Average Error: 3.4 → 3.4
Time: 39.9s
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 r815019 = x;
        double r815020 = 1.0;
        double r815021 = y;
        double r815022 = r815020 - r815021;
        double r815023 = z;
        double r815024 = r815022 * r815023;
        double r815025 = r815020 - r815024;
        double r815026 = r815019 * r815025;
        return r815026;
}
↓
double f(double x, double y, double z) {
        double r815027 = y;
        double r815028 = 1.0;
        double r815029 = r815027 - r815028;
        double r815030 = z;
        double r815031 = fma(r815029, r815030, r815028);
        double r815032 = x;
        double r815033 = r815031 * r815032;
        return r815033;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.4
Target0.3
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.8922376496639029 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.4

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

  2. Simplified3.4

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

  3. Final simplification3.4

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

Reproduce

herbie shell --seed 2020047 +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))))