Average Error: 0.5 → 0.4
Time: 35.3s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[\left(\left(d4 + d2\right) - \left(d3 + d1\right)\right) \cdot d1\]
\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)
↓
\left(\left(d4 + d2\right) - \left(d3 + d1\right)\right) \cdot d1
double f(double d1, double d2, double d3, double d4) {
        double r7057508 = d1;
        double r7057509 = d2;
        double r7057510 = r7057508 * r7057509;
        double r7057511 = d3;
        double r7057512 = r7057508 * r7057511;
        double r7057513 = r7057510 - r7057512;
        double r7057514 = d4;
        double r7057515 = r7057514 * r7057508;
        double r7057516 = r7057513 + r7057515;
        double r7057517 = r7057508 * r7057508;
        double r7057518 = r7057516 - r7057517;
        return r7057518;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r7057519 = d4;
        double r7057520 = d2;
        double r7057521 = r7057519 + r7057520;
        double r7057522 = d3;
        double r7057523 = d1;
        double r7057524 = r7057522 + r7057523;
        double r7057525 = r7057521 - r7057524;
        double r7057526 = r7057525 * r7057523;
        return r7057526;
}

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

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

  8. Applied associate-+r-0.4

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

  10. Applied *-commutative0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 +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)))