Average Error: 17.4 → 0.0
Time: 28.4s
Precision: 64
\[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, y, -z \cdot y\right) + 0 \cdot z\]
\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) + 0 \cdot z
double f(double x, double y, double z) {
        double r334090 = x;
        double r334091 = y;
        double r334092 = r334090 * r334091;
        double r334093 = z;
        double r334094 = r334091 * r334093;
        double r334095 = r334092 - r334094;
        double r334096 = r334091 * r334091;
        double r334097 = r334095 - r334096;
        double r334098 = r334097 + r334096;
        return r334098;
}
↓
double f(double x, double y, double z) {
        double r334099 = x;
        double r334100 = y;
        double r334101 = z;
        double r334102 = r334101 * r334100;
        double r334103 = -r334102;
        double r334104 = fma(r334099, r334100, r334103);
        double r334105 = 0.0;
        double r334106 = r334105 * r334101;
        double r334107 = r334104 + r334106;
        return r334107;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.4
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.4

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

  3. Applied prod-diff17.4

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

  4. Applied associate--l+17.4

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

  5. Applied associate-+l+7.9

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"
  :precision binary64

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

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