Average Error: 0.1 → 0.0
Time: 4.5s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)
double f(double x, double y, double z) {
        double r1166 = 1.0;
        double r1167 = 4.0;
        double r1168 = x;
        double r1169 = y;
        double r1170 = 0.25;
        double r1171 = r1169 * r1170;
        double r1172 = r1168 + r1171;
        double r1173 = z;
        double r1174 = r1172 - r1173;
        double r1175 = r1167 * r1174;
        double r1176 = r1175 / r1169;
        double r1177 = r1166 + r1176;
        return r1177;
}
↓
double f(double x, double y, double z) {
        double r1178 = 4.0;
        double r1179 = x;
        double r1180 = y;
        double r1181 = r1179 / r1180;
        double r1182 = z;
        double r1183 = r1182 / r1180;
        double r1184 = r1181 - r1183;
        double r1185 = 2.0;
        double r1186 = fma(r1178, r1184, r1185);
        return r1186;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{y}, 2 - 4 \cdot \frac{z}{y}\right)}\]

  4. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]

  5. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)}\]

  6. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))