Average Error: 0.2 → 0.0
Time: 21.6s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)
double f(double x, double y, double z) {
        double r33933994 = 1.0;
        double r33933995 = 4.0;
        double r33933996 = x;
        double r33933997 = y;
        double r33933998 = 0.75;
        double r33933999 = r33933997 * r33933998;
        double r33934000 = r33933996 + r33933999;
        double r33934001 = z;
        double r33934002 = r33934000 - r33934001;
        double r33934003 = r33933995 * r33934002;
        double r33934004 = r33934003 / r33933997;
        double r33934005 = r33933994 + r33934004;
        return r33934005;
}
↓
double f(double x, double y, double z) {
        double r33934006 = 4.0;
        double r33934007 = x;
        double r33934008 = y;
        double r33934009 = r33934007 / r33934008;
        double r33934010 = z;
        double r33934011 = r33934010 / r33934008;
        double r33934012 = r33934009 - r33934011;
        double r33934013 = fma(r33934006, r33934012, r33934006);
        return r33934013;
}

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

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

  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

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

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))