Average Error: 3.5 → 3.5
Time: 17.3s
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 r493451 = x;
        double r493452 = 1.0;
        double r493453 = y;
        double r493454 = r493452 - r493453;
        double r493455 = z;
        double r493456 = r493454 * r493455;
        double r493457 = r493452 - r493456;
        double r493458 = r493451 * r493457;
        return r493458;
}
↓
double f(double x, double y, double z) {
        double r493459 = y;
        double r493460 = 1.0;
        double r493461 = r493459 - r493460;
        double r493462 = z;
        double r493463 = fma(r493461, r493462, r493460);
        double r493464 = x;
        double r493465 = r493463 * r493464;
        return r493465;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.5
Target0.2
Herbie3.5
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.8922376496639029 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.5

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

  2. Simplified3.5

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

  3. Final simplification3.5

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

Reproduce

herbie shell --seed 2020042 +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))))