Average Error: 0.2 → 0.0
Time: 5.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r1387 = 1.0;
        double r1388 = 4.0;
        double r1389 = x;
        double r1390 = y;
        double r1391 = 0.75;
        double r1392 = r1390 * r1391;
        double r1393 = r1389 + r1392;
        double r1394 = z;
        double r1395 = r1393 - r1394;
        double r1396 = r1388 * r1395;
        double r1397 = r1396 / r1390;
        double r1398 = r1387 + r1397;
        return r1398;
}
↓
double f(double x, double y, double z) {
        double r1399 = 4.0;
        double r1400 = x;
        double r1401 = y;
        double r1402 = r1400 / r1401;
        double r1403 = z;
        double r1404 = r1403 / r1401;
        double r1405 = r1399 * r1404;
        double r1406 = r1399 - r1405;
        double r1407 = fma(r1399, r1402, r1406);
        return r1407;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

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, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))