Average Error: 0.4 → 0.4
Time: 55.1s
Precision: 64
\[x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - z\right)\]
↓
\[\mathsf{fma}\left(\left(y - x\right) \cdot \left(\frac{2.0}{3.0} - z\right), 6.0, x\right) + \left(6.0 \cdot \left(y - x\right)\right) \cdot \mathsf{fma}\left(-z, 1, z\right)\]
x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - z\right)
↓
\mathsf{fma}\left(\left(y - x\right) \cdot \left(\frac{2.0}{3.0} - z\right), 6.0, x\right) + \left(6.0 \cdot \left(y - x\right)\right) \cdot \mathsf{fma}\left(-z, 1, z\right)
double f(double x, double y, double z) {
        double r11491137 = x;
        double r11491138 = y;
        double r11491139 = r11491138 - r11491137;
        double r11491140 = 6.0;
        double r11491141 = r11491139 * r11491140;
        double r11491142 = 2.0;
        double r11491143 = 3.0;
        double r11491144 = r11491142 / r11491143;
        double r11491145 = z;
        double r11491146 = r11491144 - r11491145;
        double r11491147 = r11491141 * r11491146;
        double r11491148 = r11491137 + r11491147;
        return r11491148;
}
↓
double f(double x, double y, double z) {
        double r11491149 = y;
        double r11491150 = x;
        double r11491151 = r11491149 - r11491150;
        double r11491152 = 2.0;
        double r11491153 = 3.0;
        double r11491154 = r11491152 / r11491153;
        double r11491155 = z;
        double r11491156 = r11491154 - r11491155;
        double r11491157 = r11491151 * r11491156;
        double r11491158 = 6.0;
        double r11491159 = fma(r11491157, r11491158, r11491150);
        double r11491160 = r11491158 * r11491151;
        double r11491161 = -r11491155;
        double r11491162 = 1.0;
        double r11491163 = fma(r11491161, r11491162, r11491155);
        double r11491164 = r11491160 * r11491163;
        double r11491165 = r11491159 + r11491164;
        return r11491165;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.4

    \[x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - z\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\frac{2.0}{3.0} - \color{blue}{1 \cdot z}\right)\]

  4. Applied add-sqr-sqrt0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \left(\color{blue}{\sqrt{\frac{2.0}{3.0}} \cdot \sqrt{\frac{2.0}{3.0}}} - 1 \cdot z\right)\]

  5. Applied prod-diff0.4

    \[\leadsto x + \left(\left(y - x\right) \cdot 6.0\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{2.0}{3.0}}, \sqrt{\frac{2.0}{3.0}}, -z \cdot 1\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right)\right)}\]

  6. Applied distribute-rgt-in0.4

    \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{2.0}{3.0}}, \sqrt{\frac{2.0}{3.0}}, -z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6.0\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6.0\right)\right)}\]

  7. Applied associate-+r+0.4

    \[\leadsto \color{blue}{\left(x + \mathsf{fma}\left(\sqrt{\frac{2.0}{3.0}}, \sqrt{\frac{2.0}{3.0}}, -z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6.0\right)\right) + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6.0\right)}\]

  8. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{2.0}{3.0} - z\right) \cdot \left(y - x\right), 6.0, x\right)} + \mathsf{fma}\left(-z, 1, z \cdot 1\right) \cdot \left(\left(y - x\right) \cdot 6.0\right)\]

  9. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\left(y - x\right) \cdot \left(\frac{2.0}{3.0} - z\right), 6.0, x\right) + \left(6.0 \cdot \left(y - x\right)\right) \cdot \mathsf{fma}\left(-z, 1, z\right)\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))