Average Error: 3.4 → 3.4
Time: 26.4s
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 r506871 = x;
        double r506872 = 1.0;
        double r506873 = y;
        double r506874 = r506872 - r506873;
        double r506875 = z;
        double r506876 = r506874 * r506875;
        double r506877 = r506872 - r506876;
        double r506878 = r506871 * r506877;
        return r506878;
}
↓
double f(double x, double y, double z) {
        double r506879 = z;
        double r506880 = y;
        double r506881 = 1.0;
        double r506882 = r506880 - r506881;
        double r506883 = fma(r506879, r506882, r506881);
        double r506884 = x;
        double r506885 = r506883 * r506884;
        return r506885;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.4
Target0.2
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607048970493874632750554853795 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.892237649663902900973248011051357504727 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.4

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

  2. Simplified3.4

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

  3. Final simplification3.4

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

Reproduce

herbie shell --seed 2019325 +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))))