Average Error: 0.1 → 0.0
Time: 10.4s
Precision: 64
\[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x - z}{y} + 0.25, 4.0, 1.0\right)\]
1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x - z}{y} + 0.25, 4.0, 1.0\right)
double f(double x, double y, double z) {
        double r10139001 = 1.0;
        double r10139002 = 4.0;
        double r10139003 = x;
        double r10139004 = y;
        double r10139005 = 0.25;
        double r10139006 = r10139004 * r10139005;
        double r10139007 = r10139003 + r10139006;
        double r10139008 = z;
        double r10139009 = r10139007 - r10139008;
        double r10139010 = r10139002 * r10139009;
        double r10139011 = r10139010 / r10139004;
        double r10139012 = r10139001 + r10139011;
        return r10139012;
}
↓
double f(double x, double y, double z) {
        double r10139013 = x;
        double r10139014 = z;
        double r10139015 = r10139013 - r10139014;
        double r10139016 = y;
        double r10139017 = r10139015 / r10139016;
        double r10139018 = 0.25;
        double r10139019 = r10139017 + r10139018;
        double r10139020 = 4.0;
        double r10139021 = 1.0;
        double r10139022 = fma(r10139019, r10139020, r10139021);
        return r10139022;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4.0, 1.0\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{x - z}{y} + 0.25, 4.0, 1.0\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))