Average Error: 0.1 → 0
Time: 13.2s
Precision: 64
\[x + \frac{x - y}{2.0}\]
↓
\[\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\]
x + \frac{x - y}{2.0}
↓
\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)
double f(double x, double y) {
        double r15638837 = x;
        double r15638838 = y;
        double r15638839 = r15638837 - r15638838;
        double r15638840 = 2.0;
        double r15638841 = r15638839 / r15638840;
        double r15638842 = r15638837 + r15638841;
        return r15638842;
}
↓
double f(double x, double y) {
        double r15638843 = 1.5;
        double r15638844 = x;
        double r15638845 = 0.5;
        double r15638846 = y;
        double r15638847 = r15638845 * r15638846;
        double r15638848 = -r15638847;
        double r15638849 = fma(r15638843, r15638844, r15638848);
        return r15638849;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.1
Target0.1
Herbie0
\[1.5 \cdot x - 0.5 \cdot y\]

Derivation

  1. Initial program 0.1

    \[x + \frac{x - y}{2.0}\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{1.5 \cdot x - 0.5 \cdot y}\]
  3. Using strategy rm

  4. Applied fma-neg0

    \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)}\]

  5. Final simplification0

    \[\leadsto \mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3"

  :herbie-target
  (- (* 1.5 x) (* 0.5 y))

  (+ x (/ (- x y) 2.0)))