Average Error: 0.1 → 0
Time: 13.7s
Precision: 64
\[x + \frac{x - y}{2}\]
↓
\[\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\]
x + \frac{x - y}{2}
↓
\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)
double f(double x, double y) {
        double r25559185 = x;
        double r25559186 = y;
        double r25559187 = r25559185 - r25559186;
        double r25559188 = 2.0;
        double r25559189 = r25559187 / r25559188;
        double r25559190 = r25559185 + r25559189;
        return r25559190;
}
↓
double f(double x, double y) {
        double r25559191 = 1.5;
        double r25559192 = x;
        double r25559193 = 0.5;
        double r25559194 = y;
        double r25559195 = r25559193 * r25559194;
        double r25559196 = -r25559195;
        double r25559197 = fma(r25559191, r25559192, r25559196);
        return r25559197;
}

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

    \[x + \frac{x - y}{2}\]

  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 2019172 +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)))