Average Error: 3.3 → 3.3
Time: 3.7s
Precision: 64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]
↓
\[\mathsf{fma}\left(y - 1, z, 1\right) \cdot x\]
x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
\mathsf{fma}\left(y - 1, z, 1\right) \cdot x
double f(double x, double y, double z) {
        double r825191 = x;
        double r825192 = 1.0;
        double r825193 = y;
        double r825194 = r825192 - r825193;
        double r825195 = z;
        double r825196 = r825194 * r825195;
        double r825197 = r825192 - r825196;
        double r825198 = r825191 * r825197;
        return r825198;
}
↓
double f(double x, double y, double z) {
        double r825199 = y;
        double r825200 = 1.0;
        double r825201 = r825199 - r825200;
        double r825202 = z;
        double r825203 = fma(r825201, r825202, r825200);
        double r825204 = x;
        double r825205 = r825203 * r825204;
        return r825205;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.3
Target0.3
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.8922376496639029 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.3

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

  2. Simplified3.3

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

  3. Final simplification3.3

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

Reproduce

herbie shell --seed 2020020 +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))) -1.618195973607049e+50) (+ x (* (- 1 y) (* (- z) x))) (if (< (* x (- 1 (* (- 1 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1 y) (* (- z) x)))))

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