Average Error: 17.6 → 0.0
Time: 2.1s
Precision: 64
\[\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y\]
↓
\[\mathsf{fma}\left(y, x - z, 0\right)\]
\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y
↓
\mathsf{fma}\left(y, x - z, 0\right)
double f(double x, double y, double z) {
        double r454749 = x;
        double r454750 = y;
        double r454751 = r454749 * r454750;
        double r454752 = r454750 * r454750;
        double r454753 = r454751 + r454752;
        double r454754 = z;
        double r454755 = r454750 * r454754;
        double r454756 = r454753 - r454755;
        double r454757 = r454756 - r454752;
        return r454757;
}
↓
double f(double x, double y, double z) {
        double r454758 = y;
        double r454759 = x;
        double r454760 = z;
        double r454761 = r454759 - r454760;
        double r454762 = 0.0;
        double r454763 = fma(r454758, r454761, r454762);
        return r454763;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.6
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.6

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - z, 0\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, x - z, 0\right)\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, C"
  :precision binary64

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

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