Average Error: 3.4 → 1.6
Time: 3.5s
Precision: 64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]
↓
\[\mathsf{fma}\left(x \cdot z, y - 1, 1 \cdot x\right)\]
x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
\mathsf{fma}\left(x \cdot z, y - 1, 1 \cdot x\right)
double f(double x, double y, double z) {
        double r837015 = x;
        double r837016 = 1.0;
        double r837017 = y;
        double r837018 = r837016 - r837017;
        double r837019 = z;
        double r837020 = r837018 * r837019;
        double r837021 = r837016 - r837020;
        double r837022 = r837015 * r837021;
        return r837022;
}
↓
double f(double x, double y, double z) {
        double r837023 = x;
        double r837024 = z;
        double r837025 = r837023 * r837024;
        double r837026 = y;
        double r837027 = 1.0;
        double r837028 = r837026 - r837027;
        double r837029 = r837027 * r837023;
        double r837030 = fma(r837025, r837028, r837029);
        return r837030;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.4
Target0.2
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.8922376496639029 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.4

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

  2. Simplified3.4

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

  4. Applied add-cube-cbrt4.0

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(y - 1, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y - 1, z, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y - 1, z, 1\right)}\right)} \cdot x\]

  5. Taylor expanded around inf 3.4

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

  6. Simplified1.6

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

  7. Final simplification1.6

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

Reproduce

herbie shell --seed 2020100 +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))))