Average Error: 0.5 → 0.4
Time: 17.8s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 + \left(\left(d4 - d3\right) - d1\right)\right)\]
\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(d4 - d3\right) - d1\right)\right)
double f(double d1, double d2, double d3, double d4) {
        double r8683215 = d1;
        double r8683216 = d2;
        double r8683217 = r8683215 * r8683216;
        double r8683218 = d3;
        double r8683219 = r8683215 * r8683218;
        double r8683220 = r8683217 - r8683219;
        double r8683221 = d4;
        double r8683222 = r8683221 * r8683215;
        double r8683223 = r8683220 + r8683222;
        double r8683224 = r8683215 * r8683215;
        double r8683225 = r8683223 - r8683224;
        return r8683225;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r8683226 = d1;
        double r8683227 = d2;
        double r8683228 = d4;
        double r8683229 = d3;
        double r8683230 = r8683228 - r8683229;
        double r8683231 = r8683230 - r8683226;
        double r8683232 = r8683227 + r8683231;
        double r8683233 = r8683226 * r8683232;
        return r8683233;
}

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. Using strategy rm

  6. Applied associate--r+0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))