Average Error: 0.5 → 0.4
Time: 22.2s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(\left(d4 + d2\right) - \left(d1 + d3\right)\right)\]
\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(\left(d4 + d2\right) - \left(d1 + d3\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r1878812 = d1;
        double r1878813 = d2;
        double r1878814 = r1878812 * r1878813;
        double r1878815 = d3;
        double r1878816 = r1878812 * r1878815;
        double r1878817 = r1878814 - r1878816;
        double r1878818 = d4;
        double r1878819 = r1878818 * r1878812;
        double r1878820 = r1878817 + r1878819;
        double r1878821 = r1878812 * r1878812;
        double r1878822 = r1878820 - r1878821;
        return r1878822;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r1878823 = d1;
        double r1878824 = d4;
        double r1878825 = d2;
        double r1878826 = r1878824 + r1878825;
        double r1878827 = d3;
        double r1878828 = r1878823 + r1878827;
        double r1878829 = r1878826 - r1878828;
        double r1878830 = r1878823 * r1878829;
        return r1878830;
}

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

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))