Average Error: 3.3 → 3.3
Time: 29.0s
Precision: 64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]
↓
\[\mathsf{fma}\left(z, y - 1, 1\right) \cdot x\]
x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
\mathsf{fma}\left(z, y - 1, 1\right) \cdot x
double f(double x, double y, double z) {
        double r529991 = x;
        double r529992 = 1.0;
        double r529993 = y;
        double r529994 = r529992 - r529993;
        double r529995 = z;
        double r529996 = r529994 * r529995;
        double r529997 = r529992 - r529996;
        double r529998 = r529991 * r529997;
        return r529998;
}
↓
double f(double x, double y, double z) {
        double r529999 = z;
        double r530000 = y;
        double r530001 = 1.0;
        double r530002 = r530000 - r530001;
        double r530003 = fma(r529999, r530002, r530001);
        double r530004 = x;
        double r530005 = r530003 * r530004;
        return r530005;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.3
Target0.2
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607048970493874632750554853795 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.892237649663902900973248011051357504727 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.3

    \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]

  2. Simplified3.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, y - 1, 1\right) \cdot x}\]

  3. Final simplification3.3

    \[\leadsto \mathsf{fma}\left(z, y - 1, 1\right) \cdot x\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1 (* (- 1 y) z))) -1.618195973607049e50) (+ x (* (- 1 y) (* (- z) x))) (if (< (* x (- 1 (* (- 1 y) z))) 3.8922376496639029e134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1 y) (* (- z) x)))))

  (* x (- 1 (* (- 1 y) z))))