Average Error: 12.8 → 0.0
Time: 12.9s
Precision: 64
\[\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z\]
↓
\[\mathsf{fma}\left(x, y, \left(-z\right) \cdot y\right)\]
\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z
↓
\mathsf{fma}\left(x, y, \left(-z\right) \cdot y\right)
double f(double x, double y, double z) {
        double r483809 = x;
        double r483810 = y;
        double r483811 = r483809 * r483810;
        double r483812 = r483810 * r483810;
        double r483813 = r483811 - r483812;
        double r483814 = r483813 + r483812;
        double r483815 = z;
        double r483816 = r483810 * r483815;
        double r483817 = r483814 - r483816;
        return r483817;
}
↓
double f(double x, double y, double z) {
        double r483818 = x;
        double r483819 = y;
        double r483820 = z;
        double r483821 = -r483820;
        double r483822 = r483821 * r483819;
        double r483823 = fma(r483818, r483819, r483822);
        return r483823;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.8
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 12.8

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

  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. Simplified0.0

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

  7. Simplified0.0

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

  9. Applied fma-def0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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)))