Average Error: 0.5 → 0.4
Time: 25.1s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 + \left(d4 - \left(d3 + d1\right)\right)\right)\]
\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(d4 - \left(d3 + d1\right)\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r7851400 = d1;
        double r7851401 = d2;
        double r7851402 = r7851400 * r7851401;
        double r7851403 = d3;
        double r7851404 = r7851400 * r7851403;
        double r7851405 = r7851402 - r7851404;
        double r7851406 = d4;
        double r7851407 = r7851406 * r7851400;
        double r7851408 = r7851405 + r7851407;
        double r7851409 = r7851400 * r7851400;
        double r7851410 = r7851408 - r7851409;
        return r7851410;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r7851411 = d1;
        double r7851412 = d2;
        double r7851413 = d4;
        double r7851414 = d3;
        double r7851415 = r7851414 + r7851411;
        double r7851416 = r7851413 - r7851415;
        double r7851417 = r7851412 + r7851416;
        double r7851418 = r7851411 * r7851417;
        return r7851418;
}

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

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

  4. Applied associate--l+0.4

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

  5. Applied distribute-lft-in0.4

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

  7. Applied distribute-lft-out0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019135 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))