Average Error: 0.5 → 0.4
Time: 34.5s
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 r4717250 = d1;
        double r4717251 = d2;
        double r4717252 = r4717250 * r4717251;
        double r4717253 = d3;
        double r4717254 = r4717250 * r4717253;
        double r4717255 = r4717252 - r4717254;
        double r4717256 = d4;
        double r4717257 = r4717256 * r4717250;
        double r4717258 = r4717255 + r4717257;
        double r4717259 = r4717250 * r4717250;
        double r4717260 = r4717258 - r4717259;
        return r4717260;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r4717261 = d1;
        double r4717262 = d4;
        double r4717263 = d2;
        double r4717264 = r4717263 - r4717261;
        double r4717265 = r4717262 + r4717264;
        double r4717266 = d3;
        double r4717267 = r4717265 - r4717266;
        double r4717268 = r4717261 * r4717267;
        return r4717268;
}

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