Average Error: 0.5 → 0.4
Time: 10.7s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\left(d2 - \left(d3 + d1\right)\right) + d4\right)\]
\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 \left(\left(d2 - \left(d3 + d1\right)\right) + d4\right)
double f(double d1, double d2, double d3, double d4) {
        double r2796489 = d1;
        double r2796490 = d2;
        double r2796491 = r2796489 * r2796490;
        double r2796492 = d3;
        double r2796493 = r2796489 * r2796492;
        double r2796494 = r2796491 - r2796493;
        double r2796495 = d4;
        double r2796496 = r2796495 * r2796489;
        double r2796497 = r2796494 + r2796496;
        double r2796498 = r2796489 * r2796489;
        double r2796499 = r2796497 - r2796498;
        return r2796499;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r2796500 = d1;
        double r2796501 = d2;
        double r2796502 = d3;
        double r2796503 = r2796502 + r2796500;
        double r2796504 = r2796501 - r2796503;
        double r2796505 = d4;
        double r2796506 = r2796504 + r2796505;
        double r2796507 = r2796500 * r2796506;
        return r2796507;
}

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-+l-0.4

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

  6. Applied sub-neg0.4

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

  7. Applied distribute-rgt-in0.5

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

  8. Simplified0.5

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019124 +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)))