Average Error: 0.2 → 0.0
Time: 12.1s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x + \left(-z\right)}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x + \left(-z\right)}{y}, 4\right)
double f(double x, double y, double z) {
        double r150759 = 1.0;
        double r150760 = 4.0;
        double r150761 = x;
        double r150762 = y;
        double r150763 = 0.75;
        double r150764 = r150762 * r150763;
        double r150765 = r150761 + r150764;
        double r150766 = z;
        double r150767 = r150765 - r150766;
        double r150768 = r150760 * r150767;
        double r150769 = r150768 / r150762;
        double r150770 = r150759 + r150769;
        return r150770;
}
↓
double f(double x, double y, double z) {
        double r150771 = 4.0;
        double r150772 = x;
        double r150773 = z;
        double r150774 = -r150773;
        double r150775 = r150772 + r150774;
        double r150776 = y;
        double r150777 = r150775 / r150776;
        double r150778 = fma(r150771, r150777, r150771);
        return r150778;
}

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

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

  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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +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)))