Average Error: 16.3 → 0.0
Time: 29.3s
Precision: 64
\[x + \left(1 - x\right) \cdot \left(1 - y\right)\]
↓
\[\left(x \cdot y + \left(-1\right) \cdot y\right) + 1\]
x + \left(1 - x\right) \cdot \left(1 - y\right)
↓
\left(x \cdot y + \left(-1\right) \cdot y\right) + 1
double f(double x, double y) {
        double r474474 = x;
        double r474475 = 1.0;
        double r474476 = r474475 - r474474;
        double r474477 = y;
        double r474478 = r474475 - r474477;
        double r474479 = r474476 * r474478;
        double r474480 = r474474 + r474479;
        return r474480;
}
↓
double f(double x, double y) {
        double r474481 = x;
        double r474482 = y;
        double r474483 = r474481 * r474482;
        double r474484 = 1.0;
        double r474485 = -r474484;
        double r474486 = r474485 * r474482;
        double r474487 = r474483 + r474486;
        double r474488 = r474487 + r474484;
        return r474488;
}

Error

Bits error versus x

Bits error versus y

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Results

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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

    \[\leadsto \color{blue}{y \cdot \left(x - 1\right) + 1}\]
  4. Using strategy rm

  5. Applied sub-neg0.0

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

  6. Applied distribute-lft-in0.0

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

  7. Simplified0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 
(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))))