Average Error: 3.8 → 3.8
Time: 19.5s
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 r35742650 = x;
        double r35742651 = 1.0;
        double r35742652 = y;
        double r35742653 = r35742651 - r35742652;
        double r35742654 = z;
        double r35742655 = r35742653 * r35742654;
        double r35742656 = r35742651 - r35742655;
        double r35742657 = r35742650 * r35742656;
        return r35742657;
}
↓
double f(double x, double y, double z) {
        double r35742658 = z;
        double r35742659 = y;
        double r35742660 = 1.0;
        double r35742661 = r35742659 - r35742660;
        double r35742662 = fma(r35742658, r35742661, r35742660);
        double r35742663 = x;
        double r35742664 = r35742662 * r35742663;
        return r35742664;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.8
Target0.2
Herbie3.8
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607048970493874632750554853795 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.892237649663902900973248011051357504727 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.8

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

  2. Simplified3.8

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

  3. Final simplification3.8

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

  (* x (- 1.0 (* (- 1.0 y) z))))