Average Error: 0.1 → 0
Time: 25.0s
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 r28590019 = x;
        double r28590020 = y;
        double r28590021 = r28590019 - r28590020;
        double r28590022 = 2.0;
        double r28590023 = r28590021 / r28590022;
        double r28590024 = r28590019 + r28590023;
        return r28590024;
}
↓
double f(double x, double y) {
        double r28590025 = 1.5;
        double r28590026 = x;
        double r28590027 = 0.5;
        double r28590028 = y;
        double r28590029 = r28590027 * r28590028;
        double r28590030 = -r28590029;
        double r28590031 = fma(r28590025, r28590026, r28590030);
        return r28590031;
}

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 2019200 +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)))