Average Error: 0.1 → 0.0
Time: 17.2s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 2\right)
double f(double x, double y, double z) {
        double r16066 = 1.0;
        double r16067 = 4.0;
        double r16068 = x;
        double r16069 = y;
        double r16070 = 0.25;
        double r16071 = r16069 * r16070;
        double r16072 = r16068 + r16071;
        double r16073 = z;
        double r16074 = r16072 - r16073;
        double r16075 = r16067 * r16074;
        double r16076 = r16075 / r16069;
        double r16077 = r16066 + r16076;
        return r16077;
}
↓
double f(double x, double y, double z) {
        double r16078 = x;
        double r16079 = y;
        double r16080 = r16078 / r16079;
        double r16081 = z;
        double r16082 = r16081 / r16079;
        double r16083 = r16080 - r16082;
        double r16084 = 4.0;
        double r16085 = 2.0;
        double r16086 = fma(r16083, r16084, r16085);
        return r16086;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019315 +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)))