Average Error: 0.3 → 0.3
Time: 13.3s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot z\]
↓
\[\mathsf{fma}\left(\left(y - x\right) \cdot 6, z, x\right)\]
x + \left(\left(y - x\right) \cdot 6\right) \cdot z
↓
\mathsf{fma}\left(\left(y - x\right) \cdot 6, z, x\right)
double f(double x, double y, double z) {
        double r545381 = x;
        double r545382 = y;
        double r545383 = r545382 - r545381;
        double r545384 = 6.0;
        double r545385 = r545383 * r545384;
        double r545386 = z;
        double r545387 = r545385 * r545386;
        double r545388 = r545381 + r545387;
        return r545388;
}
↓
double f(double x, double y, double z) {
        double r545389 = y;
        double r545390 = x;
        double r545391 = r545389 - r545390;
        double r545392 = 6.0;
        double r545393 = r545391 * r545392;
        double r545394 = z;
        double r545395 = fma(r545393, r545394, r545390);
        return r545395;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.3
Target0.2
Herbie0.3
\[x - \left(6 \cdot z\right) \cdot \left(x - y\right)\]

Derivation

  1. Initial program 0.3

    \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot z, x\right)}\]

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot 6, z, x\right)}\]

  5. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\left(y - x\right) \cdot 6, z, x\right)\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E"

  :herbie-target
  (- x (* (* 6.0 z) (- x y)))

  (+ x (* (* (- y x) 6.0) z)))