Average Error: 0.2 → 0.0
Time: 16.8s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)
double f(double x, double y, double z) {
        double r190380 = 1.0;
        double r190381 = 4.0;
        double r190382 = x;
        double r190383 = y;
        double r190384 = 0.75;
        double r190385 = r190383 * r190384;
        double r190386 = r190382 + r190385;
        double r190387 = z;
        double r190388 = r190386 - r190387;
        double r190389 = r190381 * r190388;
        double r190390 = r190389 / r190383;
        double r190391 = r190380 + r190390;
        return r190391;
}
↓
double f(double x, double y, double z) {
        double r190392 = 4.0;
        double r190393 = x;
        double r190394 = z;
        double r190395 = r190393 - r190394;
        double r190396 = y;
        double r190397 = r190395 / r190396;
        double r190398 = fma(r190392, r190397, r190392);
        return r190398;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, 0.75 - \frac{z - x}{y}, 1\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  6. Applied sub-div0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))