Average Error: 3.6 → 3.6
Time: 40.0s
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 r856087 = x;
        double r856088 = 1.0;
        double r856089 = y;
        double r856090 = r856088 - r856089;
        double r856091 = z;
        double r856092 = r856090 * r856091;
        double r856093 = r856088 - r856092;
        double r856094 = r856087 * r856093;
        return r856094;
}
↓
double f(double x, double y, double z) {
        double r856095 = y;
        double r856096 = 1.0;
        double r856097 = r856095 - r856096;
        double r856098 = z;
        double r856099 = fma(r856097, r856098, r856096);
        double r856100 = x;
        double r856101 = r856099 * r856100;
        return r856101;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.6
Target0.2
Herbie3.6
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.8922376496639029 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.6

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

  2. Simplified3.6

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

  3. Final simplification3.6

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

Reproduce

herbie shell --seed 2020046 +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))))