Average Error: 0.5 → 0.4
Time: 12.0s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot \left(d2 - \left(d3 + d1\right)\right) + d1 \cdot d4\]
\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(d3 + d1\right)\right) + d1 \cdot d4
double f(double d1, double d2, double d3, double d4) {
        double r3772545 = d1;
        double r3772546 = d2;
        double r3772547 = r3772545 * r3772546;
        double r3772548 = d3;
        double r3772549 = r3772545 * r3772548;
        double r3772550 = r3772547 - r3772549;
        double r3772551 = d4;
        double r3772552 = r3772551 * r3772545;
        double r3772553 = r3772550 + r3772552;
        double r3772554 = r3772545 * r3772545;
        double r3772555 = r3772553 - r3772554;
        return r3772555;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r3772556 = d1;
        double r3772557 = d2;
        double r3772558 = d3;
        double r3772559 = r3772558 + r3772556;
        double r3772560 = r3772557 - r3772559;
        double r3772561 = r3772556 * r3772560;
        double r3772562 = d4;
        double r3772563 = r3772556 * r3772562;
        double r3772564 = r3772561 + r3772563;
        return r3772564;
}

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 associate-+l-0.4

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

  6. Applied sub-neg0.4

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

  7. Applied distribute-rgt-in0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied distribute-lft-in0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))