Average Error: 16.8 → 0.0
Time: 14.3s
Precision: 64
\[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, y, -z \cdot y\right) + y \cdot \left(z + \left(-z\right)\right)\]
\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, y, -z \cdot y\right) + y \cdot \left(z + \left(-z\right)\right)
double f(double x, double y, double z) {
        double r23645117 = x;
        double r23645118 = y;
        double r23645119 = r23645117 * r23645118;
        double r23645120 = z;
        double r23645121 = r23645118 * r23645120;
        double r23645122 = r23645119 - r23645121;
        double r23645123 = r23645118 * r23645118;
        double r23645124 = r23645122 - r23645123;
        double r23645125 = r23645124 + r23645123;
        return r23645125;
}
↓
double f(double x, double y, double z) {
        double r23645126 = x;
        double r23645127 = y;
        double r23645128 = z;
        double r23645129 = r23645128 * r23645127;
        double r23645130 = -r23645129;
        double r23645131 = fma(r23645126, r23645127, r23645130);
        double r23645132 = -r23645128;
        double r23645133 = r23645128 + r23645132;
        double r23645134 = r23645127 * r23645133;
        double r23645135 = r23645131 + r23645134;
        return r23645135;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original16.8
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 16.8

    \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y\]
  2. Using strategy rm

  3. Applied prod-diff16.8

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

  4. Applied associate--l+16.8

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

  5. Applied associate-+l+7.6

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"

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

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