Average Error: 16.9 → 0.0
Time: 1.0s
Precision: 64
\[x + \left(1 - x\right) \cdot \left(1 - y\right)\]
↓
\[\mathsf{fma}\left(y, x, 1 - 1 \cdot y\right)\]
x + \left(1 - x\right) \cdot \left(1 - y\right)
↓
\mathsf{fma}\left(y, x, 1 - 1 \cdot y\right)
double f(double x, double y) {
        double r800353 = x;
        double r800354 = 1.0;
        double r800355 = r800354 - r800353;
        double r800356 = y;
        double r800357 = r800354 - r800356;
        double r800358 = r800355 * r800357;
        double r800359 = r800353 + r800358;
        return r800359;
}
↓
double f(double x, double y) {
        double r800360 = y;
        double r800361 = x;
        double r800362 = 1.0;
        double r800363 = r800362 * r800360;
        double r800364 = r800362 - r800363;
        double r800365 = fma(r800360, r800361, r800364);
        return r800365;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.9
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.9

    \[x + \left(1 - x\right) \cdot \left(1 - y\right)\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(x \cdot y + 1\right) - 1 \cdot y}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 1 - 1 \cdot y\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, x, 1 - 1 \cdot y\right)\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* y x) (- y 1))

  (+ x (* (- 1 x) (- 1 y))))