Average Error: 16.3 → 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 r537962 = x;
        double r537963 = 1.0;
        double r537964 = r537963 - r537962;
        double r537965 = y;
        double r537966 = r537963 - r537965;
        double r537967 = r537964 * r537966;
        double r537968 = r537962 + r537967;
        return r537968;
}
↓
double f(double x, double y) {
        double r537969 = y;
        double r537970 = x;
        double r537971 = 1.0;
        double r537972 = r537971 * r537969;
        double r537973 = r537971 - r537972;
        double r537974 = fma(r537969, r537970, r537973);
        return r537974;
}

Error

Bits error versus x

Bits error versus y

Target

Original16.3
Target0.0
Herbie0.0
\[y \cdot x - \left(y - 1\right)\]

Derivation

  1. Initial program 16.3

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

  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 2020036 +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))))