Average Error: 0.1 → 0.0
Time: 9.7s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)
double f(double x, double y, double z) {
        double r316820 = 1.0;
        double r316821 = 4.0;
        double r316822 = x;
        double r316823 = y;
        double r316824 = 0.25;
        double r316825 = r316823 * r316824;
        double r316826 = r316822 + r316825;
        double r316827 = z;
        double r316828 = r316826 - r316827;
        double r316829 = r316821 * r316828;
        double r316830 = r316829 / r316823;
        double r316831 = r316820 + r316830;
        return r316831;
}
↓
double f(double x, double y, double z) {
        double r316832 = 4.0;
        double r316833 = x;
        double r316834 = y;
        double r316835 = r316833 / r316834;
        double r316836 = z;
        double r316837 = r316836 / r316834;
        double r316838 = r316835 - r316837;
        double r316839 = 2.0;
        double r316840 = fma(r316832, r316838, r316839);
        return r316840;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)}\]
  5. Using strategy rm

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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)))