Average Error: 3.3 → 3.3
Time: 1.1m
Precision: 64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]
↓
\[x \cdot \mathsf{fma}\left(z, y - 1, 1\right)\]
x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
x \cdot \mathsf{fma}\left(z, y - 1, 1\right)
double f(double x, double y, double z) {
        double r38356561 = x;
        double r38356562 = 1.0;
        double r38356563 = y;
        double r38356564 = r38356562 - r38356563;
        double r38356565 = z;
        double r38356566 = r38356564 * r38356565;
        double r38356567 = r38356562 - r38356566;
        double r38356568 = r38356561 * r38356567;
        return r38356568;
}
↓
double f(double x, double y, double z) {
        double r38356569 = x;
        double r38356570 = z;
        double r38356571 = y;
        double r38356572 = 1.0;
        double r38356573 = r38356571 - r38356572;
        double r38356574 = fma(r38356570, r38356573, r38356572);
        double r38356575 = r38356569 * r38356574;
        return r38356575;
}

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

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

Reproduce

herbie shell --seed 2019200 +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))))