Average Error: 12.9 → 0.0
Time: 1.5s
Precision: 64
\[\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z\]
↓
\[\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)\]
\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z
↓
\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)
double f(double x, double y, double z) {
        double r1027539 = x;
        double r1027540 = y;
        double r1027541 = r1027539 * r1027540;
        double r1027542 = r1027540 * r1027540;
        double r1027543 = r1027541 - r1027542;
        double r1027544 = r1027543 + r1027542;
        double r1027545 = z;
        double r1027546 = r1027540 * r1027545;
        double r1027547 = r1027544 - r1027546;
        return r1027547;
}
↓
double f(double x, double y, double z) {
        double r1027548 = y;
        double r1027549 = x;
        double r1027550 = z;
        double r1027551 = -r1027550;
        double r1027552 = r1027548 * r1027551;
        double r1027553 = fma(r1027548, r1027549, r1027552);
        return r1027553;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.9
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 12.9

    \[\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z\]

  2. Simplified0.0

    \[\leadsto \color{blue}{y \cdot \left(x - z\right)}\]
  3. Using strategy rm

  4. Applied sub-neg0.0

    \[\leadsto y \cdot \color{blue}{\left(x + \left(-z\right)\right)}\]

  5. Applied distribute-lft-in0.0

    \[\leadsto \color{blue}{y \cdot x + y \cdot \left(-z\right)}\]
  6. Using strategy rm

  7. Applied fma-def0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)}\]

  8. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, D"
  :precision binary64

  :herbie-target
  (* (- x z) y)

  (- (+ (- (* x y) (* y y)) (* y y)) (* y z)))