Average Error: 0.1 → 0.0
Time: 13.3s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x - z}{y}, 4, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x - z}{y}, 4, 2\right)
double f(double x, double y, double z) {
        double r207201 = 1.0;
        double r207202 = 4.0;
        double r207203 = x;
        double r207204 = y;
        double r207205 = 0.25;
        double r207206 = r207204 * r207205;
        double r207207 = r207203 + r207206;
        double r207208 = z;
        double r207209 = r207207 - r207208;
        double r207210 = r207202 * r207209;
        double r207211 = r207210 / r207204;
        double r207212 = r207201 + r207211;
        return r207212;
}
↓
double f(double x, double y, double z) {
        double r207213 = x;
        double r207214 = z;
        double r207215 = r207213 - r207214;
        double r207216 = y;
        double r207217 = r207215 / r207216;
        double r207218 = 4.0;
        double r207219 = 2.0;
        double r207220 = fma(r207217, r207218, r207219);
        return r207220;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019304 +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)))