Average Error: 0.5 → 0.4
Time: 10.1s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 + \left(\left(-d3\right) + \left(d4 - d1\right)\right)\right)\]
\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 \left(d2 + \left(\left(-d3\right) + \left(d4 - d1\right)\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r1611410 = d1;
        double r1611411 = d2;
        double r1611412 = r1611410 * r1611411;
        double r1611413 = d3;
        double r1611414 = r1611410 * r1611413;
        double r1611415 = r1611412 - r1611414;
        double r1611416 = d4;
        double r1611417 = r1611416 * r1611410;
        double r1611418 = r1611415 + r1611417;
        double r1611419 = r1611410 * r1611410;
        double r1611420 = r1611418 - r1611419;
        return r1611420;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r1611421 = d1;
        double r1611422 = d2;
        double r1611423 = d3;
        double r1611424 = -r1611423;
        double r1611425 = d4;
        double r1611426 = r1611425 - r1611421;
        double r1611427 = r1611424 + r1611426;
        double r1611428 = r1611422 + r1611427;
        double r1611429 = r1611421 * r1611428;
        return r1611429;
}

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 sub-neg0.4

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

  5. Applied associate-+l+0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019124 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))