527318;
        double r3527320 = d3;
        double r3527321 = r3527317 * r3527320;
        double r3527322 = r3527319 - r3527321;
        double r3527323 = d4;
        double r3527324 = r3527323 * r3527317;
        double r3527325 = r3527322 + r3527324;
        double r3527326 = r3527317 * r3527317;
        double r3527327 = r3527325 - r3527326;
        return r3527327;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r3527328 = d2;
        double r3527329 = d4;
        double r3527330 = r3527328 + r3527329;
        double r3527331 = d1;
        double r3527332 = r3527330 - r3527331;
        double r3527333 = d3;
        double r3527334 = r3527332 - r3527333;
        double r3527335 = r3527334 * r3527331;
        return r3527335;
}

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

    \[\leadsto \color{blue}{d1 \cdot \left(\frac{\left(d2 - d3\right)}{\left(d4 - d1\right)}\right)}\]
  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-neg0.4

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

  7. Applied associate-+l+0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied associate--r+0.4

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

  12. Final simplification0.4

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

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)))