Average Error: 0.5 → 0.4
Time: 35.6s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\left(\frac{d4}{\left(d2 - d1\right)}\right) - d3\right)\]
\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(\frac{d4}{\left(d2 - d1\right)}\right) - d3\right)
double f(double d1, double d2, double d3, double d4) {
        double r4903563 = d1;
        double r4903564 = d2;
        double r4903565 = r4903563 * r4903564;
        double r4903566 = d3;
        double r4903567 = r4903563 * r4903566;
        double r4903568 = r4903565 - r4903567;
        double r4903569 = d4;
        double r4903570 = r4903569 * r4903563;
        double r4903571 = r4903568 + r4903570;
        double r4903572 = r4903563 * r4903563;
        double r4903573 = r4903571 - r4903572;
        return r4903573;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r4903574 = d1;
        double r4903575 = d4;
        double r4903576 = d2;
        double r4903577 = r4903576 - r4903574;
        double r4903578 = r4903575 + r4903577;
        double r4903579 = d3;
        double r4903580 = r4903578 - r4903579;
        double r4903581 = r4903574 * r4903580;
        return r4903581;
}

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

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

  4. Applied sub-neg0.4

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

  5. Applied associate-+l+0.4

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

  6. Simplified0.4

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

  8. Applied associate--r+0.4

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

  10. Applied associate-+r-0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019164 +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)))