Average Error: 0.1 → 0.0
Time: 12.4s
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 r183170 = 1.0;
        double r183171 = 4.0;
        double r183172 = x;
        double r183173 = y;
        double r183174 = 0.75;
        double r183175 = r183173 * r183174;
        double r183176 = r183172 + r183175;
        double r183177 = z;
        double r183178 = r183176 - r183177;
        double r183179 = r183171 * r183178;
        double r183180 = r183179 / r183173;
        double r183181 = r183170 + r183180;
        return r183181;
}
↓
double f(double x, double y, double z) {
        double r183182 = 4.0;
        double r183183 = x;
        double r183184 = y;
        double r183185 = r183183 / r183184;
        double r183186 = z;
        double r183187 = r183186 / r183184;
        double r183188 = r183185 - r183187;
        double r183189 = fma(r183182, r183188, r183182);
        return r183189;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  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 2019350 +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)))