Average Error: 0.1 → 0.0
Time: 8.1s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)
double f(double x, double y, double z) {
        double r591811 = 4.0;
        double r591812 = x;
        double r591813 = y;
        double r591814 = r591812 - r591813;
        double r591815 = z;
        double r591816 = 0.5;
        double r591817 = r591815 * r591816;
        double r591818 = r591814 - r591817;
        double r591819 = r591811 * r591818;
        double r591820 = r591819 / r591815;
        return r591820;
}
↓
double f(double x, double y, double z) {
        double r591821 = 4.0;
        double r591822 = x;
        double r591823 = z;
        double r591824 = r591822 / r591823;
        double r591825 = y;
        double r591826 = r591825 / r591823;
        double r591827 = r591824 - r591826;
        double r591828 = 2.0;
        double r591829 = -r591828;
        double r591830 = fma(r591821, r591827, r591829);
        return r591830;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.0
Herbie0.0
\[4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right)\]

Derivation

  1. Initial program 0.1

    \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(x - \mathsf{fma}\left(0.5, z, y\right)\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{4 \cdot \frac{x}{z} - \left(4 \cdot \frac{y}{z} + 2\right)}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

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, B"

  :herbie-target
  (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))